Shapes and Geometries: Analysis, Differential Calculus, and Optimization (Advances in Design and Control)

148

Chapter 3. Relaxation to Measurable Domains

They are special cases of the general transmission problem (3.3) of section 3.1,

for ki = 1 and k% = 1/e. Some continuity of the Dirichlet problem with respect to domains is then obtained by combining a continuity result in section 7.1 for the above transmission problem with the approximation (7.5) by transmission problems in section 7.2.

7.1

Continuity of Transmission Problems

Recall that when D is a bounded open connected domain in R N , the first eigenvalue AI = \i(D) of the Laplacian —A on HQ(D] is strictly positive and hence

The following continuity will be sufficient to obtain some existence results. Proposition 7.1. Let D be a bounded open connected domain in R N . Let k\ > 0, &2 > 0, and f in L'2(D} be given such that minjfci, fc2} > & for some constant a > 0. For any sequence {Xn} and x in coX(D) such that Xn —» X € L2(D)-strong, there exists a subsequence {ynk = y(Xnk}} of {yn = y(Xn}} such that

and any strongly convergent subsequence of {yn} converges to the same limit. Proof. For each n, let kn = kiXn + ^2(1 — Xn} and let yn 6 HQ(D) be the solution of the equation

The difference zn = yn — y is the solution of the variational equation

By assumption, kn > a. Setting (p = zn, we get

7. Continuity of the Dirichlet Boundary Value Problem

149

If AI = Ai(-D), the first eigenvalue of the Laplacian —A in HQ(D], then

On the right-hand side we use the Lebesgue-dominated convergence theorem as follows: there exist some subsequences {knk — k} (resp., {Xnk — %}) which converge to 0 almost everywhere in D while at the same time

The last part of the lemma is straightforward. 7.2

Approximation by Transmission Problems

We now show that, as e —» 0, the convergence in HQ(D) of ye(x] to the solution y ( x ) of the relaxed homogeneous Dirichlet problem (7.1)-(7.2) can be controlled through the norm of some extension operator. Theorem 7.1. Let D be a bounded open connected domain in RN and 0 a measurable subset of D such that

Assume that there exists a continuous linear extension operator

where \\P\\ is the norm of P in C(H(£IC},HQ(D}}. Given x = Xjy ande, 0 < £ < 1, let ye be the solution of the variational equation (7.5) in HQ(D) and y the solution of the variational equation (7.1) in the space Hl(£l; D) defined in (7.2) for x = XnThen there exists a constant c'(D] > 0 such that

Proof. For e, 0 < £ < 1, substitute (p = y£ in (7.5) to get the first bound

150

Chapters. Relaxation to Measurable Domains

In view of the properties of f), ||V<^||/ / 2(Q C ) is a norm on H(£l°}. Set
Observe that for all (p G Hi(tt;D), V(p = 0 in £>\H. By subtracting (7.5) from (7.1)

Note that for all ^ € #o(-D)

In particular, (p = y - ye - P([y - y e ]|nc) e ^(fi; D) and

7.3

Continuity of the Dirichlet Problem

Theorem 7.2. Lei {rjn} 6e a sequence of open sets in D, lln C D, satisfying the assumptions of Theorem 7.1. Furthermore, assume that there exists M > 0 such that for all n,

Then if Xfin ~^ Xn y(Xa )} such that

?n

L2(D]-strong, there exists a subsequence {ynk} of {yn =

7. Continuity of the Dirichlet Boundary Value Problem

151

//, in addition, all the domains are Lipschitzian, then if Xnn ~^ Xn ^n L2(D)-strong, there exists a subsequence {ynk} of [yn = y(xsir,}} sucn that

Proof. The proof follows from Proposition 7.1 and Theorem 7.1.

Remark 7.1.

In Agmon, Douglis, and Nirenberg [1, 2] the extension operator is constructed from the locally uniform cone property of Definition 6.1 in Chapter 2. It can be verified that for a family of domains 0 satisfying the uniform cone property in a bounded hold-all D, the corresponding extension operators P^ and P^c are uniformly bounded with respect to all such domains 0 in D.

Remark 7.2.

In section 7 of Chapter 6 a sharper continuity of the homogeneous Dirichlet problem for the Laplacian will be given under capacity conditions. Nevertheless, the penalization technique used in this section can be applied to higher-order elliptic, and even parabolic, problems and the Navier-Stokes equation (cf. Dziri and Zolesio [6]).

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Chapter 4

bpologies Generated by bistance Functions

1 Introduction In Chapter 3 the characteristic function was used to embed the equivalence classes of measurable subsets of D into LP(D] or Lf oc (D), 1 < p < oo, and induce a metric on the equivalence classes of measurable sets. This construction is generic and extends to other set-dependent functions embedded in an appropriate function space. The Hausdorff metric is the result of such a construction, where the distance function plays the role of the characteristic function. The distance function embeds equivalence classes of subsets A of a closed hold-all D with the same closure A into the space C(D) of continuous functions. When D is bounded the C*°-norm of the difference of two distance functions is the Hausdorff metric. The Hausdorff topology has many much-desired properties. In particular, for D bounded the set of equivalence classes of nonempty subsets A of D is compact. Yet the volume and perimeter are not continuous with respect to the Hausdorff topology. This can be fixed by changing the space C(D] for the space Wl'p(D] since distance functions also belong to that space. With that metric the volume is again continuous. The price to pay is the loss of compactness even when D is bounded. But new sequentially compact subfamilies can easily be constructed. By analogy with Caccioppoli sets we introduce the sets for which the elements of the Hessian matrix of second-order derivatives of the distance function are bounded measures. They are called sets of bounded curvature. Their closure is a Caccioppoli set, and they seem to enjoy other interesting properties. Convex sets belong to that family. General compactness theorems are obtained for such families under global or local conditions. This chapter simultaneously deals with the family of open sets characterized by the distance function to their complement. They are discussed in parallel with the sets described by their distance function. The properties of distance functions and Hausdorff and Hausdorff complementary metric topologies are studied in section 2. The differentiability of distance functions is discussed in section 3 along with the notions of projections, skeleton, and cracks. W^'P-topologies are introduced in section 4. The compact families of 153

154

Chapter 4. Topologies Generated by Distance Functions

sets of bounded and locally bounded curvature are characterized in section 5. The special families of convex sets and Federer's sets of positive reach are studied in sections 6 and 7 and will be further investigated in Chapter 5. Finally, section 8 gives several compactness theorems under global and local conditions on the Hessian matrix of the distance function.

2

Hausdorff Metric Topologies

2.1

The Family of Distance Functions Cd(D]

In this section we review some properties of distance functions and present the general approach that will be followed in this chapter. Definition 2.1. Given A C RN the distance function from a point x to A is defined as

and the family of all distance functions of nonempty subsets of D, as

When D = R N the family C d (R N ) is denoted Cd. Observe that dA is finite in RN if and only if A ^ 0. We recall the following properties of distance functions. Theorem 2.1. Assume that A and B are nonempty subsets o/R N . (i) The map x i—>• dA(x) is uniformly Lipschitz continuous in R N 7

anddAeC^lc(W).1 (ii) There exists y € A such that dA(x] = \y — x and dA = d-^ in R N .

1

A function / belongs to C^(R N ) if for all bounded open subsets D of RN its restriction to D belongs to C°>l(D).

2. Hausdorff Metric Topologies (vii) d& is (Frechet) differentiable

155 almost everywhere and

Proof, (i) For all z 6 A and x, y G RN

and hence the Lipschitz continuity. (ii) Let {yn} C A be a minimizing sequence

Then {yn — x} and hence {yn} are bounded sequences. Hence there exist a subsequence converging to some y G A and cU(x) = |y — £ • Clearly by definition d^(x) < d/i(x) since A C A. So either there exists y G A such that c^(x) = \y — x and hence d-^(x) = ^(x) or there exists {yn} C A such that

and we get the identity. By the continuity of d^, d~^{ti} is closed and

Conversely, if cU(x) = 0, there exists y in A such that 0 = d^(x) = \y ~- x and necessarily x = y G A. (iii) follows from (ii). (iv) follows from (iii). (v) follows from (ii) for x G A, C^A(X) = 0, and ds(x) < d^(^) = 0 necessarily implies d#(x) — 0 and x e B. Conversely, if A C B, then

(vi) is obvious, (vii) follows from Rademacher's theorem. 2.2

Hausdorff Metric Topology

Let D be a nonempty subset of RN and associate with each nonempty subset A of D the equivalence class

since from Theorem 2.1(v) d& = ds if and only if A = B. representative of the class [A]. Consider the set

So A is the closed

156

Chapter 4. Topologies Generated by Distance Functions

By the definition of [A] the map

is injective. So ^(D] can be identified with the subset of distance functions Cd(D] in C(D). The distance function plays the same role as the set X(£>) in LP(D] of equivalence classes of characteristic functions of measurable sets. When D is bounded, C(D] is a Banach space when endowed with the norm

As for the characteristic functions of Chapter 3, this will induce a complete metric

on F(D), which turns out to be equal to the classical Hausdorff metric

(cf. Dugundji [1, p. 205, Chap. IX, Prob. 4.8] for the definition of px). When D is open but not necessarily bounded, the space C(D) of continuous functions on D is endowed with the Frechet topology of uniform convergence on compact subsets K of D, which is defined by the family of seminorms

It is metrizable since the topology induced by the family of seminorms {QK} is equivalent to the one generated by the subfamily {qKk}k>ii where the compact sets {Kf-}k>i are chosen as follows:

(cf., for instance, Horvath [1, Example 3, p. 116]). Thus the Frechet topology on C(D] is equivalent to the topology defined by the metric

In that case we use the notation C\OC(D). It will be shown below that Cd(D] is a closed subset of C\OC(D] and that this will induce the following complete metric on^(D):

which is a natural extension of the Hausdorff metric to an unbounded domain D.

2. Hausdorff Metric Topologies

157

Theorem 2.2. Let D be a nonempty open (resp., bounded open) subset o/R N . (i) The set Cd(D) is closed in C\OC(D) (resp., C(D)) and p$ (resp., p) defines a complete metric topology on J~(D). (ii) When D is a bounded open subset o/R N , the set Cd(D) is compact in C(D) and the metrics p and pn are equal. Proof, (i) We use the proof given by Dellacherie [1, p. 42, Thrn. 2 and p. 43, Remark I}. The basic constructions will again be used in Chapter 5. Consider a sequence {An} of nonempty subsets of D such that d,An converges to some element / of C\OC(D). We wish to prove that / = dA for

and that the closed subset A of D is nonempty. Fix x in D; then

Hence {yn — x} and {yn} C D are bounded, and there exists a subsequence, still indexed by n, which converges to some y 6 D,

In particular, f(y) = 0, since in the inequality

the last term is zero and d^n is Lipschitz continuous with constant equal to 1:

and both terms go to zero. By the definition of A, y £ A and A is not empty. Therefore for each a: € D, there exists y £ A such that

Next we prove the inequality in the other direction. By construction for any An C D

and

By uniform convergence the first and last terms converge to zero, and by Lipschitz continuity of d^n,

158

Chapter 4. Topologies Generated by Distance Functions

Hence for all x E D and y G A, f(y) = 0 and

This proves the reverse equality. (ii) Observe that on a compact set D and for any A C D

The compactness of Cd(D) now follows by the Ascoli-Arzela Theorem 2.1 of Chapter 2 and part (i). By construction p and ps are metrics on F(D}. By definition, for A and B in the compact D,

Conversely, for any x G D and y in A,

and there exists XB G -B such that d,B(x) = \ X - X B \ - Therefore,

Similarly and for all x in D

When D is bounded the family F(D] enjoys many more interesting properties. Theorem 2.3. Let D be a nonempty open (resp., bounded open) subset of R N . Define for a subset S of RN the sets

2. Hausdorff Metric Topologies

159

(i) Let S be a subset o/R N . Then H(S) is closed in Cioc(D) (resp., C(D}). (ii) Let S be a closed subset o/R N . Then I(S) is closed in C\OC(D) (resp., C(D}}. If, in addition, STlD is compact, then J ( S ) is closed in C\OC(D] (resp., C(D}}. (\\\) Let S be an open subset o/R N . Then J ( S ) is open in C\OC(D) (resp., C(D}). If, in addition, CtSH-D is compact, then I(S) is open in C\OC(D) (resp., C(D)). (iv) For D bounded, associate with an equivalence class [A] #([A]} — number of connected components of A. Then the map is lower semicontinuous. In particular, for a fixed number c > 0 the subset

is compact in C(D}. Proof, (i) If S <£ D, then H(S) = 0 and there is nothing to prove. Assume that S C D. From Theorem 2.1(v), 5 C A =3- d$ > &A- So for any sequence {dAn} in H(S) converging to &A in C\OC(D),

(ii) We use the same technique for I ( S ) as for H(S}, but here we need S = S to conclude. For D fl S = 0, J(S) = 0 and there is nothing to prove. Assume that D n S ^ 0 and consider a sequence {d^n} in J(S) converging to &A in Cioc(D). Assume that ~A H__S = 0. Then ~A C D implies A n [5 n D] = ~A n S = 0. By assumption, S Pi D is compact and

So for all x e D n 5

and this contradicts the fact that An e J(S). (iii) By definition, for any 5, CJ(5) = 7(05):

Since CS1 is closed, CJ(5) is closed from part (ii) and J(S] is open. Similarly, replacing S by C51 in the previous identity, 7(5) — CJ(C5). So from part (ii) if D n C5 is compact, then C/(5) is closed and I(S) is open. From part (ii) if D D C5 is compact, then C/(5) is closed and 1(5) is open.

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Chapter 4. Topologies Generated by Distance Functions

(iv) Let {An} and A be nonempty subsets of D such that d,An converges to d^ in C(D}. Assume that #([-4]) = k is finite. Then there exists a family of disjoint open sets GI , . . . , Gk such that

In view of the definitions of I(S) and J(S) in part (i),

But U is not empty and open as the finite intersection of k -f 1 open sets. As a result there exists e > 0 and an open neighborhood of [A],

Hence since d,An converges to c^, there exists n > 0 such that

and necessarily

Therefore, [A] i—>• #([A]) is lower semicontinuous. Now for #([A|) = +00, we repeat the above procedure and refine the open covering. This theorem has many interesting corollaries. For instance, part (iii) says something about the function that gives the number of connected components of A (cf. Richardson [1] for an application to image segmentation).

2.3

Hausdorff Complementary Metric Topology and C%(D)

In the previous section we dealt with a theory of closed sets, since the equivalence class of the subsets of D was completely determined by their unique closure. In partial differential equations the underlying domain is usually open. To accommodate this point of view, consider the set of open subsets 0 of a fixed nonempty open hold-all D in R N , endowed with the Hausdorff topology generated by the distance functions d^ to the complement of 0. This approach has been used in several contexts, for instance, in Zolesio [6, section 1.3, p. 405] in the context of free boundary problems. Also, Sverak [2] uses the family C%(D) for open sets 17 such that #([Cfi]) < c for some fixed c > 0. His main result is that in dimension 2 the convergence of a sequence {On} to fi of such sets implies the convergence of the corresponding projection operators {P$in : HQ(D) —» #0(^1)} to P^: HQ(D] —> #g(Q), where the projection operators are directly related to homogeneous Dirichlet linear boundary value problems on the corresponding domains {f2n} and fl In dimension 1 the constraint on the number of components can be dropped. This result will be discussed in Theorem 8.2 of section 8 in Chapter 6.

2. Hausdorff Metric Topologies

161

By analogy with the constructions of the previous section, consider for a nonempty subset D of RN the family

By definition,

and, by Lipschitz continuity,

and in C0(int D) if D is bounded. If int D = 0, then dCA = 0 in R N . If int D ^ 0, associate with each A the open set

By definition and the previous considerations

So finally, for int D ^ 0,

From this analysis it will be sufficient to consider the family of open subsets of an open hold-all.

Definition 2.2.

Let D be a, nonempty open subset of R N . Define the family of functions

corresponding to the family of open sets

For D bounded define the Hausdorff

complementary metric p°H on G(D):

Note that for QI C D and f^ C D, d^l — d^2 — 0 in CD. Therefore,

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Chapter 4. Topologies Generated by Distance Functions

Theorem 2.4. Let D be a nonempty open subset o/R N . (i) The set Ccd(D] is closed in Cioc(D}. (ii) //, in addition, D is bounded, then C^(D] is compact in Co(D) and (G(D},p°H} is a compact metric space. (iii) (Compact!vorous property) Let {fin} and ft, be sets in Q(D] such as

Then for any compact subset K C ft, there exists an integer N(K) > 0 such that Proof, (i) Let {On} be a sequence of open subsets in D such that {d^n} is a Cauchy sequence in C\OC(D}. For all n, d^n = 0 in CD and {d^n} is also a Cauchy sequence in (7ioc(R-N)- By Theorem 2.2 (i) the sequence {d^n} converges in CIOC(RN) to some distance function d&. By construction

By choosing the open set Jl = \jA in D we get

(ii) To prove the compactness we use the compactness of Cd(D] from Theorem 2.2 (ii) and the fact that C%(D) is closed in Co(D). Observe that since Cnn D CD, d^n = 0 in CD, d^n £ C0(D), and

There exists a subsequence, still indexed by n, and a set A, 0 ^ A C D such that

since d^n = 0 = <%uC£) in CD. The result now follows by choosing the open set

(iii) Define

3. Projection, Skeleton, Crack, and Differentiability

163

Since K C Q C D, there exists x E K, y E dfl such that

Necessarily, m > 0 since x E K C ft and y E $£7. Now

Notation 2.1. It will be convenient to write for the Hausdorff complementary convergence of open sets of G(D],

3

Projection, Skeleton, Crack, and Differentiability

In this section we study the connection between the gradient of dA and d^A and the projections and the characteristic functions associated with A and CA. We further relate the set of singularities of the gradients and the notions of skeleton and set of cracks.

Definition 3.1. (i) Given A C R N , 0 ^ A (resp., 0 ^ CA), and x E R N , the set of projections of x on A (resp., CA) is given by

The elements of n^(x) (resp., I\.^A(x]} are called projections onto A (resp., CA) and denoted by PA(X] (resp., PCA(X}}(ii) Given A C R N , 0 ^ A (resp., 0 ^ CA), the set of points where the projection on A (resp., CA) is not unique,

is called the exterior (resp., interior) skeleton. Since for x E dA the sets n^(x) and HQA(X) are singletons,

164

Chapter 4. Topologies Generated by Distance Functions

Figure 4.1. Nonuniqueness of the projection and skeleton. Intuitively, the smoothness of the boundary dA of A is related to the smoothness of Vd,A(x) in a small exterior neighborhood of the boundary dA of A. However, even for very smooth sets, Vd^(x] may not exist far from the boundary, as shown in Figure 4.1, where Vd^(x) exists outside A except on a semi-infinite line. The directional derivative of the square of the distance function can be computed by using a theorem on the differentiability of the min with respect to a parameter (cf. Chapter 9, section 2.3, Thm. 2.1). It is related to the support function of the set

ru(z).

Theorem 3.1. Let A, 0 ^ A C R N , and x <E R N . Define

(i) The set HA(X) is nonempty, compact, and

(ii) For all x and v in RN

and coB is the convex hull of B. (ni) The following statements are equivalent:

3. Projection, Skeleton, Crack, and Differentiability

(a) d\(x} is (Frechet) differentiable (b) d\(x] is Gateaux differentiable

165

at x, at x,

(c) n^a?) is a singleton. Henceforth

where

When d\ is differentiable

at x, 11,4(2;) = [p^(x)} is a singleton and

For all x G A, UA(X) = {x}, d\ is differentiable

at x and Vd2A(x) = 0.

(iv) For £>A j^ 0, the conclusions of parts (i)-(iii) are true with £>A in place of A and

in place of C ext (A). Proof, (i) Existence. The function

is continuous and z — x\2 —> OD as \z\ —> oo. Hence for any

the set

is nonempty and compact and

Thus HA(X) is nonempty and

166

Chapter 4. Topologies Generated by Distance Functions

is compact. By definition, UA(X) C A. If x £ dA, PA(X) = £ £ <9A If x £ int(L4, then ^A(^) > 0. Now if PA(%) £ int.4, then there exists an open ball B of radius r, 0 < r < dA(x)/2, at £u(:r), which is contained in A. Choose

Then y belongs to A and

So there exists y £ A such that

This contradicts the minimality of cU(:r). Hence a; £ <9A (ii) Differentiability of /A- It is sufficient to prove the semidifferentiability of d\; that is, for each x and v the existence of the limit

Recall that for t > 0, and consider for t > 0 the quotient

For all p 6 HA (x) and pt £ 11^ (x + tv]

and for all p £ H^ (x)

In the other direction choose a sequence t^ > 0 such that tk —> 0 and
3. Projection, Skeleton, Crack, and Differentiability

167

and we can find po G A and a subsequence of {ptk}, still denoted {ptk}, such that Ptk —> Po- But by continuity

Therefore,

Hence (iii) From part (ii) d\(x) is Gateaux differentiable at x if and only if UA(X) is a singleton. The (Frechet) differentiability is a standard part of Rademacher's theorem, but we reproduce it here for completeness as a lemma. Lemma 3.1. Let f : RN —>• R be a locally Lipschitzian function. (Frechet) differentiable at x; that is,

if and only if it is Gateaux differentiable

and the map v i—>• df(x;v)

Then f is

at x; that is,

is linear and continuous.

Proof. It is sufficient to prove that the Gateaux differentiability implies that

The function / is locally Lipschitzian and there exist 61 > 0 and c\ > 0 such that / is uniformly Lipschitzian in the open ball B(x,5i) with Lipschitz constant c\. Therefore, |V/(x)| < cl. The unit sphere 5(0,1) = {v e RN : \v\ = 1} is compact and can be covered by the family of open balls {B(v, £/(4ci)) : v € 5(0,1)} for an arbitrary e > 0. Hence there exists a finite subcover {B(vn,£/(4ci}) : I < n < N} of 5(0,1). As a result, given any v G 5(0,1), there exists n, 1 < n < N, such that \v — vn < e/(4ci). Define the function

168

Chapter 4. Topologies Generated by Distance Functions

for x G R N , v G 5(0,1), and t > 0. Since f(x] is Gateaux clifferentiable

Hence for \y — x < 6

and we conclude that f ( x ) is differentiable at x. From the above equivalences, when d\ is differentiable at x, then T[^(x) = {PA(X}} is & singleton, and from part (ii)

which yields the expression for PA(X). When x £ A, UA(X) — {x}, and by substitution, Vd^(rr) = 0. This completes the proof of the theorem. Remark 3.1. In general, for each v G R N ,

but the choice of p depends on the direction v and is not necessarily unique. The set-valued map (/C(R N ), the set of nonempty compact subsets of R N ) contains a lot of information about the set A. We now turn to the differentiability of d,A- The distance function is uniformly Lipschitzian of constant 1 and differentiable almost everywhere. When it is differentiable, d\ is also differentiable, and from Theorem 3.1, UA(X] contains a unique element PA (x). However, the smoothness of the boundary dA does not imply that HA(X) is a singleton everywhere in R N . For instance, if

then dA is C°° but H^(0) = A. However, we shall see in section 6 that for convex sets A, HA(X) is always a singleton. We now explicitly compute the gradients of dA and d$A and relate them to the characteristic functions of the closures of A and CA

3. Projection, Skeleton, Crack, and Differentiability

169

Theorem 3.2. (i) Let A, 0 ^ A c R N . // VcU(#) exists at a point x in R N , then H^(x) = {PA(X)} is a singleton,

(ii) For A, 0 ^ A C R N , and x G RN \dA, dA is differentiable d2A is differentiable at x.

at x if and only if

(iii) Given A C R N , 0 ^ A (resp., 0 ^ (M),

and for almost all x

(iv) Given x G R N 7 a G [0,1], p G 11^(re), anrf xa = p + a (x — p ] ,

In particular, i/n^(x) is a singleton, then n^(a;a) ^s a singleton and Vd2A(xa] exists for all a, 0 < a < 1. //, m addition, x ^ A, VdA(xa] exists for all 0 < a < 1. Proof, (i) If VeU(x) exists for x G A, then for all v G RN

which implies that Vd^(^) = 0. If Vd^(x) exists for x ^ A, then

exists, and by Theorem 3.1, Ilyi(x) = {PA(X}} is a singleton and Vc^(x) = 2(x — PA(X})- But for x £ A, and it is sufficient to divide both sides by 2 PA(X] — x .

1 70

Chapter 4. Topologies Generated by Distance Functions

(ii) If VdA(x] exists, then Vd\(x) = 2dA(x) VdA(x) exists. Conversely, asA(x) exists. If x G int A, then dA = 0 in some neighborhood of x and hence VdA(x) exists and is equal to zero. If x ^ A, then for t ^ 0

So VdA(x) exists and is equal to

Vd\(x)/(2dA(x)}.

(iii) From part (i) if VdA(x) exists for x G A, it is equal to 0. Hence X~A(X] — 1 = 1- \Vd exists and |VcU(#)| = 1 for all points in int (L4\Skext (A}. Hence from part (i) XA(x) = 0 = 1 - |VcU(ff)| = 0 on int k4\Skext (-4). Since m(Sing (VoU)) = 0, the above identities are satisfied almost everywhere in R N . We get the same results with CA in place of A. (iv) If x G A, 11,4(0;) = {x} and there is nothing to prove. If x £ A, then dA(x) = \x — p\ > 0 and p ^ x for all p 6 IlA(x). First

If the inequality is strict, then there exists pa 6 A such that xa — pa = dA(xa) and

This contradicts the minimality of dA(x) with respect to A. Therefore, Now for any pa e HA(xa), pa £ A and and pa G n^(o;). Hence TiA(xa} C 11^(0;). This completes the proof. Observe that for points outside of A the norm of VdA(x) is equal to 1. It coincides with the outward unit normal to A at the point p(x] when A is sufficiently smooth. When A is not smooth this normal is not always unique, as can be seen in Figure 4.2. We now complete the description of the singularities of the gradients of dA and dCA,

by giving a name to the sets C ex t (A) and Ci nt (A) introduced in Theorem 3.2.

3. Projection, Skeleton, Crack, and Differentiability

171

Figure 4.2. Nonuniqueness of the exterior normal.

Definition 3.2. (i) For A C R N , A ^ 0, the set of exterior cracks is defined as

(ii) For A C R N , CA 7^ 0, the set of interior cracks is defined as

Points of Cext (A) cannot belong to RN \dA since, from Theorem 3.2 (ii), for such points the existence of Vd2A(x] implies the existence of Vd^(x). The same is true for C int (A). Therefore,

The sets Cext (A) and C int (A} are generally not equal to all of dA, as can be seen from the next example.

Example 3.1. Let £?(0,1) be the open ball of radius 1 at the origin and let

Then

Note that in the equivalence class [A], the boundary can be different for two members of the class. In fact, dA C dA for the closed representative of A and this inclusion can be strict, as shown from this example.

172

4

Chapter 4. Topologies Generated by Distance Functions

W^-Topology and Characteristic Functions

Distance functions are locally Lipschitzian. They belong to Clo'^(RN) and hence to VFlo'^(RN) for all p > I . Thus the previous constructions can be repeated with W^(D] in place of C\OC(D] to generate new HAl>p-metric topologies on the family Cd(D). One big advantage is that the Wl^-convergence of sequences will imply the //^-convergence of the corresponding characteristic functions of the closure of the sets and hence the convergence of volumes (cf. Theorem 3.2 (iii)). The convergence of volumes and perimeters is lost in the Hausdorff topology. The next example shows that the HausdorfF metric convergence is not sufficient to get the Z/p-convergence of the characteristic functions of the closure of the corresponding sets in the sequence. The volume function is only upper semicontinuous with respect to the Hausdorff topology. The perimeters do not converge either. This is also illustrated by the example of the staircase of Example 5.1 and Figure 3.7 in Chapter 3, where the volumes converge but not the perimeters.

Example 4.1.

Denote by D = [—1,2] x [—1,2] the unit square in R2, and for each n > 1 define the sequence of closed sets

Figure 4.3. Vertical stripes of Example 4.1. This defines n vertical stripes of equal width l/2n each distant of l/2n (cf. Figure 4.3). Clearly, for all n > 1,

where S = [0,1] x [0,1] and Pp(An} is the perimeter of An. Hence

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173

But

Since the characteristic functions do not converge, the sequence {Vc^,} does not converge in LP(D)N. Theorem 4.1. Let D be an open (resp., bounded open) subset o/R N . (i) The topologies induced by W^(D) (resp., Wl>p(D)) on Cd(D) and Ccd(D] are all equivalent for p, I < p < oo. (ii) Cd(D) is closed in W^(D) (resp., Wl'p(D)) for p, 1 < p < oo, and

defines a complete metric structure on F(D}. For p, 1 < p < oo, the map

is "Lipschitz continuous": for all bounded open subsets K of D and nonempty subsets A\ and A<2 of D,

(iii) C%(D] is closed in W^(D) (resp., W^P(D}} for p, l
defines a complete metric structure on the family Q(D) of open subsets of D. For p, 1 < p < oo, the map

is "Lipschitz continuous": for all bounded open subsets K of D and open subsets QI ^ RN and f22 ^ RN of D

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Chapter 4. Topologies Generated by Distance Functions

Proof, (i) The proof is similar to the proof of Theorem 2.2 in Chapter 3. It is sufficient to prove it for D bounded open. Since D is bounded it is contained in a sufficiently large ball of radius c. Therefore,

since y 6 A C D, and by Theorem 2.1 (vii)

For all p > I the injection of W1>P(.D) into Wljl(D) is continuous since

with l/p + l/q — 1. Conversely, we have the continuity in the other direction. For p > 1 and any d& and ds in Cd(D),

Therefore, for any e > 0, pick 6 = £p/(2 maxjc, I})*3"1 and

(ii) It is sufficient to prove it for D bounded open. Let {d^.n } be a Cauchy sequence in Cd(D] which converges to some / in VFlip(.D)-strong. By Theorem 2.2 (ii), Cd(D) is compact in C(D] and there exist a subsequence, still denoted {cUn}, and 0 ^ A C Z> such that and, a fortiori, in Lp(D)-strong since D is bounded. By uniqueness of the limit in Lp(D)-strong, / = d^ and d^n converge to cU in H/1>p(D)-strong. Therefore, Cd(D] is closed in Wljp(D). For the Lipschitz continuity, recall that the distance function dA is differentiate almost everywhere in RN for A ^ 0. In view of Theorem 3.2 (iv)

Given two nonempty subsets AI and A% of D

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175

for 1 < p < oo and with the ess-sup norm for p = oo. (iii) Again it is sufficient to prove the result for D bounded. In that case CfJ 7^ 0 for all open subsets £1 of D. Let {On} be a sequence of open subsets of D such that {o?crin} *s Cauchy in VF1;P(D). By assumption £ln C D, Cf7n D C.D,

and the Cauchy sequence converges to some / G W0'P(D}. By Theorem 2.4 (ii), C%(D) is compact in C(D] and there exist a subsequence, still denoted {c^n}, and an open set 0 C D such that

and hence in L p (D)-strong, since D is bounded. By uniqueness of the limit in L P (D), / = d£n and the Cauchy sequence d^n converges to d^ in W0'P(D). The other part of the proof is similar to that of part (ii). We have the following general result. Theorem 4.2. Let D be a bounded open domain in R N . (i) // {o?^n} weakly converges in Wl'p(D) for some p, I < p < oo, then it weakly converges in Wl'p(D] for all p, 1 < p < oo. (ii) If {dAn} converges in C(D], then it weakly converges in Wl'p(D) for all p, I < p < oo. Conversely if {dAn} weakly converges in Wl'p(D) for some p, I < p < oo, it converges in C(D}. (iii) Cd(D) is compact in W1>p(D)-weak for all p, 1 < p < oo.2 (iv) Parts (i)-(iii) also apply to C^(D}. Proof, (i) Recall that for D bounded there exists a constant c > 0 such that for all dA e Cd(D]

If {dAn} weakly converges in W /1 ' P (D), then

By Lemma 2.1 (iii) in Chapter 3 both sequences weakly converge for all p > 1, and hence {dAn} weakly converges in W1'P(D) for all p > 1. 2 In a metric space the compactness is equivalent to the sequential compactness. For the weak topology we use the fact that if E is a separable normed space, then, in its topological dual E', any closed ball is a compact metrizable space for the weak topology. Since Cd(D) is a bounded subset of the normed reflexive separable Banach space Wl>p(D), 1 < p < oo, the weak compactness of C^(D] coincides with the weak sequential compactness (cf. Dieudonne [1, II, Chap. XII, section 12.15.9, p. 75]).

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(ii) If {d,An} converges in C(D), then by Theorem 2.2 (i) there exists dA e Cb(D) such that dAn -> oU in C(D) and hence in LP(D). So for all p e V(D}N,

By density of £>(£>) in L 2 (D), VeUn -»• VeU in L2(D)^-weak and hence dAn -»• cU in W1)2(D)-weak. From part (i) it converges in W1>p(.D)-weak for all p, 1 < p < oo. Conversely, the weakly convergent sequence converges to some / in W 1>P (Z)). By compactness of Cb(D) there exist a subsequence, still indexed by n, and dA such that d,An —>• rfyi in C(D] and hence in W1>p(D)-weak. By uniqueness of the limit, dA = f • Therefore, all convergent subsequences in C(D) converge to the same limit, so the whole sequence converges in C(D). This concludes the proof. (iii) Consider an arbitrary sequence {d,An} in Cd(D}. From Theorem 2.2 (ii) Cd(D] is compact and there exists a subsequence {d,An } and d,A & Cd(D) such that d,An —> dA in C(D). From part (ii) the subsequence weakly converges in W1>P(D) and hence Cd(D] is compact in W lip (Z>)-weak. Theorem 4.3. Let D be a closed domain in RN and {An} a sequence of nonempty sets converging to a nonempty set A of D in the Hausdorff topology:

Then

and for all compact subsets K of D

Corollary 1. Let D ^ 0 be a bounded open subset of R N and {Qn}, Qn ^ 0, a sequence of open subsets of D converging to an open subset Q, Q =£ 0, of D in the Hausdorff complementary topology:

Then

and

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1 77

Proof of Theorem 4.3. For all x ^ A, eU(#) > 0 and

So for all £ ^ A

Finally, for all x G A and all n > 1

and the result follows trivially by taking the limsup of each term. We conclude that limsup XT < XA, n—>oo

and by using the analogue of Fatou's lemma for the limsup we get for all compact subsets K of D

In order to get the Lp-convergence of the characteristic functions of the closure of the sets in the sequence, we need the Lp-convergence of the gradients of the distance functions which are related to the characteristic functions of the closure of the sets (cf. Theorem 3.2 (iv)). We have seen in Example 4.1 that the weak convergence of the characteristic functions is not sufficient to obtain the strong convergence of the sequence {cUn } to d,A in Wl'2(D). However, if we assume that (x^ } is strongly convergent, it converges to the characteristic function XB of some measurable subset B of D. Is this sufficient to conclude that XB — XA^ The answer is negative. The counterexample is provided by Example 5.2 in Chapter 3, where

for some B C D such that

Remark 4.1. In view of part (ii) of Theorem 4.1 an optimization problem with respect to the characteristic functions xn of open sets Q in D for which we have the continuity with respect to xn can be transformed into an optimization problem with respect to d^ in W1'l(D] since For instance, this would apply to the transmission problem (3.3)-(3.6) in section 3.1 of Chapter 3.

Chapter 4. Topologiews Generated by Distance Functions

a

5

Sets of Bounded and Locally Bounded Curvature

Going from the uniform Hausdorff metric topology to the Wl'p'-topology readily extends the applicability of the distance function to problems involving the characteristic function. However, even for a bounded open hold-all D, the families Cd(D) and Cj(D) are closed but not compact in W1'p(D)-stTong (cf. Example 4.1). The situation is similar to that encountered for the set of characteristic functions X(.D) in the I/p(D)-topology, where we have introduced the Caccioppoli sets. Their natural analogue in Cd(D] and CJ(-D) are the sets introduced by Delfour and Zolesio [17, 32], which include Ck domains, convex sets, and Federer's [2] sets of positive reach. They lead to compactness theorems for Cd(D] and C%(D) in Wljp(D).

5.1

Definitions and Properties

Definition 5.1. (i) Given a bounded open set D in R N , a subset O of D, fJ ^ 0 (resp., Ci7 ^ 0), is said to be of bounded exterior (resp., interior] curvature with respect to Dif

Denote those families as follows:

(ii) A subset O of R N , $ 7 ^ 0 (resp., Cf2 ^ 0), is said to be of locally bounded exterior (resp., interior) curvature if

where B(x,p) is the open ball of radius p > 0 in x. From Theorem 5.1 and Definition 5.1 in Chapter 3, a function belongs to BVi oc (R N ) if and only if for each x e RN it belongs to BV(B(x, p)) for some p > 0. As in Theorem 5.2 of Chapter 3 for Caccioppoli sets, it is sufficient to satisfy this condition for points of the boundary d£l. Thus property (5.2) only depends on the properties of the boundary. Theorem 5.1. Let ft, 0 ^ 0 (resp., C$1 ^ 0)), be a subset o/R N . Then Q is of locally bounded exterior (resp., interior) curvature if and only ifVda (resp., VC/QQ) belongs to BVloc(RN)N. This theorem will be proved later as part (iii) of Theorem 5.3. Since we simultaneously deal with sets O and their complement Cf2, we shall often use the notation A to cover both cases. Recall that the relaxation of the perimeter of a set was obtained from the norm of the gradient of its characteristic

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179

function for Caccioppoli sets. For distance functions X~A = 1 ~~ \^^A and the gradient of d& also has a jump discontinuity along the boundary dA of magnitude at most 1. So it is not too surprising to discover that the closure of a set A with bounded interior or exterior curvature is a Caccioppoli set. Theorem 5.2. (i) Let D be a bounded open Lipschitzian subset o/R N . For any 0 C D, 0 ^ 0 (resp., 0 7^ CQ) such that

0 (resp., CO) has finite perimeter; that is,

(ii) For any subset Q o/R N ; 0 ^ 0 (resp., 0 ^ CO) swc/i t/iat

D (resp., Cf2) /ias locally finite perimeter; that is,

Proof. Given Vd^ in BV(D) 7V , there exists a sequence {uk} in C°°(D}N such that

as /c goes to infinity, and since |VcJ^(o;)| < 1, this sequence can be chosen in such a way that This follows from the use of mollifiers (cf. Giusti [1, Thm. 1.17, p. 15]). For all V in T)(D}N

For each Uk

where T Duk is the transpose of the Jacobian matrix Duk and

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Figure 4.4. Vd^ for Examples 5.1, 5.2, and 5.3. since for W^'^Z^-functions ||V/||^i(£))N = \\^7f\\Mi(D')N• infinity

Therefore, as k goes to

where D2d,A is the Hessian matrix of second-order partial derivatives of d&- Therefore, Vxx e Ml(D}N. 5.2

Examples

It is useful to consider the following three simple illustrative examples (cf. Figure 4.4). Example 5.1 (half-plane in R2). Consider the domain

It is readily seen that

Thus Ad A is the one-dimensional Hausdorff measure of dA. Example 5.2 (ball of radius R > 0 in R 2 ). Consider the domain

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181

Clearly,

where HI is the one-dimensional Hausdorff measure. We see that Ad^ contains the one-dimensional Hausdorff measure of the boundary dA plus a term that corresponds to the volume integral of the mean curvature over the level sets of dA in CA Example 5.3 (unit square in R 2 ). Consider the domain

Since A is symmetrical with respect to both axes, it is sufficient to specify dA in the first quadrant. We use the notation Qi, Q2, Qa, and Q^ for the four quadrants in the counterclockwise order and c\, 02, 03, and 04 for the four corners of the square in the same order. We also divide the plane into three regions:

Hence

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<

Notice that the structure of the Laplacian is similar to that observed in the previous examples. The norm \\D2dA\\M1(uh(A)) IS decreasing as h goes to zero. The limit is particularly interesting since it singles out the behavior of the singular part of the Hessian matrix in a shrinking neighborhood of the boundary dA. Example 5.4. Let TV = 2. For the finite square and the ball of finite radius,

where HI is the one-dimensional Hausdorff measure. We now turn to the proof of Theorem 5.1, which is part of a more general theorem that contains several important technical results that will be used later for sets of positive reach and in Chapter 5. It is based on a slight extension of a result in Fu [1, Prop. 1.2]. We first give a few additional definitions. Definition 5.2. (i) Given any set A, 0 ^ A C R N , and a real number h > 0, the h-tubular neighborhood of A is defined as

and the closed h-tubular neighborhood of A as

(ii) A function / : U —>• R defined in a convex subset U of RN is convex if for all x and y in U and all A e [0,1],

It is concave if the function — / i s convex. A function / : U —•> R defined in a convex subset U of RN is semiconvex (resp., semiconcave] if

is convex in U.

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183

(iii) A function / : U —> R defined in a subset U of RN is locally convex (resp., locally concave) if it is convex (resp., concave) in every convex subset of U. A function / : U —>• R defined in a subset U of RN is locally semiconvex (resp., locally semiconcave) if it is semiconvex (resp., semiconcave) in every convex subset of U. When U is convex the local definitions coincide with the global ones. In general, Uh(A) C Ah, but the equality does not necessarily hold. Lemma 5.1. Given a subset A, A ^ 0, o/R N and h > 0, the function

is convex in R N . In particular,

are, respectively, locally convex in RN \Ah and Uh(A). Proof. For all p € A define the convex function

Since tp is nonnegative, $L is convex and

By subtracting the constant term p|2 + h2 and the linear term —2 x • p from ^, we get the new convex function

Then the function

is finite for each x e R N , convex in x, and

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Chapter 4. Topologies Generated by Distance Functions

If x is such that dA(x) > h, then for all p 6 A, \x — p\ > dA(x) > h and fc(a;) = \x\2 — 2hdA(x). If d,A(x] < /i, then there exists p £ A such that # — p\ < h and

Then, either for all p G A, \x — p\ < h and k(x) = \x\2 — h2 — d\(x) or, else, there exists p 6 A such that \x — p\ > /i,

and k(x) = \x 2 — d2A(x)~h2. We recover the result of Fu [1, Prop. 1.2] by observing that the restriction of k to RN \^4^ is locally convex. In addition, the function x 2 — d2A(x) is locally convex in Uh(A). This lemma has far-reaching consequences. Theorem 5.3. Let A be a nonempty subset o/R N . (i) The function fA(x) = \ (\x\2-d2A(x)) is convex mR N andVd\ €. BVioc(RN)N For all x and y in R N ,

or equivalently,

The set of projections T[A ( x ) onto A is a singleton PA (x) if and only if the gradient of fA exists. In that case

For all x and y in RN

The function d2A (x) is the difference

of two convex functions

(ii) VdA e BVioc(RN \A)N. More precisely, for all x 6 RN \A, there exists p > 0, 0 < 3p < dA(x], such that k&p is convex in B(x,1p) and hence VdA£BV(B(x,p))N.

6. Characterization of Convex Sets

185

(iii) A subset A ^ 0 of RN is of locally bounded exterior curvature if and only if VdA 0, \x\2 — d2A is locally convex in Uh(A) and hence in RN = (Jh>QUh(A). Therefore, \x\2 — d\ is convex in R N , and hence from Evans and Gariepy [1, Thm. 3, p. 240; Thm. 2, p. 239; and Aleksandrov's Theorem, p. 242], Vd2A G BVio C (R N ) ]V . Inequality (5.3) follows directly from the inequality Vx and y e R N , Vp(x) € KU(x),


since

and

Inequality (5.4) is (5.3) rewritten. Inequality (5.6) follows by adding inequality (5.3) to the same inequality with x and y permuted. (ii) For any point x in RN \A, there exists p, 0 < 3p < d,A(x) such that the open ball B ( x , 2 p ] is contained in R N \A p , where fc^ip is locally convex by Lemma 5.1. Hence k^,p is convex in J5(x, 2 p] and Vfcy^p, and a fortiori VC?A belong toBVCB^p))*. (iii) Clearly, if dA £ BVi oc (R N ) 7V , then property (5.2) is satisfied and A is of locally bounded exterior curvature. Conversely, by Theorem 5.1 of Chapter 3, it is sufficient to establish that for each x there exists p > 0 such that VdA G ~BV(B(x, p ) } N . This is true in hit A, where Vd,A = 0, and in dA by assumption. It is also true for any point x in RN \A by part (ii). Therefore, Vc^ belongs to BVi oc (R N ) Ar and A is of locally bounded exterior curvature. Similarly, for an open subset Q of RN we only need to prove that for each x there exists p > 0 such that Vd Cfi e BV(B(x,p)}N. This is true in int Cfi, where Vd Cn = 0, and in dft by assumption. It is also true for any point x in RN \CO by part (ii). Therefore, VC^QQ belongs to BVioc(RN)'/v and £7 is of locally bounded interior curvature.

6

Characterization of Convex Sets

In the convex case the squared distance function is differentiable everywhere and this property can be used to characterize the convexity of a set. Theorem 6.1. Let A be a nonempty subset o/R N with convex closure. Then

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Chapter 4. Topologies Generated by Distance Functions

(i) for each x G R N , HA(X)

=

{PA(X}} is a singleton and

(ii) d\ belongs to C^(RN) (and a fortiori to W^'C°°(RN)). Proof, (i) By definition

and since A is convex and z i—> \z — x\2 is strictly convex, there exists a unique PA(X] in A such that

(cf., for instance, Zarantonello [1, pp. 237-246] or Aubin [2, p. 24, Example 4.3]). So, for any two points x and y,

By adding up the above two inequalities,

(ii) From part (i) the map x i—> PA(X) is Lipschitz continuous and hence the map x i—>• Vd\(x) = 2[x— PA(X}} is also Lipschitz continuous. Therefore, o9A belongs toC^CR").

Theorem 6.2. (i) Let A be a nonempty subset of R N . Then (a) A convex =>• dyi convex; (b) rf^i convex =$> A convex; (c) Va; G R N , n^(a:) zs a singleton ^ A is convex.3 (ii) Let D be a nonempty open subset of R N . The subfamily

of Cd(D] is closed in C\OC(D}. It is compact in C(D] when D is bounded. The subfamily

of C%(D) is closed in C\OC(D}. It is compact in C(D] when D is bounded. 3

This part of the theorem is related to deeper results on the convexity of Chebyshev sets in metric spaces. A subset A of a metric space X is called a Chebyshev set provided that every point x of X has a unique projection PA (x) in A. The reader is referred to Klee [1] for details and background material.

6. Characterization of Convex Sets

187

(iii) For all subsets A of RN such that dA 7^ 0 and A is convex, Vd^ belongs to BVioc(R N )^> and the Hessian matrix D2d^ of second-order derivatives is a matrix of signed Radon measures that are nonnegative on the diagonal. Moreover, dA has a second-order derivative almost everywhere, and for almost all x and y in R N ,

as y —* x. Proof, (i) (a) Given x and y in R N , there exist x and y in A such that cU(#) = \x—x\ and cU(y) = \y~y\- By convexity of A, A is convex, and for all A, 0 < A < 1, A x + (I - X)y G A and

and dA is convex in R N . (b) If dA is convex, then

But x and y in A imply that
Thus Xx + (1 — A)y G A and A is convex. (c) By Theorem 6.1 (i) and for the converse, see in Valentine [1, p. 179, Prop. 7.1] the general result of Klee [1]. (ii) The set of all convex functions in C\OC(D} is a closed convex cone with vertex at 0. Hence its intersection with Cd(D] is closed. Any Cauchy sequence in Cd(D] converges to some convex dA- From part (i) A is convex. But dA = d^ and the nonempty convex set A can be chosen as the limit set. For the complementary distance function, it is sufficient to prove the compactness for D bounded. Given any sequence {^Q^n}, there exist an open subset 0 of D and a subsequence, still indexed by n, such that d^n —> d^ in C(D}. If £7 is empty, there is nothing to prove. If fi is not empty, consider two points x and y in £7 and A £ [0,1]. There exists r > 0 such that

There exists N > Q such that for all n > N, \\d^n — <^cnllc(D) < r /2 and

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Chapter 4. Topologies Generated by Distance Functions

By convexity of fin, x\ = \x + (1 — X)y 6 J7n,

Therefore, x\ 6 Q, (7 is convex, and C^(D] is compact. (iii) Prom part (i), cU is convex and continuous, and the result follows from Evans and Gariepy [1, Thm. 3, p. 240; Thm. 2, p. 239; and Aleksandrov's Theorem, p. 242].

Remark 6.1. It is important to observe that the convexity of d,A is equivalent to the convexity of the closure of A. Notice that the set A of all points of the unit open ball with rational coordinates has a convex closure but is not convex. Theorem 6.3. Let A be a nonempty subset o/R N : (i) The Fenchel transform

of d*A is given by

where O~A(X*) is the support function of A and

is the indicator function of the closed unit ball .6(0,1) at the origin. (ii) The Fenchel transform of d*A is given by

In particular, dco A is the convex envelope of d^ • The function dA and the indicator function 1-^ are both zero on A and the Fenchel transform of IA,

coincides with d*A on 5(0,1).

6. Characterization of Convex Sets

189

Proof, (i) From the definition,

(ii) We now use the property that a A — <7CO A '•

But in fact we have a saddle point. If x** € co~A, pick x* = 0 and

For x** $. co A rewrite the function as

Since p(x**) is the minimizing point of the function x — x** 2 over the closed convex set co A it is completely characterized by the variational inequality

By choosing the following special x*,

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Chapter 4. Topologies Generated by Distance Functions

we get

in view of (6.3). Therefore

Finally, from the previous theorem, dCOA is a convex function that coincides with the convex envelope since it is the bidual of d^Since the properties in Theorem 6.2 (iii) arise solely from the convexity of the function d^, the distance function d>A can be convex or semiconvex. However, it is never semiconcave in RN for nontrivial sets. Theorem 6.4. (i) Given a subset A, A ^ 0 o/R N ,

if and only if is locally convex in Uh(A). (ii) Given a subset A o/R N such that 0 ^ A ^ R N ,

Proof, (i) If fc is convex in R N , it is locally convex in Uh(A). Conversely, assume that there exists h > 0 and c > 0 such that fc is locally convex in Uh(A). From Lemma 5.1,

is locally convex in RN \Afl/2- Therefore, for c = max{c, 1/h}, the function fc is locally convex in RN and hence convex on R N . (ii) Assume the existence of a c > 0 for which fc is convex in RN. For each y 6 A the function

is also convex since it differs from fc(x) by the linear term

6. Characterization of Convex Sets

191

Since 0 / A =£ R N , there exist x E CA and p <E A such that

For any t > 0, define x t = p — t (x — p] and A = t/(l + t) G ]0,1[ and observe that

But

since by construction C[A(X} > 0 and cd^(x) < 1/2. Therefore for t, 0 < t < 1, the above quantity is strictly negative, and we have constructed two points x and Xt and a A, 0 < A < 1, such that

This contradicts the convexity of the function x i-+ Fc(x,p) and a fortiori of fc. Theorem 6.5. Let A be a nonempty subset o/R N for which the following condition is satisfied: Vx G dA, 3/9 > 0 such that d^ is semiconvex in B(x,p). Then the gradient of dA belongs to BVi oc (R' N ) 7V Proof. For all x e int A, dA = 0 and there exists p > 0 such that B(x,p) C int^4 and the result is trivial. For each x € R N \A, dA(x) > 0. Pick h = d^(x)/2. Then for all y G B(x, h) and all a G A,

and B(x,h) C RN \Ah- By Lemma 5.1 the function \x 2 /(2/i) — ^A^) is locally convex in RN \A^ and a fortiori convex in B(x,h). Finally, by assumption for all x E <9A, there exists p > 0 such that dA is semiconvex. Hence for each x G R N , VcU belongs to BV(5(x,p/2)) for some p > 0. Therefore, VdA belongs to BV loc (R N ) N .

192

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Chapter 4. Topologies Generated by Distance Functions

Federer's Sets of Positive Reach

For closed submanifolds of codimension larger than 1, £1 — <90, d$i = dg^, d^ = 0, and the gradient of d^ has a discontinuity along 0. In that case it is natural to go to the square d|fi = d^ of the function and relate the properties of d^ to the smoothness of 0. This is directly connected with the sets of positive reach as introduced by Federer [1].

Definition 7.1.

A nonempty set A C RN is said to have positive reach if there exists h > 0 such that HA(X) = {PA(X}} is a singleton for every x G Uh(A). The maximum h for which the property holds is called the reach of A and denoted reach (A). For a convex set A, reach (A) — +00. All nonempty convex sets have positive reach. Bounded domains and submanifolds of class C2 also have positive reach. A quite impressive result of Federer was to make sense of the classical SteinerMinkowski formula for that class of sets. He also showed that for all r, 0 < r < h, the boundaries of the closed tubular neighborhoods Ar are C1'1-submanifolds of codimension 1 in R N . The next theorem summarizes several characterizations of sets with positive reach. Note that condition (vii) is a global condition on the smoothness of d2A in the tubular neighborhood Uh (A) that is similar to the one of Definition 5.1(ii) on Vefo in a neighborhood of 50. Theorem 7.1. Given a nonempty subset A of R N , the following conditions are equivalent. (i) 3/i > 0 such that d& belongs to C^(Uh(A)\A). (ii) 3/i > 0 such that dA belongs to Cl(Uh(A}\A). (iii) 3h > 0 such that MX e Uh(A)\A, HA(X) is a singleton. (iv) 3h > 0 such that Vx € Uh (A), HA (x) is a singleton. (v) A has positive reach, that is, reach (A) > 0. (vi) 3/i > 0 such that PA belongs to C^(Uh(A)). (vii) 3/i > 0 such that d2A belongs to C^(Uh(A)). Proof. The elements of the proof can be found in Federer [1], (i) =>• (ii) =^> (iii) =$• (iv) => (v) are obvious. (v) =$>• (vi) For each x e Uh(A), HA(X) = {p(x)} is a singleton, and for all t > 0 such that £d^(rr) < h,

By assumption for all t > 0, td,A(x] < h, and the projection of p(x) + t(x — p(x}} onto A is unique and equal to p(x). Otherwise we could reduce d^(x), leading to a contradiction. Hence for all t > 0, tdA(x) < h,

7. Federer's Sets of Positive Reach

193

For all a G A, t/ G Uh(A) and t > 0 such that £cU(y) < h

So for any yi and y2 in RN and t > 0 such that tcUG/i) < ^ an
For any x G t/h(-A)> there exists p > 0 such that cU (#)+/» < /i. Let t = /i/(cU(:r)+p), which is strictly greater than 1. For all y £ B(x,p), d,A(y) < AA(X} + p and t d A ( y ) < h. Therefore, for all y\ and y^ in B(x,p]

The result follows from the fact that any compact subset K of Uh (A] can be covered by a finite number of neighborhoods B(xi, p i ) , Xi G K, pi > 0. (vi) => (vii) The proof follows from the identity PA(X) = x — ^Vd2A(x). (vii) =>• (i) For each x e Uh(A)\A, there exists p, 0 < /o, such that dA(x}+p < h. Therefore, for all y G 5(x,p), /i > ^A(^) + /o > cU(y) > ^A(^) — P > 0, and d^ G Cl'l(B(x,p}}. For any yi and y2 in B(x,p),

So, from the proof of (v) =4> (vi) and d A (y 2 ) > d^(x) — p > 0,

and Vd A £ C10'1 (•£?(£)/?))> d A G C 1)1 (5(x,p)). Therefore, the property holds in any compact subset of Uh(A)\A and dA G C110'c1(Vh(A)\A).

194

8

Chapter 4. Topologies Generated by Distance Functions

Compactness Theorems

Subsets of a bounded hold-all D with a positive reach greater than or equal to some h > 0 form a compact family of sets (cf. Federer [1, Thm. 4.13]). Theorem 8.1. Let D be a fixed bounded open subset o/R N . Let {An}, An ^ 0, be a sequence of subsets of D. Assume that there exists h > 0 such that

Then there exist a subsequence {Ank} and A C D, A ^ 0, such that d\ £ C&(Uh(AD and

This is a compactness theorem similar to the compactness of Cd(D) in C(D] for a bounded open subset of R N . For the family of sets with bounded interior or exterior curvature, the key result is the compactness of the embeddings

for bounded open Lipschitzian subsets D of RN and p, 1 < p < oo. It is the analogue of the compactness theorem (Theorem 5.3 of Chapter 3) for Caccioppoli sets

which is a consequence of the compactness of the embedding

for bounded open Lipschitzian subsets D of RN (cf. Morrey [1, Def. 3.4.1, p. 72; Thm. 3.4.4, p. 75] and Evans and Gariepy [1, Thm. 4, p. 176]). As for characteristic functions in Chapter 3, we give a first version involving global conditions on a fixed bounded open Lipschitzian hold-all D. In the second version the sets are contained in a bounded open hold-all D with local conditions in their tubular neighborhood or the tubular neighborhood of their boundary. 8.1

Global Conditions on D

Theorem 8.2. Let D be a nonempty bounded open Lipschitzian subset o/R N . The embedding (8.2) is compact. Thus for any sequence {&,„.}, ® ^ Qn, of subsets of D such that

8. Compactness Theorems

195

there exist a subsequence {Qnk} and a set £1, 0 ^ 0, such that Vcfo G BV(D) Af and for all

andxn GBV(£>). Proof. Given c > 0 consider the set

By compactness of the embedding (8.5), given any sequence {cfo^}, there exist a subsequence, still denoted {dnT,}, and / G EV(D}N such that Vcfon —» / in Ll(D}N. But by Theorem 2.2 (ii), C d (D) is compact in C(D) for bounded D and there exist another subsequence {dnn } and d$i G Cd(D] such that d^r, "^ ^n m C(D) and, a fortiori, in Ll(D). Therefore, dnn converges in W1'1(D) and also in Ll(D}. By uniqueness of the limit / = Vefo and d^7i converges in Wl'l(D) to cfo. For $ G Vl(D}NxN as fc goes to infinity

D2d$i \MI(D) ^ c? and Vd^ G BV(Z)) 7V . This proves the compactness of the embedding for p = I and properties (8.7). The conclusions remain true for p > I by the equivalence of the VF^-topologies on Cd(D] in Theorem 4.1(i). When D is bounded open, C^(D) is compact in C(D] and closed in W lip (D), 1 < p < oo, and we have the analogue of the previous two compactness theorems. Theorem 8.3. Let D be a nonempty bounded open Lipschitzian subset o/R N . The embedding (8.3) is compact. Thus for any sequence {£ln} of open subsets of D such that there exist a subsequence {Qnk} ana an open subset £1 of D such that V^cn £ BV(D}N and

for all p, I < p < oo.

ana?^(EBV(L>).

196

Chapter 4. Topologies Generated by Distance Functions

Proof. Given c> 0 consider the set

By compactness of the embedding (8.5), given any sequence {d^n} there exist a subsequence, still denoted {^cnn}? an<^ / ^ BV(-D)^ such that Vrffj On —> / in L1(D)N. But by Theorem 2.4 (if), C%(D) is compact in CQ(D) for bounded D and there exist another subsequence {<^rjQn } ano^ ^Cfi ^ C%(D) such that ^con "^ ^Cft in Co(D) and, a fortiori, in Ll(D}. Therefore, drjo k converges in Wo' (D) and also n

in Ll(D). By uniqueness of the limit, / = Voffj^ and rfrjnn converges in W 0 ' (Z?) to dcn. For all $ € T>1(D)NxN as k goes to infinity

II-P2^ II MI (D) < c, VrfCo e BV(J9)jv, and xCfi e BV(D). This proves the compactness of the embedding for p = 1 and properties (8.10)-(8.11). The conclusions remain true for p > 1 by the equivalence of the W1)p-topologies on C^(D] in Theorem 4.1 (i).

8.2

Local Conditions in Tubular Neighborhoods

The global conditions (8.6) and (8.8) can be weakened to a local one in a neighborhood of each set of the sequence. Simultaneously the Lipschitzian condition on D can be removed since only the uniform boundedness of the sets of the sequence is required. Theorem 8.4. Let D be a nonempty bounded open subset o/R N and {^ln}, 0 ^ Q n; be a sequence of subsets of D. Assume that there exist h > 0 andc > 0 such that

Then there exist a subsequence {Onfc} and a subset fi, 0 ^ fi, of D such that Vcfo € BVi oc (R N ) 7V , and for all p, l
Moreover, for all (p e £>° (tf/i(fi)),

and XQ belongs to BVi oc (R N ).

8. Compactness Theorems

197

Proof. First notice that since Qn is bounded and nonempty, d$ln ^ 0. Further, Uh(dQn) can be replaced by C/^(Q n ) in condition (8.12) and it is sufficient to prove the theorem for that case. Lemma 8.1. For any fJ 7 <9fJ 7^ 0 and h > 0,

The proof of the lemma will be given after the proof of the theorem. (i) The assumption fin c D implies that C/^(On) C Uh(D}. Since Uh(D}js bounded, there exist a subsequence, still indexed by n, and a set 0, 0 ^ 0 C D, such that and another subsequence, still denoted {o?nn}, such that

For all e > 0, 0 < 3e < /i, there exists N > 0 such that for all n > N and x in tfh(0)

Therefore,

From (8.12) and (8.15)

In order to use the compactness of the embedding (8.5) as in the proof of Theorem 8.2, we would need Uh-£(^i) to be Lipschitzian. To get around this, we construct a bounded Lipschitzian set between Uh-ie(ty and t/^_ e (fi). Indeed, by definition,

and by compactness, there exists a finite sequence of points {x^}"_=1 in $1 such that

Since UB is Lipschitzian as the union of a finite number of balls, it now follows by compactness of the embedding (8.5) for UB that there exists a subsequence, still denoted {cfon}, and / e EV(UB}N such that Vcfon -> / in Ll(UB)N. Since Uh(D) is bounded, Cd(Uh(D}} is compact in C(Uh(D}} and there exists another subsequence, still denoted {d^n}, and 0 ^ 0 C D such that efon —» d^ in C(Uh(D)} and, a fortiori, in Ll(Uh(D}}. Therefore, d$in converges in WI>I(UB] and also in LI(UB)- By uniqueness of the limit, / = Vc?n on UB and d^,, converges to cfo in

198

Chapter 4. Topologies Generated by Distance Functions

Wl'l(UB}- By Definition 5.2 and Theorem 5.3 (iii), Vcfo and Vcfon all belong to BVi oc (R N ) JV since they are BV in tubular neighborhoods of their respective boundaries. Moreover, by Theorem 5.2 (ii), %^ E BVi oc (R N )- The above conclusions also hold for the subset t/j l _2 £ (fi) of UB(ii) Convergence in W1'p(Ufl(D)). Consider the integral

Prom part (i) the first integral on the right-hand side converges to zero as n goes to infinity. The second integral is on a subset of Ct// l _2e(fi). From (8.16) for all n>N,

The second integral reduces to

which converges to zero since Vcfon —*• Vcfo in L2 (Uh(D))N-weak in part (i) and the fact that |Vcfo| = 1 almost everywhere in Uh(D)\Uh-2e(ty- Therefore, since dfln-^dnmC(U^(D)),

and by Theorem 4.1(i) the convergence is true in W1>p(C7/l(Z))) for all p > 1. (iii) Properties (8.14). Consider the initial subsequence {dnn} which converges to d$i in H1 (Uh(D))-weak constructed at the beginning of part (i). This sequence is independent of e and the subsequent constructions of other subsequences. By convergence of dnn to cfo in Hl(Uh(D))-weak for each $ e T>l(Uh(fy)NxN,

Each such $ has compact support in t/i^fi), and there exists e ~ £•($) > 0, 0 < 3e < /i, such that

From part (ii) there exists N(e) > 0 such that

8. Compactness Theorems

199

For n > N(e) consider the integral

By convergence of VUf4(ft))-weak, for all $ G T>l(Uh(tt))NxN

Finally, the convergence remains true for all subsequences constructed in parts (i) and (ii). This completes the proof. Proof of Lemma 8.1. First check that

It is sufficient to show that all points x of int 0 are contained in the right-hand side of the above expression. If ddn(x) < h, then x G Uh(dft); if ddti(x) > /i, then B(x,ti) C 0 and x E ft-h. For all $ e T>l(Uh(ty}NxN, $ has compact support and there exists £ > 0 , 0 < 2 e < / i , such that

But Uh-e(ty C Uh(Q) and there exists {xj G fJ_h : 1 < j < m} such that

Let i/>0 G P(C/h.(5ri)) and ^ G T>(B(xj,h}} be a partition of unity such that

Consider the integral

200

Chapter 4. Topologies Generated by Distance Functions

By construction for ||$||£7(1^(0)) < 1

But for j, 1 < j < m, B(xj,h] C intfJ, where both d^ and Vcfo are identically zero. As a result the above integral reduces to

and this completes the proof. Theorem 8.5. Let D be a nonempty bounded open subset o/R N . Let {fJn} be a sequence of nonempty open subsets of D and assume that there exist h > 0 and c > 0 such that

Then there exist a subsequence {Slnk} and an open subset J7 of D such that VC?QQ G BVi oc (R N ) 7V , and for all p, l
Moreover, for all (p 6 Z>° (£//»((&)),

and xsi belongs to BVi oc (R N )Proof. First note that <9J7n ^ 0 since £7n is bounded and nonempty. By Lemma 8.1, Uh(d£ln) can be replaced by C//j(COn) in condition (8.17). Moreover, since CfJn can be unbounded, we shall work with the bounded neighborhoods t/^(C^Jln) of C^f2n rather than with tT/^Cf^n). Lemma 8.2. Let h > 0 and let 17 be an open subset of a nonempty bounded open subset D o/R N . Then

Ifdtt^0,

8. Compactness Theorems

201

The proof of this lemma will be given after the proof of the theorem. (i) By Theorem 2.4 (ii) since D is bounded, there exists a subsequence, still denoted {dcnn}> and an open subset f2 of D such that

and another subsequence, still denoted {dfmn}, such that

For all e > 0, 0 < 3e < h, there exists TV > 0 such that for all n > N

Clearly, since fi C D and On C D and D is open, C^0n ^ 0 and C^O / 0. Furthermore,

From (8.17) and (8.20),

As in part (i) of the proof of Theorem 8.4, we can construct a bounded open Lipschitzian set UB between Uh-2e(^rjft) and [/^-^(C^O) such that

Since UB is bounded and Lipschitzian, it now follows by compactness of the embedding (8.5) for UB that there exists a subsequence, still denoted {d(jQn}, and / € BV(UB)N such that Wc^ -> / in Ll(UB}N. Since L> U Z7B is bounded, C^(D U £/#) is compact in C$(D U C/B), and there exists another subsequence, still denoted {d^n}, and an open subset Q of D such that d^n —>• ^Q^ in Co(D U t/s) and, a fortiori, in L^DUt/e). Therefore d^n converges in WI>I(UB) and also in LI(UB}- By uniqueness of the limit, / = Vdco on BD and c/Q Qj converges to d^Q in H /I ' I ([/B). By Definition 5.2 and Theorem 5.3 (iii), V^QQ and V^QQ^ all belong to BVioc(R'N)JV since they are BV in tubular neighborhoods f/^(C^Jl) and Uh(£>pQn} of their respective boundaries 90 and d£ln. Moreover, by Theorem 5.2 (ii), xcn ^ BVi oc (R N ) and, a fortiori, xn since 17 is open. The above conclusions also hold for the subset Uh-2e(^>D^ °^ UB(ii) Convergence in WQ'P(D}. Consider the integral

202

Chapter 4. Topologies Generated by Distance Functions

From part (i) the first integral on the right-hand side converges to zero as n goes to infinity. The second integral is on a subset of C^f//1_2e(C^O). From the relations (8.21) for all n > N,

The second integral reduces to

which converges to zero by weak convergence of Wcfin to Vo^ in L2(D)N and the fact that |Vdg^| = 1 almost everywhere in J D\C// l _2 £ (C^fi). Therefore, since d Cttn ->• dCfi in C0(D) and by Theorem 4.1 (i) the convergence is true in WQ'P(D) for all p > 1. (iii) Properties (8.19). Consider the initial subsequence {d^n}, which converges to rffjo in HQ(D U [/h(Cfi))-weak constructed at the beginning of part (i). This sequence is independent of e and the subsequent constructions of other subsequences. By convergence of d^ to d^ in H$(D U ?7^(CO))-weak for each

fcePH^Cn))^",

Now each $ G P 1 (C// l (Cfi))^ Vx7V has compact support in [/^(CO) and there exists e = e($) > 0, 0 < 3e < h, such that From part (ii) there exists N(e) > 0 such that

For n > N(e) consider the integral

By convergence of VdCfln to WCn in L2(JDUC//l(CO))-weak, for all $ e T>l(Uh(£ty)N>

Finally, the convergence remains true for all subsequences constructed in parts (i) and (ii). This completes the proof.

8. Compactness Theorems Proof of Lemma 8.2. The proof is similar to that of Lemma 8.1 with

since both d^ and VC^CQ are identically zero on CD.

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Chapter 5

priented Distance Function and Smoothness of Sets

1

Introduction

In this chapter we study oriented distance functions and their role in the description of the geometric properties of domains and their boundaries. They are also known as algebraic or signed distance functions. Our choice of terminology emphasizes the fact that, for a smooth domain, the associated oriented distance function defines an orientation of the normal to the boundary. They enjoy many interesting properties. For instance, they retain the nice properties of the distance functions but also generate the classical geometric properties associated with sets and their boundaries. The smoothness of the oriented boundary functions in a neighborhood of the boundary of the set is equivalent to the smoothness of its boundary. Similarly, the convexity of the function is equivalent to the convexity of the closure of the set. In addition, their respective gradient and Hessian matrix, respectively, coincide with the unit outward normal and the second fundamental form on the boundary of the set. Finally, they provide a framework for the classification of domains and sets according to their degree of smoothness, much like Sobolev spaces and spaces of continuous and Holderian functions do for functions. The first part of the chapter deals with basic definitions, constructions, and results. The second part specializes to specific subfamilies of oriented distance functions. The last part concentrates on compact families of subsets of oriented distance functions. In the first part, section 2 presents the basic properties, introduces the uniform metric topology, and shows its connection with the Hausdorff and complementary Hausdorff topologies of Chapter 4. Section 3 is devoted to the differentiability properties, the associated set of projections onto the boundary, and completes the treatment of skeletons and cracks. Section 4 gives the equivalence of the smoothness of a set and the smoothness of its oriented distance function in a neighborhood of its boundary for sets of class C1'1 or better. When the domain is sufficiently smooth the trace of the Hessian matrix of second-order partial derivatives on the boundary is the classical second fundamental form of geometry. Section 5 deals with the VF1'p-topology on the set of oriented distance functions 205

206

Chapter 5. Oriented Distance Function and Smoothness of Sets

and the closed subfamily of sets for which the volume of the boundary is zero. In the second part of the chapter, in section 6, we study the subfamily of sets for which the gradient of the oriented distance function is a vector of functions of bounded variation: the sets with global or local bounded curvature. They are rather large classes for which quite general compactness theorems will be proved in section 9. Some examples are given to illustrate the behavior of the norms in tubular neighborhoods as the thickness of the neighborhood goes to zero in section 6.2. Section 6.3 introduces Sobolev or Ws'p-domains, which provide a framework for the classification of sets according to their degree of smoothness. For smooth sets this classification intertwines with the classical classification of Ck- and Holderian domains. Section 7 extends the characterization of closed convex sets by the distance function to the oriented distance function and introduces the notion of semiconvex sets. The set of equivalence classes of convex subsets of a compact hold-all D is again compact, as it was for all the topologies considered in Chapters 3 and 4. Section 8 shows that sets of positive reach introduced in Chapter 4 are sets of locally bounded curvature and that their boundary has zero volume. In the last part of the chapter, section 9 gives compactness theorems for sets of global and local bounded curvatures from a uniform bound in tubular neighborhoods of their boundary. They are the analogues of the theorems of Chapter 4. Finally, section 10 gives the compactness of the sets of Lipschitzian domains in a bounded hold-all under the uniform cone property for the VFlip-topology associated with the oriented distance functions. We recover as a corollary the compactness of Theorem 5.9 of section 5.4 in Chapter 3 for the associated family of characteristic functions in LP(D}. Furthermore, we get the compactness of the set of characteristic functions of the complements in LP(D] and the compactness of the associated families of distance functions and distance functions of the complement in W1
2

Uniform Metric Topology

2.1

The Family of Oriented Distance Functions Cb(D)

The distance function oU provides a good description of a domain A from the "outside." However, its gradient undergoes a jump discontinuity at the boundary of A which prevents the extraction of information about the smoothness of A from the smoothness of dA in a neighborhood of dA. To get around this, it is natural to take into account the negative of the distance function of its complement (M, which somehow cancels the jump discontinuity at the boundary. Definition 2.1. Given a subset A of R N , the oriented distance function from x to A is defined as

2. Uniform Metric Topology

207

Associate with a nonempty subset D of RN the family

This new function gives a level set description of a set whose boundary coincides with the zero level set. From Chapter 4 the function bA is finite in RN if and only if 0 7^ A ^ R N , since 6^ = dA = +00 when A = 0 and 6,4 = — d^A = —oo when [>A = 0. This condition is completely equivalent to dA ^ 0, in which case bA coincides with the algebraic distance function to the boundary of A:

Noting that b^A = —bA, it means that we have implicitly chosen the negative sign for the interior of A and the positive sign for the interior of its complement. We shall see later that for sets with a smooth boundary, the restriction of the gradient of bA to dA coincides with the outward unit normal to the boundary of A. Changing the sign of bA gives the inward orientation to the gradient and the normal. Theorem 2.1. Let A be a subset o/R N . Then (i) A ^ 0 and (L4 ^ 0 <=» dA ^ 0.

(ii) Given bA and bs in Cb(D],

bA = bB in D «=> ~B = A and CU = £B <=^ ~B = A and dA = dB.

In particular, bA < bA and bA = bA 4=> dA — dA.

(iii) (iv) (v)

(vi) If dA ^ 0, the function bA is uniformly Lipschitz continuous in RN and

Moreover, bA is (Frechet) differentiable

almost everywhere and

208

Chapter 5. Oriented Distance Function and Smoothness of Sets

Proof, (i) The proof is obvious, (ii) By assumption,

Since A C D and B C D, then £A D Cl) and C# D C.D and

Also, since B C D for all x e B,
and a fortiori in D. The equality case follows from the fact that bA = 65 if and only if bA > &s and 6^
since CA D <M and_the inf over CA is smaller than the inf over its subset dA. Similarly, for x in C^4

and finally Conversely, for each x in C^4, the set of projections UA(X} C A fi C^4 = dA is not empty. Hence

and similarly for all x in A,

Therefore (6^(^)1 > ddA(x). (iv) If 6.4 = d,A — d£A > 05 then rf^ > d$A and A C C^4, and necessarily A C dA and A = <9A Conversely, if x £ 5-A, then by definition 6,4(2:) = 0. If x ^ 5A, then x e C5^ = CI = int CA and 6^(a;) = dA(x) > 0. (v) bA = 0 is equivalent to 6^ > 0 and b^A — ~bA > 0. Then we apply (v) twice. But CU = dA = A <=> int C^ = 0 = int A <^=> dA = R N . (vi) Clearly,

2. Uniform Metric Topology

209

For x e A and y e int CA = CA, cU(j/) > 0 and

By assumption B(y,dA(y}) C intCA Define the point

The argument is similar for x e int A and i/ £ C A The differentiability follows from Theorem 2.1 (vii) in Chapter 4.

2.2

Uniform Metric Topology

From Theorem 2.1 (i) the function b^ is finite at each point when dA ^ 0. This excludes A = 0 and A = R N . The zero function &A(#) = 0 Vx G RN corresponds to the equivalence class of sets A such that

This class of sets is not empty. For instance, choose the subset of points of RN with rational coordinates or the set of all lines parallel to one of the coordinate axes with rational coordinates. Let D be a nonempty subset of RN and associate with each subset A of D, dA 7^ 0, the equivalence class

and the family of equivalence classes

The equivalence classes induced by 6^ are finer than those induced by G?A since both the closures and the boundaries of the respective sets must coincide. As in the case of dA we identify ^(D) with the family Cb(D] through the embedding

When D is bounded, the space C(D] endowed with the norm H/Hc^D) is a Banach space. Moreover, for each A C D, 6^ is bounded, uniformly continuous on D, and bA G C(D}. This will induce the following complete metric:

on Fb(D).

210

Chapter 5. Oriented Distance Function and Smoothness of Sets

When D is open but not necessarily bounded, we use the space C\OC(D) defined in section 2.2 of Chapter 4 endowed with the complete metric p$ defined in (2.8) for the family of seminorms {QK} defined in (2.7). It will be shown below that Cb(D] is a closed subset of C\OC(D] and that this induces the following complete metric on K(D):

The following subfamilies:

of Fb(D) and Cb(D) will be important. We now have the equivalent of Theorem 2.2 of Chapter 4. Theorem 2.2. Let D, 0 ^ D, be an open (resp., bounded open) subset o/R N . (i) The set Cb(D] is closed in C\OC(D] (resp., C(D)), and ps (resp., p] defines a complete metric topology on Fb(D}. (ii) For a bounded open subset D o/R N , the set Cb(D) is compact in C(D}. (iii) For a bounded open subset D o/R N , the map

is continuous: for all b^ and bs in Cb(D],

Proof, (i) It is sufficient to consider the case for which D is bounded. Consider a sequence {An} C D, dAn ^ 0 such that b&n —> / in C(D) for some / in C(D). Associate with each g in C(D] its positive and negative parts

Then by continuity of this operation

By Theorem 2.2 (i) of Chapter 4, there exists a closed subset F, 0 ^ F C D, such that

2. Uniform Metric Topology

211

Moreover, F ^ RN since D is bounded. By Theorem 2.4 and the remark at the beginning of section 2.3 of Chapter 4, there exists an open subset G C -D, G ^ R N , such that

Therefore,

Define the sets

They form a partition of R N , RN = A~ U A° U A+, and

If A° was empty, R N could be partitioned into two nonempty disjoint closed subsets. Since A° is closed

Moreover, since A" and A+ are open, A" C A~ U A°, A+ C ^4+ U A°,

Let Q be the subset of points in RN with rational coordinates. Define

and notice that by density of Q and CQ in R N ,

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Chapter 5. Oriented Distance Function and Smoothness of Sets

Consider the following new partition of RN:

Define

and

But

and since dA~ U 8A+ = dA°

(ii) For the compactness, consider any sequence {&An} C Cb(D] C C0)1(.D). Since D is bounded, bAn and V6,4n are both pointwise uniformly bounded in D. From Theorem 2.2 in Chapter 2 the injection of C°'l(D) into C(D] is compact and there exist / € C(D] and a subsequence {6^n } such that 6^n —>• / in C(D}. From the proof of the closure in part (i), there exists A C D, dA -•£ 0, such that / = &A(iii) For all 6,4 and bs in Cb(D) and rr in Z),

Moreover, dA = b\ — (\b&\ + ^)/2 and C?QA = b~A = (|6^| — &A)/2, and necessarily

By combining the above three inequalities we get (2.8). We have the analogue of Theorem 2.3 of Chapter 4 and its corollary for A, CA and dA. In the last case it takes the following form (to be compared with Richardson [1, Lem. 3.2, p. 44] and Kulkarni, Mitter, and Richardson [1] for an application to image segmentation):

3. Projection, Skeleton, and Differentiability of bA

213

Corollary \. Let D be a nonempty open (resp., bounded open) subset o/R N . Define for a subset S of RN the sets

(i) Let S be a subset o/R N . Then Hb(S) is closed in C\OC(D) (resp., C(D)). (ii) Let S be a dosed subset o/R N . Then Ib(S) is closed in C\OC(D) (resp., C(D}}. If, in addition, SC\D is compact, then Jb(S) is closed in C\OC(D) (resp., C(D}}. (\\\) Let S be an open subset o/R N . Then Jb(S) is open in C\OC(D) (resp., C(D)). If, in addition, CSTl-D is compact, then Ib(S) is open in C\OC(D) (resp., C(D)). (iv) For D bounded, associate with an equivalent class [A]b the number #b([A]b) = number of connected components of dA. Then the map is lower semicontinuous. Proof. This proof is the same proof as for Theorem 2.3 of Chapter 4 using the Lipschitz continuity of the map b^ |~^ dgA — \^A from C(D) to C(D).

3

Projection, Skeleton, and Differentiability of 6^

In this section we study the connection between the gradient of 6,4 and the projection onto dA and the characteristic functions associated with dA. We further relate the set of singularities of the gradients and the notions of skeleton1 and set of cracks. Definition 3.1.

Let A be a subset of RN such that 0 ^ dA. (i) The set of projections of x onto dA is given by

The elements of HQA(X] are called projections onto dA and denoted by PQA(X).). (ii) The set of points where the projection onto dA is not unique,

is called the skeleton of A. Since TlgA(x) is a singleton for x 6 dA, then 1

Our definition of a skeleton does not exactly coincide with the one used in morphological mathematics (cf., for instance, Matheron [1] or Riviere [1]).

214

Chapter 5. Oriented Distance Function and Smoothness of Sets

(iii) The set of cracks is defined as

The set of singularities, Sing(V6^), of V&A has zero ^'-dimensional Lebesgue measures since 6^ is Lipschitz continuous and hence differentiable almost everywhere. The function b^ enjoys properties similar to the ones of &A and d§A since (6^4 = ddA and we have the analogue of Theorem 3.1 in Chapter 4. Theorem 3.1. Let Abe a subset o/R N such that 0 ^ dA, and let x 6 R N . Define

(i) The set HQA (x) is nonempty, compact, and

(ii) For all x and v in RN

where &B is the support function of the set B,

and coB is the convex hull of B. (iii) The following statements are equivalent: (a) b\(x) is (Frechet) differentiable atx, (b) b\(x) is Gateaux differentiable at x, (c) HgA.(x) is a singleton. Henceforthh

(iv) When b\ is differentiable

at x, 110,4(a;) = {PQA(X}} is a singleton, and

3. Projection, Skeleton, and Differentiability of 6^

For all x G dA, H.QA(X] = {x}, b\ is differentiable Hence

215

at x and Vb2A(x] = 0.

Proof. The result follows from Theorem 3.1 of Chapter 4 using the identity b2A = dlA,

Remark 3.1. The uniqueness of the projection pdA(%) at x is equivalent to the existence of V6^(x). In both cases the identity

is satisfied, and all the properties of PQA can be obtained from those of b2A or JQAWe now compute the gradient of bA and relate it to the characteristic functions of 0A Theorem 3.2. Let A be a subset o/R N such that 0 ^ dA. (i) // Vb^(x) exists at a point x in R N , then HQA(X) — {POA(X)} is a singleton,

In particular, for almost all x € dA, V6^(x) = Vdg^(x) = 0,

and the set

has zero N-dimensional Lebesgue measure. (ii) For all x G dA, V6^(:c) exists and is equal to 0. For all x £ QA, bA is differentiable at x if and only if b\ is differentiable at x. In particular,

and the last identity is satisfied almost everywhere in R N .

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Chapter 5. Oriented Distance Function and Smoothness of Sets

(iii) Given x G R N , a G [0,1], p G T\.dA(x), and xa = p + a (x — p), then

In particular if HQA(X) is a singleton, then HdA(xa) is a singleton and Vb\(xa) exists for all a, 0 < a < 1. //, in addition, x ^ dA, then V6^(x a ) exists for allO
and POA(X) € dA is unique. Similarly, for x G int CA, 6^ = d^-,

and POA(X] 6 dA is unique. Finally, when x 6 dA, PQA(X) = x. The distance function ddA is differentiable almost everywhere. From Theorem 3.2 (i) in Chapter 4, whenever it is differentiable at a point x G dA, Vda^(o;) = 0. But for all v in RN and t > 0

since |6^| = rig A, and for almost all x € dA, V6^(x) = \7dgA(x) = 0. Since 6^ is a Lipschitzian function, it is differentiable almost everywhere in R N , and in view of the previous considerations, when it is differentiable

Therefore, the set dbA has at most a zero measure. (ii) For x G dA and any y G R N , consider the differential quotient

Then

Hence

3. Projection, Skeleton, and Differentiability of 6^

217

If x ^ dA and Vb\(x} exists, then bA(x) ^ 0 and

To see this consider the new differential quotient

Then

Since b\ is differentiate at x, the first term goes to zero as y goes to x. As for the second term it is bounded by

and hence goes to zero as y goes to x. So the term

and since 6^(2^) 7^ 0, q(y] ~+ 0 as y —> x and b^ is differentiable at x. In the other direction the result is trivial. (iii) The proof of (iii) is straightforward. Remark 3.2. For sufficiently smooth domains, the subset d^A of the boundary dA coincides with the reduced boundary of finite perimeter sets. Remark 3.3. In general, VdA(x) and Vdrj A (x) do not exist for x € dA. This is readily seen by constructing the directional derivatives for the half-space

at the point (0,0). Nevertheless, V6^(0,0) exists and is equal to (1,0), which is the outward unit normal at (0,0) € dA to A. Note also that for all x G <9A, |V6^(^)| = 1- This is possible since mjv(<9A) = 0. If Vb^(x) exists, then it is easy to check that \VbA(x)\ < 1.

218

4

Chapter 5. Oriented Distance Function and Smoothness of Sets

Boundary Smoothness and Smoothness of bA

The smoothness of the boundary dA is directly related to the smoothness of the function bA in a neighborhood of dA. The next two theorems establish the equivalence for domains with a C1'1- or smoother boundary. Their proofs will use part (iv) and (vii) of the following specialization of Theorem 7.1 of Chapter 4 to dgA = \bA\Tubular neighborhoods are defined in Definition 5.2 of Chapter 4. Theorem 4.1. Given a subset A o/RN such that dA ^ 0, the following conditions are equivalent. (i) 3/i > 0 such that bA belongs to C^(Uh(dA)\dA). (ii) 3/i > 0 such that bA belongs to Cl(Uh(dA}\dA). (iii) 3/i > 0 such that Vx G Uh(dA)\dA, 110,4 (#) is a singleton. (iv) 3h > 0 such that Va: 6 Uh(dA), HQA(x) is a singleton. (v) dA has positive reach, that is, reach (dA) > 0. (vi) 3/i > 0 such that PQA belongs to C^c(Uh(dA)). (vii) 3/i > 0 such that b2A belongs to C^(Uh(dA)}. In the first direction the smoothness of bA in a neighborhood of dA implies the smoothness of the domain. Theorem 4.2. Let k > 1 be an integer, £, 0 < i < I , be a real number, and A be a subset of RN such that 0 ^ dA ^ R N . // for each x in dA there exists an open neighborhood V(x] of x where bA & Ck^(V(x}}, then A is a set of class Ck^ and m(dA) = 0. Moreover, UgA(y) — {pdA(y}} is a singleton in the open neighborhood

of dA, and

Proof. By construction V is an open neighborhood of dA arid 0 ^ dA ^ V since 0 ^ dA ^ R N . By assumption, bA <E Cfc'(V) and hence b'2A belongs to Cfc*(V). By Theorem 3.1 (iii), 110^(y) — {pdA(y}} is a singleton in V. By Theorem 4.1, b A £ ciol(y}- P™111 Theorem 3.2 (i), \VbA\ = 1 in V\dA and VbA = 0 almost everywhere in dA. But \VbA\ e C(V) and hence either |V6^| = 0 or |V6^| = 1 in V. In the first case this would imply that 6^ = 0 in V since 6^ = 0 on dA. As a result, dA C V C dA and dA would be open and closed, that is, equal to 0 or R N , which is not possible by assumption. Hence |V6^| = 1 in V. So the set A is characterized by the level sets of the function 6^ and |V6^| = 1 on dA — 6^1{0}. By Theorem 4.2 in Chapter 2 A is of class CM, and by Theorem 3.2 (i) m(dA) = 0 since |V6^| = 1 on dA.

4. Boundary Smoothness and Smoothness of 6^

219

In the other direction the smoothness of the domain implies the smoothness of bA in a neighborhood of dA when the domain is at least of class C1'1. A counterexample will be given in dimension 2. Theorem 4.3. Let A be a subset o/R N such that dA ^ 0. (i) If A is of class C1'1, then for each x G dA, there exists a neighborhood W(x] of x such that &A G Cl'l(W(x}}. Moreover, V6^ = nopdA in W(x). (ii) If A is of class Ck^ for an integer k, k > 2, and a real number i, 0 < i < 1, then for each x G dA, there exists a neighborhood W(x] of x such that 6^4 G Ck'\W(x}} and b\ G C^l(W(x)). Proof, (i) From the definition of a set of class C1'1 for each x G dA, there exists a bounded open neighborhood U(x) of x where the set A can be locally described by the level sets of the C1'^function

since by definition

Denote by cx the Lipschitz constant of V/ in U(x). The boundary dA is the zero level set of / and the gradient

is normal to that level set. Thus the outward normal to A on dA is given by

There exists a bounded open neighborhood V(x) of x, and a. > 0 such that V(x) C U(x),

By the characterization of dA for all y G V(x),

We know that the set of minimizers

220

Chapter 5. Oriented Distance Function and Smoothness of Sets

is nonempty and compact. By using the Lagrange multiplier theorem for the Lagrangian

the elements p of Hg^d/)

are

characterized by the existence of a X(p] G R such that

since V/(jp) ^ 0 on dA. The next question is the uniqueness of p. Construct the function

and show that, for y sufficiently close to dA, F(y,p) has a unique fixed point p. Consider for z2 and z\ in V(x) the difference

In particular, the outward unit normal is Lipschitzian

Coming back to (4.3),

Choose the new neighborhood of x:

So for all y € W(x)

4. Boundary Smoothness and Smoothness of 6^

221

is a contraction and there is a unique p(y) G V(x) such that

By construction of W(o;)

Since HdA(y) 7^ 0 and p(y] is unique, II^ (y) = [p(y)} is necessarily a singleton. By Theorem 4.1, 6^ e Cl'l(W(x)) and pa^ e C^V^)). By Theorem 3.1 (iii) for y e W(x)\dA, V&^y) exists and

where

Both N and p are Lipschitzian and hence the composition N o p is Lipschitzian. Moreover, since &A is differentiable almost everywhere and m(dA) — 0

and necessarily VbA 6 C0'1^^))^, &A E Cl>l(W(x)). In view of (4.2), V6A = n o p. This proves the theorem when A is of class C1'1. (ii) When A is of class C fe '^ for an integer fc, /c > 2, and a real number £, 0 < t < 1, it is C1'1, and the previous constructions and conclusions remain true. Consider the map

From the first part, there exists a unique p(y) G V(x) such that

where

By assumption, / is at least C2 in V(x) and hence bounded in some smaller neighborhood V'(x), V'(x) C V(x). By putting a smaller bound on |6>i(y)| there exists a smaller neighborhood W'(x) of x in W(x) such that DzG(y,z] is invertible for all (y,z) G W'(x) x V'(x). So the conditions of the implicit function theorem are

222

Chapter 5. Oriented Distance Function and Smoothness of Sets

met. There exists a neighborhood Y(x] C V'(x) of x and a unique Cl- (and hence Ck~lji-} mapping

(cf., for instance, Lang [1, Prop. 5.3, p. 15] and Schwartz [3, Thm. 31, p. 299]). In view of the fact that (4.4) is satisfied everywhere in Y(x),

then V6^ belongs to Ck~1'e(Y(x))N as the composition of two Ck~^^ maps. Therefore, bA is Ck^ in Y(x).

Example 4.1. Consider the two-dimensional domain

defined as the epigraph of the function

for some arbitrary integer n > 1. We claim that fJ is a set of class C1:1~1/fn and

In view of the presence of the absolute value the point (0,0) is the point where the smoothness of d£l will be minimum. Therefore, there exists no neighborhood of (0,0) where the derivative Vfo exists and &Q is not Cl and a fortiori not C1'1"1/™. Indeed, for each (0, y)

The point (0,7/), y > 0, belongs to Sk(il) if there exist two different points x that minimize the function

Since / is symmetric with respect to the y-axis it is sufficient to show that there exists a strictly positive minimizer x > 0. A locally minimizing point x > 0 must satisfy the conditions

Equation (4.6) can be rewritten

4. Boundary Smoothness and Smoothness of 6^

223

and x = 0 is a solution. The second factor can be written as an equation in the new variable X = xl/n and

It has exactly one solution X since

g(0) = — y and g(x) goes to infinity as X goes to infinity. In particular, X > 0 for y > 0 and X = 0 for y = 0. For y > 0 and x = Xn

and

Therefore, x is a local minimum. To complete the proof for y > 0 we must compare F(0, y} = y2 for the solution corresponding to x = 0 with

for the solution x > 0. Again using identity (4.8),

This proves that for y > 0, x > 0 is the minimizing positive solution. Therefore, for all n > 1, f(x] = \x 2 ~ 1 / n yields a domain fi for which the strictly positive part of

224

Chapter 5. Oriented Distance Function and Smoothness of Sets

the y-axis is the skeleton. It is interesting to note that, for each point x, the points (x, /(#)) G <90 are the projections of the point

and that the square of the distance function is equal to

Finally, for all n > 1, / £ C1'1. For n = 1, / £ C0'1, and for n > 1, / € Cl'l~l/n. This gives an example in dimension 2 of a domain 0 of class <71)A, 0 < A < 1, with skeleton converging to the boundary. D From Theorem 4.3 (i) additional information can be obtained on the flow ofVbn. Theorem 4.4. Let 0 be a subset of RN such that F == d£l ^ 0 and assume that 0 is of class C1'1. (i) For each x € F £/iere ermts a neighborhood W(x] ofx such that 6^ G C rl)1 (VF(rr)) ;

(ii) Let Tt(x) be the flow ofVbn in W(x), that is,

Then for t in a neighborhood of 0 we have the following properties:

(iii) Almost everywhere in W(x)

5. W1'P(D)-Topology and the Family Cg(£>)

225

Proof. For simplicity, we use the notation b = b^ and p — pr(i) The result follows from Theorem 4.3 (i). (ii) By assumption from part (i) the function b belongs to Cl>l(W(x)) and hence p e C°'1(W(x)\ R N ). Thus, almost everywhere in W(x),

Now

But since p and Tt belong to C0jl(W(x}]'RN}1 the identity necessarily holds everywhere in W(x). Moreover,

Finally

In particular, almost everywhere in W(x),

5

W1'p(D)-Topology and the Family C£(D)

From Theorem 2.1 (vi) bA is locally Lipschitzian and belongs to W^(R N ) for all p > 1. So the previous constructions for d& and d^A can be repeated with the space W\QC (R N ) m place of Cioc(-D) to generate new Wl'p metric topologies on the family Cb(D}. Moreover the other distance functions can be recovered from the map

and the characteristic functions from the maps

One of the advantages of the function 6^ is that the Wl^-convergence of sequences will imply the I/p-convergence of the corresponding characteristic functions of int A, int CA, and dA, that is, continuity of the volume of these sets. In this section we give the analogues of Theorems 4.1 and 4.2 in Chapter 4.

226

Chapter 5. Oriented Distance Function and Smoothness of Sets

Theorem 5.1. Let D be an open (resp., bounded open] subset o/R N . (i) The topologies induced by W^(D) (resp., Wljp(D}) on Cb(D) are all equivalent for p, 1 < p < oo. (ii) Cb(D) is closed in W^(D) (resp., W1'P(D)) forp, l
defines a complete metric structure on J-~b(D}.

(Hi) For p, 1 < p < oo, the map

is "Lipschitz continuous": for all bounded open subsets K of D and bAl and bA2 inCb(D),

In particular, the family of sets whose boundary has a zero N-dimensional measure,

is a closed subset of Cb(D] for the WljJ}-topology. (iv) Let D be a bounded open subset o/R N . For each \)A £ Cb(D)

The map

is continuous. (v) Let D be an open (resp., bounded open) subset o/R N . For all p, 1 < p < oo, the map

is continuous.

5. Wl'p(D)-Topo\ogy and the Family Cg(D)

227

Proof. The proof of (i) and (ii) is essentially the same as the proof of Theorem 4.1 of Chapter 4 using the properties established for Cb(D] in C(D) of Theorem 2.2. (iii) From Theorem 3.2 (i) for any two subsets A\ and A^ of D with nonempty boundaries and for any open U in D

for p, I < p < oo, and with the ess sup norm for p = oo. The closure of C®(D] follows from the continuity of the map

for all bounded open subsets U of D. (iv) First observe that

and since V6^ = VcU = Vd^ = 0 almost everywhere on dA — A n CA, then for almost all x in R . From this we readily get (5.4). However, this is not sufficient to prove the continuity since the map (5.5) is not linear. To get around this consider a sequence {b^n} C Cb(D) converging to bA in Wl'p(D). From Theorem 2.2 (iii), |6^n = ddAn, b'A = d,An, and 6^ = ^An converge to |&A = dgA, b\ — d^, and 6^ = d^A in C(D] and hence in W/1'p(Z))-weak. To prove that the convergence is strong, consider the L 2 -norm

by weak L2-convergence of Vc?An and Vd^An to VdA and V^Q^. So both sequences dAn and d^An converge in V^ ll2 (-D)-strong, and hence from part (i) in VF 1>p (D)-strong for all p, 1 < p < oo. (v) It is sufficient to prove the result for D bounded open. From part (iii) it is true for XdA, and from part (iv) and Theorem 4.1 (ii) and (iii) of Chapter 4 for the other two.

228

Chapter 5. Oriented Distance Function and Smoothness of Sets

Theorem 5.2. Let D be a bounded open domain in R N . (i) // {bAn} weakly converges in WI!P(D) for some p, 1 < p < oo, then it weakly converges in W1'P(D) for all p, 1 < p < oo. (ii) If {bAn} converges in C(D], then it weakly converges in Wljp(D) for all p, 1 < p < oo. Conversely, if {bAn} weakly converges in Wl'p(D] for some p, 1 < p < oo, it converges in C(D], (iii) Cb(D] is compact in Wljp(D}-weak for allp, 1 < p < oo. Proof, (i) Recall that for D bounded there exists a constant c > 0 such that for all bA £ Cb(D]

If {bAn} weakly converges in Wl'p(D) for some p > 1, then {bAn} weakly converges in LP(D), {V&A n } weakly converges in LP(D}N. By Lemma 2.1 (iii) in Chapter 3 both sequences weakly converge for all p > I and hence {bAn} weakly converges in W1'P(D] for all p>l. (ii) If {bAn} converges in C(D], then by Theorem 2.2 (ii) there exists bA € Cb(D) such that bAn -> bA in C(D) and hence in LP(D). So for all (p e T>(D)N

By density of V(D] in L2(D), VbAn -»• VbA in L2(jD)7v-weak and hence bAn -> bA in W1'2(D}-weak. From part (i) it converges in W1'p(D)-weak for all p, 1 < p < oo. Conversely, the weakly convergent sequence converges to some / in Wl'p(D). By compactness of Cb(D), there exist a subsequence, still indexed by n, and bA such that &An —* OA in C(D) and hence in W1'p(D}-weak. By uniqueness of the limit, bA = /• Therefore, all convergent subsequences in C(D] converge to the same limit. So the whole sequence converges in C(D). This concludes the proof. (iii) Consider an arbitrary sequence {6^n} in Cd(D). From Theorem 2.2 (ii) Cb(D] is compact and there exists a subsequence {bAn } and 6^ € Cb(D] such that \)An —> bA in C(D). From part (ii) the subsequence weakly converges in W1'P(D] and hence Cb(D) is compact in WAl>p(D)-weak.

6

Sets of Bounded and Locally Bounded Curvature

We introduce the family of sets with bounded or locally bounded curvature which are the analogues for 6^ of the sets of exterior and interior bounded curvature associated with dA and d^A m section 5 of Chapter 4. They include (71'1-domains, convex sets, and the sets of positive reach of Federer [2]. They lead to compactness theorems for Cb(D] in Wl>p(D).

6. Sets of Bounded and Locally Bounded Curvature

6.1

229

Definitions and Main Properties

Definition 6.1. (i) Given a bounded open nonempty subset D of R N , a subset A of D, dA ^ 0, is said to be of bounded curvature with respect to D if

This family of sets will be denoted as follows:

(ii) A subset A of R N , dA ^ 0, is said to be of locally bounded curvature if

where B(x,p) is the open ball of radius p > 0 in x. From Theorem 5.1 and Definition 5.1 in Chapter 3, a function belongs to BVi oc (R N ) if and only if for each x G RN it belongs to BV(B(x, p)} for some p > 0. As in Theorem 5.2 of Chapter 3 for Caccioppoli sets, it is sufficient to satisfy this condition for points of the boundary. Theorem 6.1. Let A, dA / 0, be a subset o/R N . Then A is of locally bounded curvature if and only ifVb^ belongs to BVi oc (R N )^ v . Corollary 2. All sets A, dA ^ 0, in RN of class C1'1 are of locally bounded curvature. The theorem will be proved as part (iii) of Theorem 6.3, and the corollary follows from Theorem 4.3 (i). Theorem 6.2. (i) Let D be a nonempty bounded open Lipschitzian subset of R N . For any subset A ofD, dA ^ 0, such that

dA has finite perimeter, that is,

(ii) For any subset A o/R N , dA ^ 0, such that

dA has locally finite perimeter, that is,

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Chapter 5. Oriented Distance Function and Smoothness of Sets

Proof. Given V6^ in BV(D)N, there exists a sequence {uk} in C°°(D)N such that

as k goes to infinity, and since |V6>i(^)| < 1> this sequence can be chosen in such a way that This follows from the use of mollifiers (cf. Giusti [1, Thm. 1.17, p. 15]). For all V in T>(D)N

For each Uk

where *Duk is the transpose of the Jacobian matrix Du^ and

since for VF1>:1(Z))-functions, ||V/||£,i(£>)jv = HV/Ujv/^D)^- Therefore, as k goes to infinity,

Therefore, VXdA e Ml(D}N. Theorem 6.3. Let A be a subset o/R N such that dA ^ 0. (i) The function fdA(x) = \ (\x\2-b2A(x)) is convex mR N andVb2A e BVi oc (R N ) A For all x and y in RN

or equivalently

The set of projections Tig A (x) onto dA is a singleton PQA (x) if and only if the gradient oj JQA exists. In both cases

6. Sets of Bounded and Locally Bounded Curvature

231

For all x and y in RN

Vp(z) G iWz), Vp(y) G n aj4 (y),

(p(y) - p(x)) • (y - x) > 0.

(6.6)

The function b\(x] is the difference of two convex functions

(ii) VbA 6 BVi oc (R N \^) Jv . More precisely, for all x & RN \#A tfiere exists p > 0, 0 < 3/? < cfo^x), suc/i i/iai ^9^4^ is convex in B(x^2p) and hence VbAeBV(B(x,p))N. (ui) A subset A of R N , dA ^ 0, is of locally bounded curvature if and only if V^eBVioclR^. Proof. The proof follows from Lemma 5.1 and Theorem 5.3 in Chapter 4 by applying it to A\- Use the fact that bA — dA and bA = — d^A outside of dA. 6.2

Examples and Limits of the Tubular Norms as h Goes to Zero

It is informative to compute the Laplacian of bA for a few examples. The first three are illustrated in Figure 5.1.

Figure 5.1. VbA for Examples 6.1, 6.2, and 6.3. Example 6.1 (half-plane in R2; cf. Example 5.1 in Chapter 4). Consider the domain

It is readily seen that

and

232

Chapter 5. Oriented Distance Function and Smoothness of Sets

Example 6.2 (ball of radius R > 0 in R2; cf. Example 5.2 in Chapter 4). Consider the domain

Clearly,

Also

Again AbdA contains twice the boundary measure on dA. Example 6.3 (unit square in R2; cf. Example 5.3 in Chapter 4). Consider the domain

Since A is symmetrical with respect to both axes, it is sufficient to specify 6^ in the first quadrant. We use the notation Qi, Q^, Qs, and Q± for the four quadrants in the counterclockwise order and c\, C2, 03, and 04 for the four corners of the square in the same order. We also divide the plane into three regions:

Hence for 1 < i < 4

and for the whole plane

6. Sets of Bounded and Locally Bounded Curvature

233

where D\ n D^ is made up of the two diagonals of the square where V6^ has a singularity, that is, the skeleton. Moreover, for 1 < i < 4

and

Notice that the structures of the Laplacian are similar to the ones observed in the previous examples except for the presence of a singular term along the two diagonals of the square. All C2-domains with a compact boundary belong to all the categories of Definition 6.1 and Definition 5.1 of Chapter 4. The /i-dependent norms \\D2bA\\M1(uh(dA))i \\D2d£A\\Mi(UhfiA^, and \\D2dA\\Mi(uh(A)) are au decreasing as h goes to zero. The limit is particularly interesting since it singles out the behavior of the singular part of the Hessian matrix in a shrinking neighborhood of the boundary dA.

Example 6.4.

If A C RN is of class C2 with compact boundary, then

Example 6.5.

Let A = {xi}f_1 be / distinct points in R N . Then dA = A and

Example 6.6.

Let A be a closed line in RN of length L > 0. Then dA — A, and

Example 6.7. Let N = 2. For the finite square and the ball of finite radius,

where HI is the one-dimensional Hausdorff measure (cf. Examples 5.2 and 5.3 in Chapter 4).

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Chapter 5. Oriented Distance Function and Smoothness of Sets

Also, looking at A6^ in Uh(A) as h goes to zero provides information about Sk(A).

Example 6.8.

Let A be the unit square in R2. Then

where HI is the one-dimensional Hausdorff measure and Sk(A) = Skjnt(A) is the skeleton of A made up of the two interior diagonals (cf. Example 6.3). It would seem that in general

where [V&AJ is the jump in V&A and n is the unit normal to Sk(^4) (if it exists!). D

6.3

(m,p)-Sobolev Domains (or Wm'p-Domains)

The TV-dimensional Lebesgue measure of the boundary of a Lipschitzian subset of RN is zero. However, this is generally not true for sets of locally bounded curvature, as can be seen from the following example.

Example 6.9. Let B be the open unit ball centered in 0 of R and define A = [x 6 B : x with rational coordinates}. Then dA = £, bA = dB, and for all h > 0

The next natural question that comes to mind is whether the boundary or the skeleton have locally finite (N — l)-Hausdorff measure. Then come questions about the mean curvature of the boundary. For the function 6^, A6^ = trD 2 6^ is proportional to the mean curvature of the boundary, which is only a measure in RN for sets of bounded local curvature. This calls for the introduction of some classification of sets that would complement the classical Holderian terminology introduced in Chapter 2, would fill the gap in between, arid would possibly say something about sets whose boundary is not even continuous. This seems to have been first introduced in Delfour and Zolesio [32].

Definition 6.2 (Sobolev domains).

Given m > 1 and p > 1, a subset A of RN is said to be an (m,p)-Sobolev domain or simply a Wm'p-domain if dA ^ 0 and there exists h > 0 such that

6. Sets of Bounded and Locally Bounded Curvature

235

This classification gives an analytical description of the smoothness of boundaries in terms of the derivatives of bA. The definition is obviously vacuous for m = 1 since bA G ^'c°°(RN) for any set A such that dA / 0. For 1 < m < 2 we shall see that it encompasses sets that are less smooth than sets of locally bounded curvature. For m — 2 the VT2'p-domains are of class CI:I~N/P for p > N. For m > 2 they are intertwined with sets of class Ck^. Theorem 6.4. Given any subset A o/R N , dA ^ 0,

Proof. Since \VbA(x)\ < I almost everywhere, VbA G BVi oc (R N ) jV n L£ C (R N ) N and the theorem follows directly from Theorem 5.8 in Chapter 3. It is quite interesting that a domain of locally bounded curvature is a W2~£^domain for any arbitrary small e > 0. So it is almost a VT^-domain, and domains of class W2~£'1 seem to be a larger class than domains of locally bounded curvature. In addition, their boundary does not generally have zero measure. Now consider the boundary case m = 2. Theorem 6.5. Given an integer N > I , let A be a subset o/R N such that dA ^ 0. (i) // there exist p > N and h > 0 such that

then

that is, A is a Holderian set of class Cl'l~N/p. Moreover, the function b2A belongs to Clo'c (Uh(dA)), dA has positive reach, and the N-dimensional Lebesgue measure of dA is zero. (ii) In dimension N = 2 the condition bA G Wj 'p(Uh(dA}) is equivalent to AbA £ Llc(Uh(dA}}. Proof, (i) Consider the function VbA\2. Since \VbA\ < 1, then

In particular, for all

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Chapter 5. Oriented Distance Function and Smoothness of Sets

For p > TV, from Adams [1, Thm. 5.4, Part III, and Remark 5.5 (3), p. 98],

From Theorem 4.1, b\ e C^(Uh(dA)) and dA has positive reach. From Theorem 4.2 A is of class Cl>l~N'p and m(dA) = 0. (ii) From part (i), |V6^(o;)|2 = 1 in Uh(dA), and D2bA(x) VbA(x) = 0 almost everywhere. Hence in dimension N = 2,

As can be seen, a certain amount of work would be necessary to characterize all the Sobolev domains and answer the many associated open questions.

7

Characterization of Convex and Semiconvex Sets

We have seen in Chapter 4 that the convexity of dA is equivalent to the convexity of A. We shall see that this characterization remains true with b^ in place of dA — dA. However, the next example shows that the convexity of bA is not sufficient to get the convexity of A.

Example 7.1.

Let B be the open unit ball in RN and A the set B minus all the points in B with rational coordinates. By definition,

By Theorem 6.2 (i) of Chapter 4 the function d,B and, a fortiori bA, are convex, but A is not convex and dA ^ dA.

Theorem 7.1.

7. Characterization of Convex and Semiconvex Sets

237

(iii) Let A be a subset o/R N such that dA ^ 0. Then

(iv) For all convex sets A such that dA ^ 0, VbA belongs to BVi oc (R N ) 7V and the Hessian matrix D2bA of second-order derivatives is a matrix of signed Radon measures which are nonnegative on the diagonal. Moreover, bA has a second-order derivative almost everywhere, and for almost all x and y in R N ,

Proof, (i) Denote by x\ the convex combination Ax + (1 — X)y of two points x and y in A for some A G [0,1]. By convexity of bA,

and A is convex. (ii) It is sufficient to prove the result for A closed. In the first direction the result follows from part (i). In the other direction we consider three cases. The first one deserves a lemma. Lemma 7.1. Let A be a subset o/R N such that dA ^ 0 and A be convex. Then

Proof. Again we can assume that A is closed. Associate with x and y in ,4, the radii rx = d,£A(x}, ry = d^A(y], r\ = \rx + (1 — A)r y , and the closed balls Bx of center x and radius r x , By of center y and radius ry, and B\ of center x\ and radius r\. By the definition of C?QA, Bx C A and By C A since A is closed. Associate with each z € B\ the points zx = x and zy — y if r\ = 0, and if r\ > 0 the points

Obviously z = x\ if r\ = 0. Therefore,

since A is closed and convex. In particular

and this proves the lemma.

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Chapter 5. Oriented Distance Function and Smoothness of Sets

For the second case consider x and y in CA By definition

By Theorem 6.2 (i) of Chapter 4, d,A is convex when A is convex and

The third and last case is the mixed one for x £ CA. and y (~ A. Define

Since A is convex, denote by H the tangent hyperplane to A through p\, by H+ the closed half-space containing A, and by H~ the other closed subspace associated with H. By the definition of H,

and by convexity of A, A C H+ and H~ C CA The projection onto H is a linear operator and

By convexity any y belongs to A C H+ and since H

C CA, we readily have

First consider the case x\ G CA. Then d,A(x\) > 0, x\ 6 H~, and necessarily xeH~. From (7.5)

But since x € H~ and A C H+

and

7. Characterization of Convex and Semiconvex Sets

239

Next consider the case x\ G A for which x\ G H+ and If x G H~, then since A C H+,

and from (7.5)

If, on the other hand, x G ff + , then x, y G #+ and XA G A C # + . So from (7.4)

This covers all cases and concludes the proof. (iii) Consider two cases. For any convex set with a nonempty interior, int A = int A and hence CCA = CCA. Therefore, CA = CA and d^A = d^A. For a convex set with an empty interior, A = <9A, CA = R N , and A is contained in an affine subspace of R N of dimension strictly less than N. Therefore, CA = R N = CA and d^A = d$A. Thus in both cases bA — bA since dA = dA and dA = A n CA = A D CA = 5A. Finally, since A is convex by Theorem 2.8 in Chapter 3 so is A, and from part (ii) so is bA = bA. (iv) Same argument as in the proof of Theorem 6.2 (iii) of Chapter 4. We have seen in Theorem 7.1 (iii) that for all convex subsets A of D such that dA 7^ 0, 6^ = bA and hence A is the unique closed representative in the equivalence class. We now prove that this subfamily of Cb(D) is closed in C(D). If we want to work with open convex subsets rather than closed subsets, it is necessary to add a constraint. Theorem 7.2. Let D be a nonempty open (resp., bounded open) subset o/R N . (i) The family

(ii) Let E be a nonempty open subset of D. Then the family

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Chapter 5. Oriented Distance Function and Smoothness of Sets

(iii) The conclusions of parts (i) and (ii) remain true with W^(D)- (resp., Wl'p(D)-) strong in place of C\OC(D] (resp., C(D}}. (iv) For D bounded, the families Cd(D] and C^(D] as defined in Theorem 6.2 (ii) of Chapter 4 and the corresponding subfamilies Cd(D',E) and C%(D;E) are also compact in Wl>p(D], 1 < p < oo. Moreover,

Proof. It is sufficient to prove the result for D bounded. Furthermore, by compactness of Cb(D] in C(D), it is sufficient to prove that Cb(D) and Cb(D;E) are closed in C(D}. (i) Let {bAn} be a Cauchy sequence in Cb(D). It converges to some b^ £ Cb(D). By Theorem 7.1 (iii), the &A n ' s are convex and so is the limit 6^, and by Theorem 7.1 (i), A is convex, and by Theorem 7.1 (ii), b^ is convex. Consider the following two cases: int A = 0 and int A ^ 0. (1) In the first case

Therefore

and b&n —> ^A — bA, where A is convex: bA — b^ e Cb(D}. (2) For the second case, intA^0 and A convex imply that int A = A by Theorems 5.4 and 5.3 in Chapter 2. Consider the following two subcases: int A = A and int A ^ A. (2a) If int A = int A = A, then, using the fact that Cint A — C^4, bint A — dint A ~ ^Cint A ~ °^A ~ d$A = 6^,

and dint A = dA ^ 0. Thus 6^n —> b^ = &mtA> where int A is convex by Theorem 6.2 (ii). (2b) We show that int^l = A leads to a contradiction and cannot occur. If int A ^ A, then either int A = 0 or int A ^ 0. If int A = 0, then dA = A, and since int A ^ 0, there exist x and r > 0 such that B(x,r) C dA. If mtA^0

7. Characterization of Convex and Semiconvex Sets

241

is open (convex) and not empty. Moreover, since B(a, d) n int A = 0,

and there exists a ball B(x,r), r > 0, such that B(x,r) C dA. Thus in both cases there exist x and r > 0 such that J5(x,r) C dA. We prove that this leads to a contradiction. Choose N such that for all n > N

We shall prove that for all n > N there exists xn G B(x,r] such that dAn(xn) > 3r/8. Hence xn G B(x,r) C dA C A and from (7.8)

which yields a contradiction and proves that necessarily int A = A, and we are back to case (2a). Let pn G dAn be a projection of x onto dAn and choose

since An is convex and bounded. Choose

Let

H+ d= {y € RN : (y -Pn) • vn > 0} , H~ d= {y G RN : (y -

Pn)

• vn < 0} .

We claim that An C Hn . This is obvious if x G CA n . For x G intA n we can show by contradiction that if there exists z G H+ such that z E An, then (z— pn) -vn > 0, and since B(x,d^An^(x}) C An

From this it is easy to show that pn G int An. But this would contradict the fact that by definition pn G dAn. As a result

By definition of H+, xn G H+ and

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Chapter 5. Oriented Distance Function and Smoothness of Sets

However, An C Hn and necessarily

and this concludes the proof of part (i). (ii) Prom part (i), given a Cauchy sequence {bnn} in Cb(D;E), there exists a convex subset A of D such that 6^ —» bA. Moreover, since d^n —»• d$A, E C fin => C£ D C0n =*> C£ D Cl - Cint A => 0 ^ E C int A Since ^4 is convex and int A ^ 0, int A — A and 6^4 = 6int A- Thus O = int A can be chosen as the limit set. (iii) Let {bAn} C Cb(D] be _a Cauchy sequence in W1'P(D). From Theorem 5.2 (ii) it is also Cauchy in C(D). Hence from part (i) there exist a convex set A, dA i- 0, such that bAn -> bA in C(D] and, a fortiori, in W1'P(D). This shows that Cb(D] is closed in Wl>p(D}. To show that it is compact consider a sequence {bAn} C Cb(D}. From part (i) there exists a subsequence, still indexed by n, and a convex subset A, dA ^ 0, of D such that bAn —> 6^ in C(D). Since D is bounded it also converges in LP(D}. Since |V6^n| is pointwise bounded by one, there exists another subsequence of {bAn}, still indexed by n, such that V6^n converges to VbA in Wl-p(D)N-weak. But An and A are convex. Thus m(dA) = 0 = m(5^n) and |V&A n | = 1 = |V6yi| almost everywhere in R N . As a result, for p = 2

Hence we get the compactness in W^1;2(.D)-strong and by Theorem 5.1 (i) in Wljp(D)strong p, 1 < p < oo. (iv) The result follows from the continuity of the map (5.5) in Theorem 5.1 (iv). We have seen in Theorem 7.1 (iv) that in view of the convexity of the distance (resp., oriented distance) functions of a closed convex set A, closed convex sets are of locally bounded exterior curvature and of locally finite perimeter (resp., locally of bounded curvature). When A is convex and of class C2, then for any X e dA, there exists a strictly convex neighborhood N(X) of X such that

or, since D2bA(x) VbA(x) = 0 in N(X), then for all x € N(X), This is related to the notion of strong elliptic midsurface in the theory of shells: there exists c > 0 All this motivates the introduction of the following notions.

8. Federer's Sets of Positive Reach

243

Definition 7.1. Let A be a closed subset of RN such that dA ^ 0. (i) The set A is locally convex (resp., locally strictly convex) if for each X G dA there exists a strictly convex neighborhood N(X) of X such that

(ii) The set A is semiconvex if

(iii) The set A is locally semiconvex if for each X S dA there exists a strictly convex neighborhood N(X] of X and

Remark 7.1. When a closed set A has a compact C2 boundary, D2bA is bounded in a bounded neighborhood of <9A and A is necessarily locally semiconvex. When A is a locally semiconvex set, for each X G d A there exists a strictly convex neighborhood N(X) of X such that V&A £ BV(N(X))N. If, in addition, dA is compact, then there exists h > 0 such that VbA e BV(C/'/ l (9A)) N . Remark 7.2. Given a fixed constant /3 > 0, consider all the subsets of D that are semiconvex with constant 0 < a < (3. Then this set is closed for the uniform and the V^llp-topologies, 1 < p < oo.

8

Federer's Sets of Positive Reach

The distance function dAr of the dilated set Ar = {x € RN : dA(x) < r}, r > 0, of A provides a uniform approximation of cU, as can be seen from the following theorem. Theorem 8.1. Assume that A is a nonempty subset o/R N . For r > 0 and, as r goes to zero,

Proof. For any q e Ar, d,A(q) < r, and for p e H^(g),

When, in addition, A has positive reach, reach(A) > r, and Ar ^ RN for some r > 0, the dilated sets Ar are of class C1'1 and 6^ can be approximated by bAr in the W^-P-topology.

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Chapter 5. Oriented Distance Function and Smoothness of Sets

Theorem 8.2. Let A be a nonempty subset o/R N . Assume that there exists h > 0 such that (i) For all r, 0 < r < h, dAr ^ 0 and Ar is a set of class C1'1 such that

and for each x £ dAr there exists a neighborhood V(x) of x in Uh(A)\A such that

(ii) Moreover, dA ^ 0, and for all r, 0 < r < h,

In particular,

(iii) As r goes to zero, 6^r —> b^ uniformly in R N ; and for p > I ,

(iv) Furthermore, A is of local bounded curvature and dA has zero volume

The proof of the theorem requires the following two lemmas. Lemma 8.1. Assume that A is a nonempty subset o/R N . For r > 0 such that Ar^RN

Proof of Lemma 8.1. (i) First inequality. For all y such that cU(y) > f and all P£ A,

8. Federer's Sets of Positive Reach

245

Second inequality. Given x G A, by definition of d^A(x]^ Bx = B(x,d^(x}} is a subset of A. For r > 0 there exists pr £ [>Ar such that

Define

By construction,

and d^Ar(x] > d^(x] + r. Lemma 8.2. Let A be a nonempty subset o/R N and assume that there exists h > 0 such that d\ e C^(Uh(A)). For all x e C// l (A)\A and 0 < r < h,

Proof of Lemma 8.2. In the region Uh(A)\A the gradient Vd/i is locally Lipschitz. For any point x, 0 < dA(x) < h, consider the flow

There exists a unique local solution through x. Moreover,

Therefore, the solution exists and is unique for s, 0 < s < h —
It is readily seen that cU(x(s)) < s + eU(x) and that

As a result

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Chapter 5. Oriented Distance Function and Smoothness of Sets

and by uniqueness of solution y(s) = x ( s ) , 0 < s < r - dA(x). z(0) = y(0) = x

In particular,

and finally

Hence for all r and x such that 0 < dA(x) < r < h,

From Theorem 3.2 in Chapter 4, VdA(x) = (x — pA(x))/dA(x}, equivalent form of the above identity.

and we get the

Proof of Theorem 8.2. (i) First observe that for 0 < r < h, 0 ^ A C Ar C Ah ^ RN implies that dAr ^ 0 and dAr = Ar fi CAr C d~^~{r}. By assumption d\ G C^l(Uh(A}}. So by Theorem 7.1 (iv) of Chapter 4 for all x G Uh(A), ILA(x) is a singleton. By part (i) of the same theorem dA G C'1^(C//l(A)\yl), and by Theorem 3.2 (i), \VdA(x)\ = 1 in C^(A)\A Define

Therefore, / is Lipschitz continuous in R N ,

and Uh(A)\A is a neighborhood of /-1{0} = d^{r} since d~^{r} C Uh(A)\Ar/4 C [7/l(A)\J4. By Theorem 4.2 of Chapter 2,

is a set of class C1'1 with boundary

Hence by Theorem 5.3 of Chapter 2, Ur(A) = Ar and dAr = dUr(A) = {x E RN : dA(x) = r} . By Theorem 4.3 (i), for each x G dAr, there exists a neighborhood V(x] of x such that bAr G C 1 ' 1 (F(rr)) and this neighborhood can be chosen in Uh(A)\A. (ii) By assumption, 0 ^ A C Ah ^ RN implies that dA ^ 0 and 6^ is finite in R N . For all x G RN such that dA(x) > h and 0 < r < h,

8. Federer's Sets of Positive Reach

247

and from inequality (8.1) in Theorem 8.1,

For all x G RN such that r < 6^4(2) < /i, d^A(x) = d§A (x) = 0. Moreover, from Lemma 8.2 and part (i)

From inequality (8.1) in Theorem 8.1,

For all x G RN such that 0 < d ^ ( x ) < r, d^A(x) = 0 = dAr(x). From Lemma 8.2 and part (i)

From the first inequality in Lemma 8.1

Finally, for all x G RN such that 0 = ^(x), that is, x G A, there exists p = PQA(X) G dA such that d c ^(x) = \p — x\. But, by assumption, A ^ 0, (L4 D CA^ ^ 0 and, by Theorem 2.1 (i), dA 7^ 0. There exists a sequence {yn} C CA, 0 < d^(y n ) < ^j such that 7/n —> p. By Lemma 8.2 and part (i), associate with each yn

for which pA(Qn) — PAfajn)- By Theorem 7.1 in Chapter 4, p^ belongs to (^{^(^(A)). In particular, PA G C 0ll (5(p, (r + h}/2)} and there exists c > 0 such that

Since {qn} is bounded there exists q and a subsequence, still indexed by n, such that

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Chapter 5. Oriented Distance Function and Smoothness of Sets

Therefore,

Henceforth, for each x £ A we can associate with p £ Ila^(x) a q 6 dAr such that PA(Q) =P and

From the second inequality in Lemma 8.1,

and since dA(x) = 0 = dAr(x),

(iii) The uniform convergence of bAr to bA in RN is a direct consequence of part (ii). For all bounded open subsets D of Uh(A) as r —> 0,

since from part (ii), VbAr = V6^ in Uh(A). The convergence of dAr = &J[ , d^Ar = bAr, and rfaAr = I&/U to 6± = dA, 6~ = d c ^, and \bA\ = ddA in W^t/^A)) is now a consequence of Theorem 5.1 (iv). (iv) From part (i) for all 0 < r < h, Ar is a set of class C1'1, and for each x € dAr there exists a neighborhood V(x) of a; in Uh(A)\A such that 6^r G C 1 ' 1 (V r (x)). By Theorem_6.1, VbAr 6 BVi oc (R N ) JV . Therefore, by Definition 5.1 in Chapter 3, for all x € dA,

since from part (ii) V6^r = VbA in Uh(A). By Theorem 6.1 (ii), A is of locally bounded curvature and, by Theorem 6.1, in BVi oc (R- N ) JV - Finally, by continuity of the map

9

Compactness Theorems for Sets of Bounded Curvature

For the family of sets with bounded curvature, the key result is the compactness of the embedding

9. Compactness Theorems for Sets of Bounded Curvature

249

for bounded open Lipschitzian subsets of RN and p, 1 < p < oo. It is the analogue of the compactness Theorem 5.3 of Chapter 3 for Caccioppoli sets

which is a consequence of the compactness of the embedding

for bounded open Lipschitzian subsets of RN (cf. Morrey [1, Def. 3.4.1, p. 72, Thm. 3.4.4, p. 75] and Evans and Gariepy [1, Thm. 4, p. 176]). As for characteristic functions in Chapter 3, we give a first version involving global conditions on a fixed bounded open Lipschitzian hold-all D. In the second version the sets are contained in a bounded open hold-all D with local conditions in the tubular neighborhood of their boundary.

9.1

Global Conditions on D

Theorem 9.1. Let D be a nonempty bounded open Lipschitzian subset o/R N . The embedding (9.1) is compact. Thus for any sequence {An}, dAn ^ 0, of subsets of D such that

there exist a subsequence {Ank} and a set A, dA ^ 0, such that VbA e EV(D}N and

Proof. Given c > 0 consider the set

By compactness of the embedding (9.3), given any sequence {bAn} there exist a subsequence, still denoted {bAn}, and / e EV(D)N_such that VbAn -> / in L1(D}N. But by Theorem 2.2 (ii), Cb(D) is compact in C(D) for bounded D and there exist another subsequence {bAn } and bA £ Cb(D} such that bAn —> bA in C(D) and, a fortiori, in Ll(D}. Therefore, bAn converges in Wljl(D] and also in Ll(D). By uniqueness of the limit, / = VbA and bAn converges in W1>l(D} to bA. For $ G Vl(D)NxN as k goes to infinity

250

Chapter 5. Oriented Distance Function and Smoothness of Sets

||-^)2^A||M1(D) < ci and V6^ G BV(D)N. This proves the sequential compactness of the embedding (9.1) for p = 1 and properties (9.5). The conclusions remain true for P > 1 by the equivalence of the W^lip-topologies on Cb(D] in Theorem 5.1 (i). D

9.2

Local Conditions in Tubular Neighborhoods

The global condition (9.4) is now weakened to a local one in a neighborhood of each set of the sequence. Simultaneously, the Lipschitzian condition on D is removed since only the uniform boundedness of the sets of the sequence is required. Theorem 9.2. Let D be a nonempty bounded open subset o/R N and {An}, 0 ^ dAn, be a sequence of subsets of D. Assume that there exist h > 0 and c > 0 such that

Then there exist a subsequence {Ank} and a subset A, 0 ^ dA, of D such that V&A e BVioc(R N ) jV , and for allp, l
Moreover, for all (p e T>°(Uh(dA)),

and XdA belongs to BVi oc (R N )Proof, (i) By assumption, An C D implies that Uh(An) C Uh(D). Since Uh(D) is bounded, there exist a subsequence, still indexed by n, and a subset A of D, dA ^ 0, such that

and another subsequence, still indexed by n, such that

For all e > 0, 0 < 3e < h, there exists N > 0 such that for all n > N and all x € Uh(D),

Therefore,

9. Compactness Theorems for Sets of Bounded Curvature

251

From (9.6) and (9.10) In order to use the compactness of the embedding (9.3) as in the proof of Theorem 9.1, we would need Uh-e(dA) to be Lipschitzian. To get around this we construct a bounded Lipschitzian set between Uh-2e(dA) and Uh-e(dA). Indeed by definition, and by compactness there exists a finite sequence of points {xi}"=1} in dA such that

Since UB is Lipschitzian as the union of a finite number of balls, it now follows by compactness of the embedding (9.3) for UB that there exists a subsequence, still denoted {bAn}, and / G BV(UB)N such that VbAn -> / in Ll(UB)N. Since Uh(D) is bounded, Cb(Uh(D}} is compact in C(Uh(D}} and there exists another subsequence, still denoted {bAn}, and A C D, dA ^ 0, such that bAn —> bA in C(Uh(D}) and, a fortiori, in Ll(Uh(D}}. Therefore, bAn converges in W1'1(C/s) and also in Ll(Us)- By uniqueness of the limit, / = VbA on UB and bAn converges to bA in Wl'l(UB}. By Definition 6.1 and Theorem 6.3 (iii), V6A and V6An all belong to BVioc(RN)]V since they are of bounded variation in tubular neighborhoods of their respective boundaries. Moreover, by Theorem 6.2 (ii), XdA G BVioc(RN). The above conclusions also hold for the subset Uh-2e(dA) of UB(ii) Convergence in Wl'p(Uh(D)}. Consider the integral

Prom part (i) the first integral on the right-hand side converges to zero as n goes to infinity. The second integral is on a subset of ^>Uh-ie(dA). From (9.11) for all n>N,

The second integral reduces to

which converges to zero by weak convergence of VbAn to VbA in the space L2(Uh(D)) in part (i) and the fact that \VbA — I almost everywhere in Uh(D}\Uh-is(dA}. Therefore, since bAn —> bA in C(Uh(D}}

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Chapter 5. Oriented Distance Function and Smoothness of Sets

and by Theorem 5.1 (i) the convergence is true in W1>p(Uh(D)) for all p > I. (Hi) Properties (9.8). Consider the initial subsequence {bAn} which converges to bA in H1(Uh(D))-weak constructed at the beginning of part (i). This sequence is independent of E and the subsequent constructions of other subsequences. By convergence of bAn to bA in Hl(Uh(D))-weak for each $ e T>1(Uh(dA))NxN,

Each such <£» has compact support in Uh(dA), and there exists e = e(3>) > 0, 0 < 3e < h, such that From part (ii) there exists N(s) > 0 such that

For n > N(e) consider the integral

By convergence of VbAn to VbA in the space L2(Uh(D))-weak, then for all 3> 6

Vl(Uh(dA)}NxN

Finally the convergence remains true for all subsequences constructed in parts (i) and (ii). This completes the proof.

10

Compactness and Uniform Cone Property

In Theorem 5.9 of section 5.4 in Chapter 3, we have seen a compactness theorem for the family of subsets of a bounded hold-all D satisfying the uniform cone property. In this section we give a direct proof of the compactness for the C(D}- and Wl'p(D)topologies associated with the oriented distance function b^. As a consequence we get the compactness for the C(D}- and W/1'p(Z))-topologies associated with d^ and d,£ft. Furthermore, we recover the compactness of Theorem 5.9 of Chapter 3 for Xfi in LP(D). Finally we also get the compactness for xcn m LP(D}. Recall the following notation and definition

10. Compactness and Uniform Cone Property

253

We have seen in Theorem 6.1 of Chapter 2 that for a Lipschitzian set fi, 90 ^ 0, m(<90) = 0, intO ^ 0, intCO ^ 0. Moreover, if the properties of Definition 6.1 of Chapter 2 are satisfied for a set 0, they are satisfied for all sets in the equivalence class

Theorem 10.1. Let D be a nonempty bounded open subset o/R N and 1 < p < oo. For r > 0, u; > 0, and A > 0, t/ie family

is compact in C(D] and Wl'p(D}. As a consequence the families

are compact in C(D] and W 1)P (D), and the families

are compact in LP(D). The proof of this theorem is similar to the proof of Theorem 5.9 in Chapter 3 and uses a key lemma. Proof of Theorem 10.1. (i) Compactness in C(D}. Consider an arbitrary sequence {fin} in L(D,r,u,\). For D compact Cb(D) is compact in C(D) and there exists 0 C D and a subsequence {Onfc} such that

It remains to prove that 0 e L(D, r, a;, A). This requires the equivalent of Lemma 5.1 in Chapter 3. Lemma 10.1. Given a sequence {ban} C Cb(D] such that b^n —> 6^ in C(D] for some b$i 6 Cb(D), we have the following properties:

and for all x £ CO,

254

Chapter 5. Oriented Distance Function and Smoothness of Sets

Moreover,

and Proof. We proceed by contradiction. Assume that

So there exists a subsequence {J7nfc}, n& —•> oo, such that

which contradicts the fact that 6nn —> &n- Of course, the same assertion is true for the complements and for all x G CO,

when x € 50, x € 0 fl CO, and we combine the two results. For the last result use the fact that the open ball cannot be partitioned into two nonempty disjoint open subsets. Coming back to the proof of the theorem, we wish to show that 0 has the uniform cone property

Since On is Lipschitzian, so is COn, and from the second part of the lemma for each x G 5O, Vfc > 1, 3nk > k such that

Denote by Xk an element of that intersection:

By construction Xk —> x. Next consider y e B(x, r) D 0. From the first part of the lemma, there exists a subsequence of {Onfc}, still denoted {Onfc}, such that

For each k > 1 denote by yk a point of that intersection. By construction

10. Compactness and Uniform Cone Property

255

There exists K > 0 large enough such that

To see this, note that y G B(x, r) and that

Now

Since r/p > I the result is true for

So we have constructed a subsequence {rinfc} such that for k > K

For each fc, 3dfc G R N , \dk = 1, such that

Pick another subsequence of {f2 nfc }, still denoted {f£ nfc }, such that

Now consider z G C y (A,u;,d). Since z is an interior point

and there exists K' > K such that

This proves that £7 C D satisfies the uniform cone property and f) G L(D, A,o;,r). (ii) Compactness in Wl'p(D). From Theorem 5.1 (i) it is sufficient to prove the result for p = 2. Consider the subsequence {Qnk} C L(D, A,o;,r) and let f2 G L(D, A,o;,r) be the set previously constructed such as b$in —» 6^ in C(D). Hence 6^n -^ b& in L2(D}. Since D is compact for all fi C D

256

Chapter 5. Oriented Distance Function and Smoothness of Sets

and there exists a subsequence, still denoted {6^n }, which converges weakly to 6^. Since all the sets are Lipschitzian, m(<9J7 nfc ) = 0 = m(<90) and |V&n| = 1 = |V6nn | almost everywhere in D, and

Therefore 6on —>• fo in Wl'2(D)-strong, since the convergence was already established in L2(£))-strong. (iii) The compactness of the other families follows from the continuity of the maps

in Theorem 2.2 (iii) and

in Theorem 5.1 (iv) and (v) and the fact that m(dO) = 0 implies X'mtn = Xfi Xint Cft = XCft almost everywhere for fi e L(D, A,a;, r).

11

an

d

Compactness and Uniform Cusp Property

The compactness theorem, Theorem 10.1, is no longer true when the uniform cone property is replaced by a uniform segment property, that is, when the cone C(X, u>, d) is replaced by the segment (0, Xd). This is readily seen by considering the following example. Example 11.1. Given an integer n > 1, consider the following sequence of open domains in R :

They satisfy the uniform segment property of Definition 7.1 (ii) of Chapter 2 by choosing A = r = 1/4. The sequence {fin} converges to the closed set

in the uniform topologies associated with d^n and ban or in the I/p-topologies associated with xnn and Xcnn • However, the segment property is not satisfied along the line L and the corresponding family of subsets of the hold-all D — B(Q, 4) satisfying the uniform segment property with r = A = 1/4 is not closed and, a fortiori, not compact.

11. Compactness and Uniform Cusp Property

257

This example shows that a uniform segment property is too meager to make the corresponding family compact. Looking back at the proof of Theorem 10.1, everything goes through with the uniform segment property except in the last seven lines of the proof of part (i), where the fact that the cone is an open set is critically used. This suggests that the cone could be replaced by an open cusp or horn that would yield larger families than those of Lipschitzian domains considered in the previous section. For instance, the cone C(A,o;,d) can be replaced by the cuspidal region

where B#(0,p) is the open ball of radius p > 0 and center 0 in the hyperplane H through 0 orthogonal to the direction d and h : [0, p] —->• R is a continuous function such that

This will be referred to as a uniform cusp property. Hence, we can now choose functions h : [0, p] —>• R of the form

The special case of the cone corresponds to

The following theorem is now a corollary to Theorem 10.1. Theorem 11.1. Let the assumptions of Theorem 10.1 be verified with the cone C(\,ui,d) replaced by the cuspidal region H(X,h,d) defined in (11.1) under the conditions (11.2) on the continuous function h. Then the compactness properties of Theorem 10.1 remain true. In some applications, it might be interesting to relax the uniform cusp property by permitting the axis of the cuspidal region to bend: this makes the region look like a horn and the corresponding property becomes a horn condition or property. Horn-shaped domains have been studied in several contexts in the literature. In particular, conditions on domains have been introduced in the context of extension operators and embedding theorems: the domains of F. John;2 the (e, 5)-domains of P. W. John;3 and domains satisfying a flexible horn condition (which is a broader notion than the previous two) by O. V. Besov.4 2

F. John, Rotation and strain, Comm. Pure Appl. Math. 14 (1961), 391-413. P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), 71-88. 4 O. V. Besov, Integral representations of functions in a domain with the flexible horn condition, and embedding theorems (Russian), Dokl. Akad. Nauk SSSR 273 (1983), 1294-1297; English translation: Soviet Math. Dokl. 28 (1983), 769-772. O. V. Besov, Embeddings of an anisotropic Sobolev space for a domain with a flexible horn condition (Russian), in "Studies in the Theory of Differentiable Functions of Several Variables and Its Applications, XII," Trudy Mat. Inst. Steklov 181 (1988), 3-14; 269; English translation: Proc. Steklov Inst. Math. (1989), 1-13. 3

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Chapter 6

Optimization of Shape Functions

1

Introduction and Generic Examples

In Chapter 3 we have seen several examples of optimization problems involving the Lp-topology on measurable sets via the characteristic function. In this chapter we consider optimization problems where the underlying topology is specified by the distance functions of Chapters 4 and 5. We give a brief account of two generic optimization problems: the minimization of a quadratic objective function which depends on the solution of the homogeneous Dirichlet boundary value problem associated with an elliptic operator, and the optimization of the first eigenvalue of the same operator. In both problems the strong continuity of solutions of the elliptic equation or the eigenvector equation with respect to the underlying domain is the key element in the proof of the existence of optimal domains. To get that continuity, some extra conditions have to be imposed on the family of open domains. Those questions have received a lot of attention for the Laplace equation with homogeneous Dirichlet boundary conditions. To a sequence of domains in a fixed hold-all D, associate a sequence of extensions by zero of the solutions in the fixed space HQ(D}. The classical Poincare inequality is uniform for that sequence as the first eigenvalue of the Laplace equation in each domain is dominated by the one associated with the larger hold-all D. By a classical compactness argument, the sequence of extensions converges to some limiting element y in HQ(D). To complete the proof of the continuity, two more fundamental questions remain. Is y the solution of the Laplace equation for the limit domain? Does y satisfy the Dirichlet boundary condition on the boundary of the limit domain? The first question can be resolved by assuming that the Sobolev spaces associated with the moving domains converges in the Kuratowski sense, i.e., with the property that any element in the Sobolev space associated with the limit domain can be approached by a sequence of elements in the moving Sobolev spaces. That property is obtained in examples where the compactivorous property1 follows from the choice of 1

A sequence of open sets converging to a limit open set in some topology has the compactivorous property if any compact subset of the limit set is contained in all sets of the sequence after a certain rank: the sequence eats up the compact set after a certain rank.

259

260

Chapter 6. Optimization of Shape Functions

the definition of convergence of the domains. If the domains are open subsets of D, then the complementary Hausdorff topology has that property. For the second question it is necessary to impose constraints at least on the limit domain. The stability condition introduced by Rauch and Taylor [1] and used by Dancer [1] and Daners [1] precisely assumes that the limit domain is sufficiently smooth, just enough to have the solution in HQ (Q). Of course assuming such a regularity on the limit domain makes things easier. The more fundamental issue is to identify the families of domains for which this stability property is preserved in the limit. The uniform cone property has been used in the context of shape optimization by Chenais [1, 2] in 1973. Capacity conditions have been introduced in 1994 by Bucur and Zolesio [3, 5, 6, 8, 9] in order to construct compact subfamilies of domains with respect to the Hausdorff complementary topology (see section 2.3 of Chapter 4). Furthermore, Bucur [1] proved that the condition given in 1993 by Sverak [1, 2] in dimension 2 involving a bound on the number of connected components of the complement of the domain can be recovered from the more general capacity conditions that are sufficient in the case of the Laplacian with homogeneous Dirichlet boundary conditions. In a more recent paper, Bucur [5] proved that they are almost necessary. Intuitively those capacity conditions are such that, locally, the complement of the domains in the family under consideration has enough capacity to preserve and retain the homogeneous Dirichlet boundary condition in the limit. As a first example consider the minimization of the objective function

over the family of open subsets 17 of a bounded open hold-all D of R N , where g in L2(D), un € HQ($I) is the solution of the variational problem

A e L°°(D; £(R N , R N )) is a matrix function on D such that

for some coercivity and continuity constants 0 < a < /?, and /|n denotes the restriction of / in H~l(D] to H~1(tt). The second example is the optimization of the first eigenvalue of the associated differential operator

We shall not provide a parallel treatment of the above problems for the homogeneous Neumann boundary conditions. However, the techniques are similar and most results remain true. In section 5 we introduce the fundamental //^-density

2. Embedding of #<}(Q) into H^(D) and First Eigenvalue

261

property to establish the continuity of the solution of the Neumann boundary value problem with respect to the domain. This property is satisfied when the segment property is verified. Such domains are C°-epigraphs. Then it is a matter of constructing compact families of subsets of a bounded hold-all with that property (for instance, the uniform cone property of section 10 and the uniform cusp property of section 11 in Chapter 5).

2

Embedding of H-J(n) into H*(D) and First Eigenvalue

Let Q and D be two open subsets of R N such that fi C D. Denote by eo(
By definition, for k > 1,

and e0 extends by continuity and density to a linear isometric map

Denote by #£(Q; D) the image of Hft($l) by e0. Theorem 2.1. Let £1 and D be two bounded open subsets o/R N . The linear subspace HQ($I; D) of HQ (D) is closed and isometrically isomorphic to HQ(£I] with the following properties: for all i/> £ HQ(^ D}

Any convergent sequence in HQ (ft)-weak converges in HQ~I(^})-strong. Proof, (i) It is sufficient to prove that HQ(£!; D) is closed. The other properties are easy to check. Pick a sequence {(pn} in H0(fJ) such that {eo((pn}} is Cauchy in HQ(D) and denote by <£ its limit in HQ(D}. Since CQ is an isometry, then {(f>n} is also Cauchy in HQ($I). Denote by (p its limit in HQ(^}. Then

and there exists (p e #J(ft) such that $ = e0((f>) in H$(D). Thence $ £ H$(0; D}. (ii) Since f2 is bounded, there exists a sufficiently large open ball B such that Q C B. By the embedding of HQ(^} into HQ(B], if a sequence (f>n converges to (f> in #Q (fi)-weak, then e^((pn] converges to e.$i((p} in HQ (B)-weak. Since the ball is sufficiently smooth, by Rellich's theorem, the sequence converges in H0~~l(B)-strong, and in view of the linear isometric isomorphism, (f>n converges to (f> in ^Q~ 1 (^)-strong. D

262

Chapter 6. Optimization of Shape Functions Going back to the restriction of an element / € H~l(D] to 17 c .D, define

which makes sense since

and by density and continuity /(^ readily extends to a continuous linear form on the subspace jF/g(17;.D) of HQ(D) and a fortiori on #0(17). Furthermore, the homogeneous Dirichlet boundary value problem (1.2) in 17 is completely equivalent to the variational problem: to find u £ HQ(^; D) such that

and u = efi(ufi). Define the following continuous symmetrical linear operator on D and its restriction to 17:

With the above notation, equation (1.2) reduces to

The following technical lemma will be useful. Theorem 2.2. Given a bounded open domain 17 in R N , the minimization problem (1.4) has nonzero solutions in H(j(£l) which are solution of the eigenvector equation

Given a bounded open nonempty domain D, there exists A^ > 0 (which only depends on the diameter of D and A) such that for all open subsets 17 of D,

Proof, (i) For any bounded open 17, the infimum is bounded below by 0 and hence finite. Let {(pn} be a minimizing sequence such that ||<^n||i,2(f2) = 1- Then the sequence ||Vv?n||L 2 (n) is bounded. Hence {
2. Embedding of #o(0) into H^(D) and First Eigenvalue

263

and, by definition of A"4(17), 0 ^ ip £ #g(Q) is a minimizing element. Since ip ^ 0, the Rayleigh quotient is difFerentiable and its directional derivative in the direction if} is given by

A nonzero solution ip of the minimization problem on 0 is necessarily a stationary point, and for all ip £ HQ(£I),

Conversely any nonzero solution of (2.3) is necessarily a minimizer of the Rayleigh quotient. (ii) Let diam (D) be the diameter of D and x £ RN be a point such that D C Bx = B(x, diam (£>)). Therefore from Theorem 2.1,

with the associated isometrics. Hence, by definition,

If \A(BX] = 0, we repeat the above construction with HQ(BX] in place of HQ(Q; Bx] and end up with an element

which is impossible in HQ(BX). Finally \A(BX) is independent of the choice of x since the eigenvalue is invariant under a translation of the domain: it only depends on the diameter of D. Remark 2.1. For A = I denote by A(Q), A(D), and AD, the quantities A A (Q), A A (D), and A^ of Theorem 2.2. For any open subset 0 of D and any

As a consequence, in what follows, the spaces HQ(£I) and HQ(D) will be endowed with the respective equivalent norms

264

3

Chapter 6. Optimization of Shape Functions

Hausdorff Complementary Topology

One way to prove the existence of extremal domains is to construct compact families in some topology of the space of domains and to prove that the map 0 i—>• J(0) is continuous. If we use a very strong topology on the family of domains, we easily get the continuity of the function, but the compact sets will very likely be trivial. In this chapter we use the Hausdorff complementary topology denned in section 2.3 of Chapter 4, but this will not be sufficient and some extra conditions will have to be imposed on the family of domains, for instance, the uniform cone property of section 6 in Chapter 2 for Lipschitzian domains or the more general capacity conditions which will be introduced in the next sections. The latter include the flat cone condition which generalizes the uniform cone property for Lipschitzian domains to a much larger class of bounded open domains. Consider for a bounded open domain D in RN the Hausdorff complementary metric (2.12) in section 2.3 of Chapter 4:

on the family Q(D] = {0 c D : Vf7 open} of all open subsets of D. It was proved in Theorem 2.4 of section 2.3 of Chapter 4 that p°H induces a complete metric topology called the Hausdorff complementary topology, denoted Hc. The Hausdorff rrc complementary convergence is denoted as O n —»0. For convenience we recall that theorem below. Theorem 3.1. Let D be a nonempty open subset o/R N . (i) The set C°d(D] is closed in C\OC(D}. (ii) //, in addition, D is bounded, C%(D) is compact in Co(D) and the space (Q(D),p°H) is a compact metric space. (iii) (Compactivorous property) Given a sequence {17n} and a set 17 in Q(D] such that for any compact subset K C 17, there exists an integer N(K) > 0 such that

4

Continuity of u^ with Respect to XI for the Dirichlet Problem

Consider a sequence of open subsets of a bounded open nonempty hold-all D of R N . By Theorem 3.1 (ii) there exists a subsequence {17 n} and 17 in Q(D] such that ^Cfi

""* ^Cfi

m

TTC

C(D]

(that is, £7n —> 0).

4. Continuity of w^ with Respect to 0 for the Dirichlet Problem

265

Denote by unn the solution in #o(Q n ) of the variational equation

and by u$i the solution in ^(0) of the variational equation (1.2). Denote by un = en n (wn n ) and UQ = e^(ufi) the extensions by zero in HQ(D] of u$in and UQ. They are the respective solutions of the variational equations

where ( • , • ) denotes the duality pairing between H~l(D) and HQ(D}. Lemma 4.1. Assume that D is a bounded open nonempty domain in R N . For any open subset £1 of D,

where u^ is the solution of (1.2) in &, Proof. Let u be the solution in Ho(Q;.D) of the variational equation (4.3). By assumption on A

and this completes the proof. In view of Lemma 4.1 for all n

where un is the solution of (4.2) in HQ ($ln; D). Therefore there exists a subsequence, still indexed by n, and u € HQ(D) such that

def

Given tp G £>(^), its support K = supp(p is a compact subset of 0. By Theorem 3.1 (iii) there exists TV > 0 such that, for all n > N, K C On and

By going to the limit in the above equation,

By density this extends to all (p in -fiT0(0; D}. We have proved the following result.

266

Chapter 6. Optimization of Shape Functions

Theorem 4.1. Let D be a bounded open nonempty domain in R N . Assume that {Qn} is a sequence of open subsets of D which converges to fi in the Hausdorff complementary topology and denote by un the solution of (4.2) in #o(fi n ; D). Then there exists a subsequence of {0,n}, still indexed by n, and u e HQ(D] such that

or equivalently

If, in addition, u 6 #o(fi;Z)) 7 then the convergence is strong in HQ(D). condition is satisfied for locally Lipschitzian domains.

This

Remark 4.1. In dimension TV = 1, u 6 Ho(fy and the convergence is strong. This follows from the fact that HQ (D) C C(D] and 0 is at most the union of a countable number of open intervals. Hence d$l Pi D is made up of at most a countable number of isolated points which are the limit of points of int CO where u is zero. Proof, (i) The first part of the theorem follows from the preceding discussion. If u G HQ(&,; D), then by letting (p = u in (4.5),

and the strong convergence follows:

(ii) Recall that By Theorem 2.1, the subsequence {un} converges in L2(Z))-strong and un = 0 almost everywhere in D\0n:

5. Continuity of the Neumann Problem

267

from Corollary 1 to Theorem 4.3 in Chapter 4. Therefore u G HQ(D), u = 0 almost everywhere in D\fL For Lipschitzian domains the trace of u is well defined on 90. It is zero since u and Vw are zero almost everywhere in the locally Lipschitzian domain CO by using the Gauss-Green formula. To get the continuity of w^ with respect to 0, it would remain to show that u belongs to #o(0; D). However, this is generally not true and u is not necessarily the solution of (4.3) in 0. To get around this difficulty the attention shifts to finding a family O of open subsets of D for which the following continuity property holds:

This problem is directly related to the sequential continuity of the projection P^v of an element v G HQ(D) onto HQ(£I; D) defined as

Therefore Theorem 4.1 applies with

and there exist VQ, an open subset 0 of D, and a subsequence {On} of open subsets of D such that

If VQ G HQ(£I; D), then VQ = P^v and the convergence is strong.

5

Continuity of the Neumann Problem

In the case of the Neumann problem associated with the Laplacian, the most important condition for obtaining the continuity with respect to the domain is the following density property. Definition 5.1. An open set Q is said to have the Hl -density property if the set

is dense in Hl(£l). Notice that this property is not verified for a domain Q with a crack in R2, since elements of H1 (0) have different traces on each side of the crack. Consider a sequence {On} of domains and a domain 0 satisfying the Hldensity property. Assume that the sequence converges to 0 in both the complementary Hausdorff metric and the Lp-topology of their characteristic functions.

268

Chapter 6. Optimization of Shape Functions

Given / e I/2 ($7), let yn and y be the respective solutions of the Neumann problems on On and 17:

Let y^ and (Vyn)° be the respective extensions by zero of yn and Vyn to RN as elements of L 2 (R N ) and L 2 (R N ) Ar . Then we get the following continuity:

where y° and (Vy)° are the respective extensions by zero of y and Vy to R N . The proof is based on the assumption that 17n and (7 satisfy the //^-density property. This implies that

Obviously

Then there exist z E L 2 (R N ) and Z E L 2 (R N ) 7V , and subsequences, still indexed by n, such that

From the compactivorous property (cf. Theorem 2.4 (iii) in Chapter 4), it follows that Z = (Vy)°. Going to the limit as n goes to infinity in the previous integral, we get

As we have also assumed that 17 had the f/"1-density property, the variational equation yields z — y. This is the approach followed by Liu and Rubio [1] in 1992. We know that the //^-density condition holds for the family of domains satisfying the segment condition of Definition 7.1 in Chapter 2 and, not only for H 1-spaces, but also VFm'p-spaces (cf. Theorem 7.5 in Chapter 2), thus extending the property and the results to a broader class of partial differential equations. We have shown in Theorem 7.3 of Chapter 2 that a domain satisfying the segment property is locally a C°-epigraph in the sense of Definition 5.2 of Chapter 2. We have seen in section 11 of Chapter 5 that the uniform segment property of Definition 7.1 (ii) in Chapter 2 is not sufficient to get the compactness of the corresponding family of subsets of a bounded hold-all D. However, from Theorem 11.1 in section 11 of Chapter 5, we have the compactness for the family of subsets of D verifying a uniform cusp property for which the segment property is also satisfied.

6. Optimization over Families of Lipschitzian or Convex Domains

269

In an unpublished report Tiba [2], and in a note Liu, Neittaanmaki, and Tiba [1], introduce a subfamily of domains which are locally a C°-epigraph in the sense of Definition 5.2 of Chapter 2. A compactness result is obtained by introducing the following conditions on the local C°-epigraphs: the boundaries of the domains belong (after a rotation and a translation) to a family F of uniformly equicontinuous functions denned in a fixed neighborhood V of 0 in R/^"1. This means that there exists a modulus of continuity /i(e) > 0 such that

The modulus JJL effectively controls the singularities of the boundary. Then the compactness follows by the Ascoli-Arzela theorem, Theorem 2.1 of Chapter 2. By Theorem 7.3 (ii) of Chapter 2 such domains satisfy the segment property and even the uniform segment property. Hence they satisfy the //^-density property and we have the continuity of the solutions of the Neumann problems with respect to domains in that family. Nevertheless, it is important to recall that the subsets of a bounded holdall verifying a uniform segment property do not include the very important domains with cracks for which some specific results have been obtained by Bucur and Zolesio [2, 4] (and by Bucur and Varchon [1] in dimension 2).

6

Optimization over Families of Lipschitzian or Convex Domains

Consider a family of bounded open Lipschitzian domains satisfying the uniform cone property of Definition 6.1 in section 6 of Chapter 2, for which we recall several compactness properties in Theorem 10.1 of section 10 in Chapter 5. Theorem 6.1. Let D be a bounded open subset o/R N and 1 < p < oo. For r > 0. (jj > 0, and A > 0, the family

is compact in C(D) and Wl>p(D}. As a consequence the families

are compact in C(D) and Wljp(D} and the families

are compact in LP(D).

270

Chapter 6. Optimization of Shape Functions

Therefore, we have the strong convergence in all the above topologies. In particular the characteristic functions converge in Ll(D) and the volume function is continuous. Also recall from Theorem 7.2 in Chapter 5 that the above properties associated with L(D,r,uj,\) are also true for

where E is some nonempty open subset of D.

6.1

Minimization of an Objective Function under State Equation Constraint

Let {£ln} be a sequence of open subsets of L(D,r, a;, A) such that the objective function J(O) defined by (1.1) converges to its infimum with respect to L(D, r, uj, A). In view of Theorem 6.1 the elements of the sequence {fin} are bounded Lipschitzian domains in D satisfying the corresponding uniform cone property. Moreover, there exists a subsequence, still indexed by n, and 0 in L(D,r,uj,X) such that 6^n —>• 6^ in C(D] and %Qn —>• xn in LP(D), 1 < p < oo. Then by Lemma 2.1 in Chapter 3 d^n -* den,

dnn ->• dn in C(D), and xnn ->• Xfi in £°°(£>)-weak*.

Since fi is Lipschitzian, 50 = <9intCQ by Theorem 6.1 in Chapter 2. Therefore from Theorem 4.1, u e f/o(fi;.D) and the convergence of the corresponding un to u is strong in HQ(D) and u = en(un). Theorem 6.2. Let D be a bounded open domain in R N . The infimum of the function (1.1) over L(D, r, cu, A) subject to the state equation constraint (1.2) /ias a minimizer fi m L(D,r,a;, A) anc? w^ satisfies equation (1.2). TTie conclusions remain true when L(D,r,w,X) is replaced by CE(D) for some nonempty open subset E ofD. Proof, (i) It is sufficient to show that the limiting element u is a minimizer of (1.1). For n

Take the liminf of both sides using Corollary 1 to Theorem 4.3 in Chapter 4 for xtin and the strong convergence for un:

By choice of the sequence, Q is a minimizing element in L(D, r, ui. A) and UQ satisfies the state equation by Theorem 4.1 for Lipschitzian domains such that intO ^ 0 and 0 ^ R N . (ii) Convex case. This proof has the same argument using the fact that CE(D] = I(£E)nC%(D), where

6. Optimization over Families of Lipschitzian or Convex Domains

271

The result follows from the compactness of C^(D] (Theorem 6.2 (ii) in section 6 of Chapter 4) and the fact that J(C£) is closed in C(D) (Theorem 2.3 (ii) of Chapter 4). Since E C 0 C -D, O is not empty and not equal to R N . Therefore 0 is locally Lipschitzian and the last condition of Theorem 4.1 is satisfied.

6.2

Optimization of the First Eigenvalue

Consider the first eigenvalue \A(£l] associated with the operator A in (2.2). By Theorem 2.2 for any open subset Q of D,

and any minimizer is solution of equation (2.3). Theorem 6.3. Let D be a bounded open domain in RN and A a matrix function satisfying assumption (1.3). The optimization problems

have solutions in L(D,r, a;, A). The conclusions of the theorem remain true when I/(D,r,u;, A) is replaced by CE(D] defined in (6.1). Proof, (i) Maximization. By the definition of L(£>,r, a;, A), for all ft C -D, there exists x 6 d£l and d x , \dx\ = 1, such that Cx = x + C(\,u),dx) C Q and

for any d, \d\ = I since the inf over HQ (Cx; D) only depends on the small cone and not its origin or direction. This provides an upper bound \c and

Prom that point on the proof is similar to what we have done before for the previous minimization problem. Let {£ln} be a maximizing sequence in L(D,r, u;, A). By Theorem 6.1 there exists a subsequence, still indexed by n, f2 G L(D,r, u;, A), A*, \A(D] < A* < Ac, u e H£(D), such that bnn -> 6n, and

272

Chapter6. Optimization of Shape Functions

By the same argument as for equation (1.2) applied to the eigenvector equation (2.3), •

and it is readily seen that u G HQ (0; D] satisfies the equation

Therefore, A* = \A($l} and 0 is a maximizer. (ii) Minimization. Same proof using the fact that, for all f2 £ L(D,r,u;, A), \A(£l) is bounded below by the strictly positive constant \A(D] > 0. (iii) Convex case. Same argument as in the proof of Theorem 6.2. Corollary 1. Under the assumptions of Theorem 6.3, the maximization and minimization problems (6.2) also have solutions in L(D,r,u>, A) under an equality or an inequality constraint of the form

provided that there exists 0' £ Z/(D,r, o>, A) such that

The conclusions remain true when L(D,r, u;, A) is replaced by CE(D) defined in (6.1). Proof. This follows from Theorem 6.1, where it is shown that the convergence is strong not only in the Hausdorff topologies, but also for the associated characteristic functions in Ll(D], thus making the volume function continuous with respect to open domains in L(D,r,u;, A). For the case of CE(D), recall from Theorem 7.2 (iii) of Chapter 5 that Cb(D; E] is compact in Wl'1(D), and hence the volume functional is continuous.

7

Elements of Capacity Theory

We have seen in the previous section that the key element in the proof of the continuity of the solution of (1.2) with respect to its underlying domain is that the limiting element satisfies the homogeneous Dirichlet boundary condition for the limiting domain. This section introduces capacity constraints which generalize the property to a broad class of open domains.

7.1

Definition and Basic Properties

The following definition of capacity and a number of basic technical results can be found in Hedberg [1] and are summarized below.

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273

Definition 7.1. Let D be a fixed bounded open subset of R N . (i) The capacity2 (with respect to D] is defined as follows: — for a compact subset K C D

— for an open subset G C D

— for an arbitrary subset E C D

(ii) A function / on D is said to be quasi-continuous if

(iii) A set E in RN is said to be quasi-open if

It can easily be shown that for a quasi-open set there exists a decreasing sequence {£ln} of open sets, such that On D E and c&pD(£ln\E) goes to zero as n goes to infinity. We say that a property holds quasi-every where (q.e.) in D if it holds in the complement D\E of a set E of zero capacity. A set of zero capacity has zero measure, but the converse is not true. The capacity is a countably subadditive set function, but it is not additive even for disjoint sets. Hence the union of a countable number of sets of zero capacity has zero capacity.

7.2

Quasi-Continuous Representative and H^Functions

Lemma 7.1 (Hedberg [1]). Any f in Hl(D] has a quasi-continuous representative: there exists a quasi-continuous function f i defined on D such that /i = / almost everywhere in D (hence f i is a representative of f in H1(D}). Any two quasi-continuous representatives of the same element of Hl(D) are equal quasieverywhere in D. The following key lemma completes the picture. 2

This definition is also referred to as the exterior capacity.

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ChapterG. Optimization of Shape Functions

Lemma 7.2 (Hedberg [1]). Lettl andD be two bounded open subsets o/R N such that Q C D and consider an element u of HQ(D). Then u\n 6 #o(^) tf ana onty if Ui = 0 quasi-every where on D\£l, where u\ is a quasi-continuous representative ofu. A function in Hl(D] is said to be zero quasi-everywhere in a subset E of D if there exists a quasi-continuous representative of (p which is zero quasi-everywhere in E. This makes sense since any two quasi-continuous representatives of an element (p of H1(D) are equal quasi-everywhere. For any (p e HQ(D] and t € R, the set {x G D : (f>(x) > t} is quasi-open. Moreover, the subspace HQ(^D] introduced in section 2 can now be characterized by the capacity. Corollary 2. Under the assumptions of Lemma 7.2,

The characterization of jF/Q (£);£)) requires the notion of capacity in a very essential way. It cannot be obtained by saying that the function and its derivatives are zero almost everywhere in D\Q. Recall the definition of the other extension #o(0; D} of Hl(£l) to a measurable set D containing 0 (cf. Chapter 3, section 2.5, Theorem 2.9, identity (2.29)):

By definition HQ(Q-,D) C H*(ft;D), but the two spaces are generally not equal, as can be seen from the following example. Denote by Br, r > 0, the open ball of radius r > 0 in R2. Define O = B2\dBi and D = B$. The circular crack dB\ in Jl has zero measure but nonzero capacity. Since dB\ has zero measure, H^(^l\D} contains functions i\) € H^B^} whose restriction to B2 are not zero on the circle dBi and hence do not belong to HQ (O). Yet for Lipschitzian domains 0 the two spaces are indeed equal. The following terminology is due to Rauch and Taylor [1]. Definition 7.2. il is said to be stable with respect to 7.3

Transport of Sets of Zero Capacity

Any Lipschitz continuous transformation of RN which has a Lipschitz continuous inverse transports sets of zero capacity onto sets of zero capacity. Lemma 7.3. Let D be an open subset of RN and T an invertible transformation of D such that both T and T~l are Lipschitz continuous. For any E C D

7. Elements of Capacity Theory

275

Proof, (a) E = K, K compact in D. Observe that, in the definition of the capacity, it is not necessary to choose the functions (p in C^(D}. They can be chosen in a larger space as long as

So the capacity is also given by

But the space 8 = HQ(D] fi C(D) is stable under the action of T:

Consider the matrix

By assumption on T, the elements of the matrix A belong to L°° (D) and

Define E(K) = & e E : v? > 1 on K}. For any (p e E(T(K}}, (p o T 6 £(#) and

Let K be a compact subset such that capD(K) = 0. For each e > 0 there exists (f> G E(K) such that

and we can repeat the proof with T""1 in place of T. (b) E = G, G open. Let

Therefore cap D (X) = 0, which implies that capD(T(K)) — 0 and hence

(c) General case. Let

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Chapter 6. Optimization of Shape Functions

Therefore for all e > 0, there exists G D E open such that

since T(E] C T(G] is open. By definition of capD(G) we have

Finally, for all £ > 0, capD(T(E)) < ae and hence capD(T(£)) = 0.

8

Continuity under Capacity Constraints

In order to preserve the homogeneous Dirichlet boundary condition for u on the boundary of the limit domain 17 in section 4, it is necessary to restrict our attention to smaller families of open domains. We have to be careful since the relative capacity of the complement of the domains near the boundary must not vanish. In order to handle this point, we introduce the following concepts and terminology related to the local capacity of the complement near the boundary points. The following definition is due to Heinonen, Kilpelainen, and Martio [1]. Definition 8.1. For r > 0 and a compact K C R N , the capacity condenser of K in the ball B(x,r) is defined as The reader is referred to Hedberg [1] and Bucur and Zolesio [5] for detailed properties. Definition 8.2. (i) Given r > 0, c > 0, and an open set, fi is said to satisfy the (r, c)-capacity density condition if

(ii) For 0 < r < 1 define the following family of open subsets of D:

Remark 8.1. There are different definitions of the capacity condenser. The definition used here is given in Hedberg [1]. The following slightly different definition is given by Heinonen, Kilpelainen, and Martio [1]:

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277

For a given compact set K we generally have

but the following two conditions are equivalent:

and

and so from the Oc^r families viewpoint, the two definitions are equivalent.

Definition 8.3. Let re be a point on d£l. The set CO is said to be thick at x if the complement of £1 locally has "enough" capacity, that is, if

Remark 8.2. If 0 satisfies the (r, c)-capacity density condition, the complement is thick in any point of the boundary. Recall Theorem 4.1 and the notation un for the solution of equation (4.2) in HQ(Q;D) and u for the weak limit in HQ(D) which satisfies equation (4.6). The following theorem gives the main continuity result. Theorem 8.1. Assume that D is a bounded open nonempty subset o/R N and that D is of class C2 for N > 3. Assume that A is a matrix function that satisfies the conditions (1.3) and that the elements of A belong to Cl(D}. Let {Hn} be a sequence in Oc^r(D) which converges in the Hc-topology to an open set 17. Then un —> u in HQ(D)-strong and MQ = u Q e //o(Q) is the solution of equation (1.2). Proof. When the dimension TV of the space is equal to 1 or 2, H1(D) C C(D) and no approximation of / is necessary. When N > 3 we first prove the result for / G HS(D), s > N/2 — 2 (since for D of class C2 the corresponding solution will belong to HS+2(D] C C(D}} and then, by approximation of /, we prove it for ftH-\D}. (i) First consider / e HS(D], s > N/2 - 2 for TV > 3 (s = -I for N = 1 or 2). The proof will make use of the following results from Heinonen, Kilpelainen, and Martio [I]. Lemma 8.1. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisfied. Let v be an A-harmonic function, that is, the weak solution of equation (1.2) in the open set 0 for f — 0. Then v can be redefined on a set of zero measure, so that it becomes continuous in 0.

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Chapter 6. Optimization of Shape Functions

Note that the continuous representative from Lemma 8.1 is in fact a quasicontinuous #0(0) representative. Indeed, let v\ = v almost everywhere and v\ be continuous on 0. We want to show that v\ is a quasi-continuous representative of v. There exists a quasi-continuous representative v% of v, which is equal to v almost everywhere. So v\ is continuous, v% is quasi-continuous, and v\ = i>2 almost everywhere. Using Heinonen, Kilpelainen, and Martio [1, Thm. 4.12], we get v\ = v% quasi- everywhere. Lemma 8.2. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisfied. Let Q belong to Oc,r(D}. If 9 G Hl(Sl] fl C(£l] and if h is an A-harmonic function in 0 such that h — 0 G HQ(£I), then

Note that the fact that £1 belongs to OCjr(D) involves the notion of thickness in any point of its boundary, which necessarily occurs in the proof of the lemma (cf. Definition 8.3 and Hedberg [1]). Returning to Theorem 8.1, it will be sufficient to prove the continuity for a subsequence of {On}. By Theorem 4.1 there exists a subsequence of {On}, still indexed by n, such that un weakly converges to u in HQ(D] and u satisfies equation (4.6) in 0. We now prove that under our assumptions, WQ = u\fi G HQ(£I), which implies that u — en(wn). For that purpose we use Lemma 7.2 which says that it is sufficient to prove that u = 0 quasi-everywhere in D\Q for some quasi-continuous representative u. From Theorem 4.1 we already know that u = 0 quasi-everywhere in D\£l. So it remains to show that u = 0 quasi-every where in <9Q n D. From the Banach-Saks theorem (cf. Ekeland and Temam [1]), there exists a sequence of averages

such that ipn —•> u in HQ(D}. Because of the strong convergence of {i/)n} to u in HQ(D), we have for a subsequence of {^n}, still indexed by n. Let GQ be the set of zero capacity on which {i^n(x)} does not converge to u(x). Given x G D\(£t U GO), we prove that, for all e > 0, u(x)\ < e. We have

There exists Ne>x > 0 such that for all n > N£jX,

It remains to show that there exists N' such that for all n > A7"', un(x}\ < e/2 implies i/Jn(x)\ < e/2. Denote by UD the solution of (1.2) in D. By assumption

8. Continuity under Capacity Constraints

279

on D and /, the solution up is continuous in D. Subtracting the corresponding equations, we obtain

Define

Therefore, the restriction hn of hn to $ln is .4-harmonic in Qn and, from Lemma 8.1, continuous in f£ n . Moreover, we have the continuity of u$in in the closure f) n , and utin is zero on the boundary. To show that, we use Lemma 8.2 with hn and 9n. By definition, hn — 9n = — u^ n belongs to #o(D n ). From the continuity of UD we obtain that the continuous extension hn of hn is equal to up on 90n. Hence the extension un of u^ n to the boundary is zero. Using Heinonen, Kilpelainen, and Martio [1, Thm. 6.44], we obtain that, if hn is 5-H61derian on d£ln, then there exists <$i, 6 < 5\ < 1, such that it is ^i-Holderian on all Q n . We have that U£> is (s + 2 - AT/2)-H61derian on D (with s + 2 - aV/2 > 0), with a constant M, because of the assumption on / and D. Finally, we get

So there exists 61 = 6i(N,/3/a,c) and

such that

By a simple argument we obtain that this inequality holds in D, and hence there exists 62, Si < 62 < 1, such that for all x, y 6 D we have

Choose R > 0 such that M^R5'2 < s/2. Because of the £Tc-convergence of fi n to fi there exists an integer HR > 0 such that for all n > HR we have (D\Q n ) n £?(x, ^) ^ 0. For x n e (D\fi n ) n S(x, /?),

because w n (x n ) = 0. So

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Chapter 6. Optimization of Shape Functions

Finally we obtain \u(x)\ < E. Because £ was arbitrary, we have u(x) = 0 quasieverywhere on D\0, which implies that u G HQ (fi; D] and u$i = U|Q 6 #0(17). The strong convergence of {un} to u now follows from Theorem 4.1 and u = e^(u^). (ii) In the next step we prove that the continuity result is preserved for / G H~l(D}. The main idea is to use the continuous dependence of the solution WQJ G HQ(£I\D} with respect to /, which is uniform in 0. Indeed, let 0 C D and /,g G H~l(D}. Then, by a simple subtraction of the equations, we get

So, let / G H~1(D) and f£ — f * p£ for some mollifier p£. Letting e —» 0 we have II/£ — f\\H~1(D) -* 0- Let {On} C OCjr(D) be a sequence that converges in the #c-topology to an open set 0. Then we have

because f£ G HS(D] and, from the previous considerations,

uniformly in (ln. Given 6 > 0, we get

Choose e sufficiently small such that ||/£ — f\\H~1(D) < ^/4j and for each E choose n£js > 0 such that

As S was arbitrary, the proof is complete. We can now recover the result of Sverak [2] in dimension N = 2. Theorem 8.2. Let the assumptions of Theorem 8.1 on the matrix function A be satisfied. Let N = 2 and I > 0 be a positive integer. Define the set

where # denotes the number of connected components. Then the set Oi is compact in the Hc-topology and the map

is continuous.

9. Flat Cone Condition and the Compact Family Oc>r(D}

281

A proof of the result of Sverak [2] is given in Bucur [1] as a consequence of Theorem 8.1. The main idea of the proof is to consider / > 0 (because of the decomposition of / = /+ — / ~ ) and a sequence {£ln} C Oi such that On converges in the #c-topology to an open set Q, and u$in —x u in HQ(D}. He constructs extensions £7+ of the domains £ln with the following properties:

where c and r are suitable constants. In fact it can be proved that 0+ satisfy this capacity density condition, because in an open connected set any two points are linked by a continuous curve that lies in the set, and in the bidimensional case a curve has a positive capacity. From the previous theorem we get u^+ —^ UQ+ . We easily obtain that M^+ — u^ > u > 0, which will imply that u = 0 quasi-everywhere on CS7, and this concludes the proof. For details, see Bucur [1].

9

Flat Cone Condition and the Compact Family Oc,r(D)

In this section we introduce families of domains which have a simpler geometric description and satisfy the capacity density condition for some constants c and r. For a weaker result in that direction see also Bucur and Zolesio [1]. We next prove that the set Oc^r(D} is compact in the Hc-topo\ogy. From this we prove existence of optimal solutions for several shape functions which continuously depend on the solution of the state equation.

Definition 9.1.

Let x e R N , 0 < uj < 7T/2, A > 0, and i/, d e R N such that z/| = \d = I. The flat cone is denned as

where Cx(X,uj,d)

is the cone defined in Notation 6.1 of Chapter 2 and

The cone CXV(X, u;, d} is said to be flat since it is contained in the hyperplane H x(v], and u;, A, and d are, respectively, the aperture, the height, and the orientation of the axis of symmetry of the cone. So its TV-dimensional Lebesgue measure is zero.

Definition 9.2 (flat cone condition). Let 0 < uj < 7T/2 and A > 0 be real numbers. (i) An open set !T£ in D, is said to satisfy the (a;, X)-flat cone condition (or simply (a;, A)-f.c.c.) if, for each x E <90, there exist unit vectors d x , vx € RN such that

(ii) The family of open subsets of D which satisfy the (u;, A)-f.c.c. will be denoted O(u,\,D}.

282

Chapter 6. Optimization of Shape Functions

Theorem 9.1. There exist constants c > 0 and r > 0 such that

Proof. This readily follows with r = A and for each x G d£l

and the properties of the capacity with respect to translation and similarity. Theorem 9.2. The family O(u,\,D) is compact in the Hc -topology. Proof. It is sufficient to prove that O(u>, A, D) is closed. For that, consider a sequence {£7n} C O(u,\,D) and n n -^fi. We shall prove that ft G 0(u;,A,D). Given a: G dfi, there exists a sequence of points, xn G £7n such that #n —>• x. As Qn G C?(u;, A,D), there exists a flat cone CXnvn(\u,dn} C C0n. It is easy to see that there exists subsequences {dnk} and {vnk} of {rfn} and {Vn}, respectively, such that dnk —> d and z/nfe —* ZA Because of the properties of the H°-topology we obtain that CXv(X,u),d) C CH. Finally O(ui,X,D) is closed and hence compact. Corollary 3. Given a sequence {£ln} of open sets in O(uj,\,D}, then

Theorem 9.3. The family Oc^r(D] of Definition 8.2 is compact. Proof. It is sufficient to prove that

ZJ-C

Pick a sequence {f2n} C Gc,r(D) such that O n —>£l. Define

Let x G d£l. We now prove that the capacity density condition for 17 is satisfied in the point x. Because of the #c-convergence,

Given e > 0 pick n e /2 and xn G d$ln such that x — xn\ < £/2. Denote by r the translation by the vector xn — x. As rB(x,r] = B(xn,r), and the capacity condenser is invariant under the translation of the two arguments,

Because Kn C ^e/2 and \x — xn\ < £/2 we have rKe D Kn. Then, from the monotonicity in the first argument, we get

9. Flat Cone Condition and the Compact Family Oc,r(D]

283

Using the capacity density condition for Qn we have ca

Ps(x,2r 0 )(^e nB(x,r 0 )) > c cap B ( X i 2 r ( ) ) (B(x, r 0 )).

Letting e —» 0, and using the continuity of the capacity for decreasing sequences of compact sets and the fact that

we get the capacity density condition for 17 in x. Hence 17 £ Oc,r(D). A first example of extremal domain follows directly from Theorem 9.3. Theorem 9.4. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisfied. Further assume that E is a nonempty open subset of D. Further assume that A satisfies conditions (1.3). Then the maximization problem

has solutions. In some sense the maximizing solution is an OC)T.(D)-approximation of the set E. The result follows from the upper semicontinuity of the map 17 i—>• AA(17) in the ./f c-topology and the fact that the family of admissible domains is compact (Theorem 2.3 in Chapter 4 and Theorem 9.3) for the Hausdorff complementary topology. We now give the following general existence theorem. Theorem 9.5. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisfied. Let un be the solution of (1.2) for 17 in Oc^r(D}. If h is continuously defined from HQ(D] into R, then J(17) = ^(CQ(UQ)) is continuously defined from OCjr(D] into R and reaches its extremal values on that set. Example 9.1. Consider the following shape function for which we can get the existence of minimizing domains even if the assumptions of Theorem 9.5 on h are not satisfied. Given a > 0,

where u^ is the solution of equation (1.2) and z e I/ 2 (D;R N ). From Theorem 2.1 u — eti(un) and VUQ are zero almost everywhere in D\f2, and the function can be rewritten as

The last term is not of the form h(u$i) for some h, but simply of the form h(£l). It turns out that J is not continuous but only lower semicontinuous for the Hctopology. Hence it can be minimized.

284

Chapter 6. Optimization of Shape Functions More precisely, we need the following result.

Lemma 9.1. Let IJL be a positive finite measure on D. The map 17 >—>• //(17) is lower semicontinuous with respect to the Hc-topology. Proof. Let {17n} be a sequence of open sets in £?, Kn = Z)\17n, K — D\17, and rrC

assume that 17n —> 17. Given £ > 0, we have K£ D _K"n for all n > n£. Then for all n > n£, n(K£] > (jL(Kn} and

But K£ is a monotonically decreasing sequence and then (JL(K£} —>• fJ-(K) as e —> 0, and the result follows. The minimization problem of Example 9.1 can now be formulated as follows on D. Let D be a bounded open nonempty domain in R N , / an element of H~1(D), and B a ball containing D. Let // be a positive measure on D and a a positive constant such that 0 < a < /t/(D) < +00. Let un be the solution of equation (1.2), and u = en(wn), its extension by zero in HQ(D}. For J(17) defined in (9.2) consider the following problem:

Theorem 9.6. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisfied. For any constants a > 0, c > 0 and r, 0 < r < 1, (9.4) has at least one solution. Proof. It is sufficient to notice that the first term in (9.3) is continuous under the assumptions of Theorem 9.5. The second term is lower semicontinuous from Lemma 9.1 (with the measure of density \z\2). To complete the proof, recall that, if 17n C D and 17n—>17, then 17 C D. As a > 0, a minimizing sequence cannot converge to 0. In fact, for any admissible domain 17o and any optimal solution WQ O , rrc

10

Examples with a Constraint on the Gradient

Let D be a fixed bounded smooth open domain in R N , / in H~l(D), and g in L2(D). For any open subset 17 of D, let u$i be the solution of the Dirichlet problem (1.2) in #0(17). Given a > 0, M > 0, and an open subset E of D, consider the following minimization problem:

10. Examples with a Constraint on the Gradient

285

It is understood that if |Vn| is not in L°°(D), then the esssup|Vu| is +00. In a first step it is easy to check that the problem

has minimizing solutions. Let {On} be a minimizing sequence. By assumption there exists £1 and a subsequence, still indexed by n, such that On —>• 0 in the #c-topology and Then |Vu n | converges in L2(D] to |Vw from the previous sections. For any (p £ L?+(D] we have

and then, in the limit, we get the same inequality with u. Then, the function being lower semicontinuous with respect to that topology, we get the existence. Lemma 10.1. For any f £ H~l(D) and any r > 0 and c> 0, problem (10.2) has solutions in Oc^r(D} In fact, the presence of the constraint on the gradient of u is helpful to get the continuity of the application 0 i—> u = e^(wn), and we directly get the existence of solutions to the original problem (10.1). In the proof of Theorem 8.1 we only needed the equicontinuity of the family of solutions {un = esin(unn}} and the fact that if x £ CO then u(x] = 0 for a quasi-continuous representative. If 0 6 (9C]T.(D), those two assumptions are readily satisfied. Notice that the boundedness of the gradients in (10.1) implies the equicontinuity of any minimizing sequence {un}. So, in order to obtain a continuity result with constraints on the boundedness of the gradient, we only have to notice that if u £ Hg(0), esssup|Vu| < M. Then u £ Wli°°(£l) and u is Lipschitzian with the constant M, or more exactly, there exists a Lipschitzian function almost everywhere equal to u, which is also a quasi-continuous representative of u. To obtain that u(x) — 0 in any point of the complement of 0, we have to introduce the following capacity constraints on 0: we require that 0 be capacity extended (see Bucur [1]), i.e., that 0 = O*, where

Indeed, let u be continuous on D, u = 0 quasi-everywhere on CO, and 0 = 0*. Then MX £ CO, Ve > 0 we have capD(B(x, ex] n CO) > 0 and there exists a point x£ in B(X,EX) Pi CO where u(xe] — 0. By continuity we get u(x) = 0. We recall from Bucur [1] the main properties of the capacity extension. Proposition 10.1. For any open 0 C D, the set 0* is open,

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Chapter 6. Optimization of Shape Functions

As cap£>(17*\f2) = 0 we get e.£i(u$i) — e^(w^*) and any minimizing sequence can only be made up of capacity-extended domains. Theorem 10.1. Given /, g G L2(D), a > 0, M > 0, problem (10.1) has minimizing solutions. Proof. From the boundedness of the gradient we obtain that {w^n } are uniformly Lipschitz continuous, and we can apply the same arguments as in Theorem 8.1, avoiding the capacity conditions. Consider the penalized version of problem (10.1). Given M > 0 and (3 > 0, define

and consider the existence of solutions to the following problem: Theorem 10.2. Given f and g in L2(D), a > 0, j3 > 0, M > 0, problem (10.5) has minimizing solutions. Proof. Let {On} be a minimizing sequence for the function J^, fin = Q*. We have esssup|VunJ < max{M,/3~ 1 J(^i)} = M' for all n > 1. Then, from the previous considerations we have the strong convergence in HQ(D] of un = e^ n (wo n ) to u = en(wn), for some subsequence {Qn} that converges to f2 in the #c-topology. Hence O is a minimizing domain. To complete the proof note that the map

is not lower semicontinuous from the #c-topology to R. Nevertheless, if {fin} is a minimizing sequence for problem (10.5), the required property is satisfied. We can assume that the sequence is chosen such that

For any e > 0 there exists n£ > 0 such that for n > n e , supD ess | Vwjin | < c + e. Because of the HQ (Z?)-strong convergence of un to w, we get

Remark 10.1. We can change the L°°-norm for a differentiate one; that is, we need By the continuous inclusion W1'P(D) C WE'°°(D), e small, the result still holds.

Chapter 7

^ansformations vers s I Flows of Veloci ti

1

Introduction

In the previous chapters we have constructed examples of metric spaces of sets, which, of course, are not topological vector spaces. Studying continuity and semidifferentiability of a function denned on such spaces is analogous to studying continuity and semidifferentiability on a manifold. The two notions can be associated with the behavior of the function along continuous one-dimensional curves or along flows generated by the solutions of a differential system. Such flows are easy to generate not only in the Euclidean space but also in smooth submanifolds where line paths between two points can no longer be used. The generic result of this chapter is that continuity of a shape function with respect to the Courant metric is equivalent to its continuity along the flows of velocity vector fields in the class of transformations associated with the Courant metric. In practice this condition is much easier to satisfy on specific examples. In the next chapter flows of velocities will be adopted as the natural framework for defining shape semiderivatives. In section 2 we motivate and adapt constructions and definitions of Gateaux and Hadamard semiderivatives in topological vector spaces to shape functions defined on shape spaces. The analogue of the Gateaux semiderivative for sets is obtained by the method of perturbation of the identity operator, while the analogue of the Hadamard semiderivative comes from the velocity (speed] method. In section 3 the velocity and transformation viewpoints will be emphasized through a series of examples of commonly used families of transformations of sets. They include Cfc-domains, Cartesian graphs, polar coordinates, and level sets. In section 4 we establish the equivalence between deformations obtained by a family of transformations and deformations obtained by the flow of a velocity field. Section 4.1 gives the equivalence under relatively general conditions. Section 4.2 shows that Lipschitzian perturbations of the identity operator can be generated by the flow of a nonautonomous velocity field. In section 4.3 the conditions of section 4.2 are sharpened for the special families of velocity fields in Co(R N ,R N ), C fc (RN, R N ), and CM(R~N, R N ). The constrained case where the family of domains 287

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Chapter 7. Transformations versus Flows of Velocities

are subsets of a fixed hold-all or universe is studied in section 5. In both sections 4 and 5 we show that under appropriate conditions, starting from a family of transformations is equivalent to starting from a family of velocity fields. This is a key result, which bridges the gap between the two points of view. Section 6 contains the central generic result of this chapter: the equivalence of the continuity of a shape function with respect to the Courant metric and its continuity alon^the flows of all velocity fields for the families Cj(R N , R N ), Ck(R™, R N ), and C' fc ' 1 (R N ,R N ) associated with the metric. This result is of both intrinsic and practical interest since it is generally easier to check the continuity along flows than with respect to the metric. In view of this equivalence the subsequent developments of this book will be based on the velocity (speed) method.

2

Shape Functions and Choice of Shape Derivatives

Recall that for a real function / : E —> R defined on a topological vector space E, the Gateaux semiderivative at x G E in the direction v G E is given as the limit (when it exists)

and the Hadamard semiderivative as

The analogue of the Gateaux semiderivative for shape functions will be obtained by the method of perturbation of the identity operator, while the analogue of the Hadamard semiderivative will come from the velocity (speed) method. The Hadamard notion of semiderivative is important in situations where the shape is parametrized and the chain rule is necessary. These basic notions and constructions will now be formally extended to deal with shape functions. We first give the definition of a shape function. Definition 2.1. Given a nonempty subset D of R N , consider the set P(D] — {ft : £1 C D} of subsets of D. The set D will be referred to as the underlying hold-all or universe. A shape function is a map

from some admissible family A of domains in P(D] into a topological space E such that for any homeomorphism T of D such that T(Q) = ft, J(ft) - J(T(ft)) for all elements 0 € A. This universe D can represent some physical or mechanical constraint, a submanifold of R N , or some mathematical constraint. In many instances it can be

2. Shape Functions and Choice of Shape Derivatives

289

chosen large and as smooth as necessary for the analysis. In the unconstrained case D is equal to R N . Consider a real-valued shape function

defined over a family A in P(R N ). All the spaces of domains considered in the previous chapters are nonlinear and nonconvex and their elements are equivalence classes of domains or transformations. So it is important to a priori specify the equivalence class under consideration and make sure that the value of the shape function J(fi) is well denned: it has the same value for all elements in the equivalence class. Even if A does not generally have a vector space structure, it is possible to consider differential quotients and their limit along one-dimensional paths around ft. Consider perturbations of the identity operator

for some family of vector fields 0 : RN —> R N . The semiderivative
as t> 0 goes to Zero , where

When 9 is restricted to some topological vector space, the continuity and the linearity of the map

can be studied, and the results are analogous to the ones in Banach spaces. Of course this approach is limited by the fact that perturbations occur along lines between 1 + 9 and the identity operator /. It is generally not applicable when D is a smooth submanifold of RN with nonzero curvature. This first approach to the definition of a shape semiderivative is not completely satisfactory since the perturbations of the identity are nonlocal transformations: the velocity field dx(t)/dt = 9(X) at the point x ( t ) — Tt(X) depends on the point X instead of x(i). In general, this will not affect first-order semiderivatives of J, but will introduce an "acceleration term" in the definition of second-order semiderivatives in Chapter 8. This can be easily fixed, and a more natural approach followed. Go back to our initial choice of deformation in (2.3) and consider t > 0 as an artificial time. Rewrite expression (2.3) as a difference

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Chapter 7. Transformations versus Flows of Velocities

Figure 7.1. Transport o/O by the velocity field V. and in differential form

In the limit this yields a new local deformation Tt : RN —>• R N ,

defined by the solution x(t] of the differential equation

This is the basis of the velocity (speed) method. Here the velocity of the point x(t) at time t is equal to the velocity field 6 evaluated at x(t): it is now a local deformation. The construction readily extends to transformations

defined by the flow of the differential equation

for nonautonomous velocity fields V(t)(x) = V(t,x) (cf. Figure 7.1). The choice of the terminology "velocity" to describe this method is accurate but may become ambiguous in problems where the variables involved are themselves "physical velocities": this situation is commonly encountered in continuum mechanics and material sciences. In such cases it may be useful to distinguish between the "artificial velocity" and the "physical velocity." This is at the origin of the terminology speed method which has often been used in the literature. The latter terminology is convenient, but not as accurate and descriptive as velocity method. We shall keep both terminologies and use the one that is most suitable in the context of the problem at hand.

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291

The shape semiderivative of J at 0 in the direction V will be defined as

(when the limit exists), where

and Tt is the transformation of RN defined by (2.10)-(2.11). In this book we have chosen to give all the basic definitions, constructions, and theorems within the context of the velocity method. This choice is motivated by the fact that most results based on perturbations of the identity or asymptotic developments can be readily recovered by direct application of the velocity method with nonautonomous velocity fields V(t,x). To support this assertion we now construct the time-dependent velocity field V(£,x) associated with the perturbation of the identity

The field V is to be chosen such that the function x in (2.14) is the solution of the differential equation

It is readily seen that the appropriate choice is V(t) = 9 o T^1 or

Under appropriate continuity and differentiability assumptions:

where D6(x] is the Jacobian matrix of 9 at the point x. In compact notation,

This last computation shows that at time 0, the points of the domain £7 are simultaneously affected by the velocity field V(0) = 9 and the acceleration field V"(0) — —[D9]9. Under suitable assumptions the two methods will produce the same first-order semiderivative. However, second-order semiderivatives will differ by an acceleration term which will appear in the expression obtained by the method of perturbation of the identity.

292

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Chapter 7. Transformations versus Flows of Velocities

Families of Transformations of Domains

In section 2 we have given a formal presentation of the main steps and options leading to the definition of semiderivatives of a shape function. We have emphasized perturbations of the identity operator and flows of velocity fields. Before proceeding with a more abstract treatment, we present several examples of definitions of shape semiderivatives that can be found in the literature. We consider special classes of domains (C°°, C fc , Lipschitzian), Cartesian graphs, polar coordinates, and level sets that provide classical examples of parametrized and/or constrained deformations. In each case we construct the associated underlying family (not necessarily unique) of transformations {Tt : Q
3.1

C°°-Domains

Let 0 be an open domain of class C°° in R N . Recall from section 5 of Chapter 2 that in any point x G F the unit outward normal is given by

where

When F is compact it is possible to find a finite sequence of points [xj : 1 < j < J} in F such that

As in the definition of the boundary integral, associate with {Uj} a partition of unity {TJ}:

for some neighborhood UQ of F such that

For the C°°-domain fi the normal satisfies

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293

since

Given any p e C°°(T} and t > 0, consider the following perturbation Tt of T along the normal field n:

We claim that for r sufficiently small and alH, 0 < t < r, the set Tt is the boundary of a C^-domain £lt by constructing a transformation Tt of RN which maps fi onto fit and F onto Tt. First construct an extension TV e D(R N , R N ) of the normal field n on F. Define

By construction

m ^ 0 on F, and there exists a neighborhood U\ of F contained in UQ where m ^ 0 since m is at least Cl. Now construct a function r0 in V(Ui), 0 < r 0 (x) < 1, and a neighborhood V of F such that Define the vector field

Hence TV belongs to £>(R N ,R N ) since supp/V C V is compact. Moreover,

For each j, po hj G C°°(BQ) and the extensions

belong, respectively, to C°°(B} and C°°(Uj). Then

is an extension of p from F to RN with compact support since supp r3 cUj CU.

294

Chapter 7. Transformations versus Flows of Velocities Define the following transformation of R N :

By construction, pN is uniformly Lipschitzian in R N , and by Theorem 4.2 there exists 0 < r such that Tt is bijective and bicontinuous from RN onto itself. As a result from Dugundji [1] for 0 < t < T,

Since the domain 0^ is specified by its boundary I\, it only depends on p and not on its extension p. The special transformation Tt introduced here is of class C°°, that is, Tt £ C°°(RN, R N ), and \ [Tt - I] is proportional to the normal field n on F, but it is not proportional to the normal nt on Tt for t > 0. In other words, at t = 0 the deformation is along n, but at t > 0 the deformation is generally not along nt. If J(fl) is a real-valued shape function defined on C^-domains in .D, the semiderivative (if it exists) is defined as follows: for all p £ C°°(F)

It turns out that this limit only depends on p and not on its extension p.

3.2

Cfe-Domains

When J7 is a domain of class Ck with boundary F, the normal field n belongs to 6 yfc ~ 1 (r,R N ). Therefore, choosing deformations along the normal would yield transformations {Tt} mapping Cfc-domains 0 onto Cfc-1-domains ftt = Tt(0). The obvious way to deal with Cfe-domains is to relax the constraint that the perturbation pN be carried by the normal. Choose vector fields 0 in £> fc (R N ,R N ) and consider the family of transformations

This is a generalization of the family of transformations (3.8) in section 3.1 from pN to 0. For k > 1, G is again uniformly Lipschitzian in R N , and by Theorem 4.2 there exists r > Q such that Tt is bijective and bicontinuous from RN onto itself. Thus for 0 < t < T,

A more restrictive approach to get around the lack of sufficient smoothness of the normal n to F would be to introduce a transverse field p on F such that

Given p and p £ C fc (F) define for t > 0

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295

Choosing Cfc-extensions p and p of p and p we can go back to the case where

For any 6 G D f c (R N ,R N ) the semiderivative is denned as

3.3

Cartesian Graphs

In many applications it is convenient to work with domains Q which are the hypograph of some positive function 7 in Cartesian coordinates. Such domains are typically of the form

where U is a connected open set in R and 7 G C(U', R+) is a positive function. Many free boundary and contact problems are formulated over such domains. Usually the domains Q (and hence the functions 7) will be constrained. The function 7 can be specified on U\U or not. In some examples the derivative of 7 could also be specified, that is, $7/dv = g on dU, where U is smooth and v is the outward unit normal field along dU. When 7 (resp., d^/dv] is specified along dU, the directions of deformation /z are chosen in C(U; R + ) such that

For small t > 0 and each such p, define the perturbed domain

and the obvious family of transformations

which can be extended to a neighborhood D of £7 containing the perturbed domains ^t, 0 < t < ti, for some small t\ > 0. In general, D will be such that D — U x [0, L] for some L > 0. This construction is also appropriate for domains that are Lipschitzian or of class Ck (1 < k < oo), depending on whether 7 is a Lipschitzian or a Cfc-function. Again the transformation Tt is equal to / + tO, where

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Chapter 7. Transformations versus Flows of Velocities

But of course this 0 is not the only choice for which £lt = 7^(0). For instance, let A : R+ —> R+ be any smooth increasing function such that A(0) = 0 and A(l) = 1. Then we could consider the transformations

This example illustrates the following general principle: the transformation Tt of D such that J7t = Tt(J7) is not unique. This means that, at least for smooth domains, only the trace Tt rt on r$ is important, while the displacement of the inner points does not contribute to the definition of f^. Nevertheless this statement is to be interpreted with caution. Here we implicitly assume that the objective function J(17) and the associated constraints are only a function of the shape of fi. However, in some problems involving singularities at inner points of fi (e.g., when the solution y(0) of the state equation has a singularity or when some constraints on the domain are active), the situation might require a finer analysis. One such example is the internal displacement of the interior nodes of a triangularization T^ when the solution y(£l) of a partial differential equation is approximated by a piecewise polynomial solution over the triangularized domain fi^ in the finite element method. Such a displacement does not change the shape of f^, but it does change the solution yh of the problem. When the displacement of the interior nodes is a priori parametrized by the boundary nodes the solution yh will only depend on the position of the boundary nodes, but the interior nodes will contribute through the choice of the specified parametrization.

3.4

Polar Coordinates

In some examples domains are star-shaped with respect to a point. Since a domain can always be translated, there is no loss of generality in assuming that this point is the origin. Then such domains 17 can be parametrized as follows:

where SN-I is the unit sphere in R N ,

and / : SN-I —* R+ is a positive continuous mapping from SN-I such that

Given any g G C(SN-I) and a sufficiently small t > 0 the perturbed domains are defined as

For example, choose t, 0 < t < ti, for some

3. Families of Transformations of Domains

297

and define the transformation Tt as

As in the previous example Tt is not unique, and for any continuous increasing function A : R + —> R+ such that A(0) — 0 and X(t) = 1 the transformation

yields the same domain fi t .

3.5

Level Sets

In sections 3.1-3.4 the perturbed domain 0^ always appears in the form £lt = Ti(Q), where Tt is a bijective transformation of RN and Tt is of the form I+t®. In some free boundary problems (e.g., plasma physics, propagation of fronts) the free boundary F is a level curve of a smooth function u defined over an open domain D. Assume that D is bounded open with smooth boundary 3D. Let u G C2(D) be a positive function on D such that

If m = max {u(x) : x G D}, then for each t in [0,ra[ the level set

is a C2-submanifold of R N in D, which is the boundary of the open set

By definition, OQ = D, for all t\ > £2, ^ti C Qt 2 , and the domains £lt converge in the Hausdorff topology to the point xu. The outward unit normal field on Tt is given by

This suggests introducing the velocity field

which is continuous everywhere but at x = xu. If V was continuous everywhere, then for each X, the trajectory x(t; X) of the differential equation

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Chapter 7. Transformations versus Flows of Velocities

would have the property that

since formally

This means that the map X i—» Tt(X] = x(t; X) constructed from (3.33) would map the level sets

onto the level set

and eventually OQ onto £lf Unfortunately, it is easy to see that this last property fails on the function u(x) = 1 — x2 defined on the unit disk. To get around this difficulty, introduce for some arbitrarily small £, 0 < £ < ra/2, an infinitely differentiate function p£ : RN —> [0,1] such that

and the velocity

As above, define the transformation

where x(t\ X] is the solution of the differential equation

For 0 < t < m — 2e, Tt maps FQ onto F f ; for 0 < s < m — E such that s +1 < m — £, Tt maps Fs onto Tt+s. However, for s > m — £, Tt is the identity operator. As a result, for 0 < t < m — 2£

Of course s > 0 is arbitrary and we can make the construction for t's arbitrary close to m. This is an example that can be handled by the velocity (speed) method and not by a perturbation of the identity. Here the domains f^ are implicitly constrained

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299

to stay within the larger domain D. We shall see in section 5 how to introduce and characterize such a constraint. Another example of description by level sets is provided by the oriented distance function for some open domain £7 of class C2 with compact boundary F. We have seen in Chapter 5 that there exists h > 0 and a neighborhood

such that &Q G C2 (Uh(F)). Then for 0 < t < h the flow corresponding to the velocity field V = V&Q maps £7 and its boundary F onto

4

Unconstrained Families of Domains

In this section we study equivalences between the velocity method (cf. Zolesio [12, 8]) and methods using a family of transformations. In section 4.1 we give some general conditions to construct a family of transformations of RN from a nonautonomous velocity field. Conversely, we show how to construct a nonautonomous velocity field from a family of transformations of R N . This construction is applied to Lipschitzian perturbations of the identity in section 4.2. In section 4.3 the equivalences of section 4.1 are specialized to velocities in Cj +1 (R N ,R N ), C f c + 1 (RN,R N ), andC f c ' 1 (RN,R N ), k > 0. 4.1

Equivalence between Velocities and Transformations

Let the real number r > 0 and the map V : [0, r] x RN —> R N be given. The map V can be viewed as a (time-dependent) nonautonomous velocity field {V(t) : 0 < t < T} defined on R N :

Assume that

where V(-,rr) is the function t H^ V(£,:r). Note that V is continuous on [0,r] x R N . Hence it is uniformly continuous on [0,r] x D for any bounded open subset D of RN and

Associate with V the solution x(t; V] of the vector ordinary differential equation

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Chapter 7. Transformations versus Flows of Velocities

and define the transformations

and the maps (whenever the inverse of Tt exists)

Notation 4.1. In what follows we shall drop the V in Ty(t,X), Tyl(t,x), and Tt(V) whenever no confusion arises. Theorem 4.1. (i) Under assumption (V) the map T specified by (4.4)-(4.6) has the following properties:

(Tl) (T2) (T3) (ii) Given a real number r > 0 and a map T : [0,r] x RN —>• R N satisfying assumptions (Tl) to (T3), the map

satisfies conditions (V), where T^1 is the inverse of X H-> T t ( X ) — T(t,X). If, in addition, T(0, •) = I, then T(-,X) is the solution of (4.4) for that V. (\\i) Given a real number r > 0 and a map T : [0, r] x R N —» RN satisfying assumptions (Tl) and (T2) and T(0, •) = /, then there exists r' > 0 such that the conclusions of part (ii) hold on [0, r'}. A more general version of this theorem for constrained domains (Theorem 5.1) will be given and proved in section 5.1. Proof, (i) Conditions (Tl) follow by standard arguments. Conditions (T2): Associate with X in RN the function

Then

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301

For each x G R N , the differential equation

has a unique solution in C 1 ([0,t];R N ). The solutions of (4.11) define the map

such that

In view of (4.10) and (4.11)

To obtain the other identity, consider the function

where y(-,x) is the solution of (4.11) for some arbitrary x in R N . By definition

and necessarily

Conditions (T3). The uniform Lipschitz continuity in (T3) follows from (4.12). and we only need to show that

Given t in [0,r] pick an arbitrary sequence {tn}, tn —>• t. Then for each x G R N there exists X G RN such that

from the first condition (Tl). But

By the uniform Lipschitz continuity of T^1

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Chapter 7. Transformations versus Flows of Velocities

and the last term converges to zero as tn goes to t. (ii) The first part of condition (V) is satisfied since, for each x 6 R N and t, s in [0,r],

Thus from (T3) and (Tl) t i—>• V(t,x) is continuous at s = £, and hence for all x in R N , V(.,x) e C([0, r];R N ). The Lipschitzian property follows directly from the Lipschitzian properties (Tl) and (T3): for all x and y in R N ,

This proves that V satisfies condition (V). (iii) From (Tl) and (T2) for f ( i ) = Tt - I and t > s,

For r' = min{r, l/(2c 2 )} and 0 < t < T', c(/(f» < 1/2,

and the second condition (T3) is satisfied on [0,r']. The first one follows by the same argument as in part (i). Therefore the conclusions of part (ii) are true on [0,r'].

This equivalence theorem says that we can start from either a family of velocity fields (V(£)} on RN or a family of transformations {Tt} of RN provided that the map V, V(t,x) = V(t)(x), satisfies (V) or the map T, T(t,X) = Tt(X), satisfies (Tl) to (T3). Starting from V, the family of homeomorphisms {Tt(V}} generates the family

of perturbations of the initial domain fi. Interior (resp., boundary) points of fi are mapped onto interior (resp., boundary) points of fV This is the basis of the velocity method which will be used to define shape derivatives.

4. Unconstrained Families of Domains

4.2

303

Perturbations of the Identity

In examples it is usually possible to show that the transformation T satisfies assumptions (Tl) to (T3) and construct the corresponding velocity field V defined in (4.9). For instance, consider perturbations of the identity to the first (A = 0) or second order: for t > 0 and X G R N ,

where U and A are transformations of R N . It turns out that for Lipschitzian transformations U and A, assumptions (Tl) to (T3) are satisfied in some interval [0,r]. Theorem 4.2. Let U and A be two uniform Lipschitzian transformations o/R N : 3c> 0 such that for all X, Y e R N 7

There exists r > 0 such that the map T given by (4.14) satisfies conditions (Tl) to (T3) on [0,r]. The associated velocity V given by

satisfies condition (V) on [0, T] .

Remark 4.1. Observe that from (4.14) and (4.15)

where DU is the Jacobian matrix of U. The term V"(0) is an acceleration at t = 0 which will always be present even when A = 0, but can be eliminated by choosing A = [DU}U.

Proof, (i) By the definition of T in (4.14), t H-> T(t, X] and t^3£(t,X) = U(X] + tA(X) are continuous on [0, oo[. Moreover, for all X and Y,

and

Thus condition (Tl) is satisfied for any finite r > 0. To check condition (T2), consider for any Y e RN the mapping h(X) =f Y - [Tt(X) - X}. For any Xl and X2

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Chapter 7. Transformations versus Flows of Velocities

For T = min{l,l/(4c)}, and any t, 0 < t < r, tc [1 + t/2] < 1/2 and h is a contraction. So for all 0 < t < r and Y £ R N , there exists a unique X G RN such that

and Tt is bijective. Therefore, (T2) is satisfied in [0,r']. The last part of the proof is the uniform Lipschitzian property of T^1. In view of (4.17), for alH, 0 < t < T, tc[I + t/2] < 1/2 and

In view of condition (T2) for all x and y,

To complete our argument we prove the continuity with respect to t for each x. Let X = T~l(x}. For any s in [0,r]

and in view of (4.18)

The continuity of T~l(x] at s = t now follows from the continuity of TS(X) at s — t. Thus condition (T3) is satisfied.

4.3

Equivalence for Special Families of Velocities

In this section we specialize Theorem 4.1 to velocities in C M (RN, R N ), CQ +I (R N , R N ), and C fc+1 (RN,R N ), k > 0. The following notation will be helpful:

whenever T^1 exists and the identities

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305

Recall also for a function F : R N —> RN the notation

Theorem 4.3. Let k > 0 be an integer. (i) Given T > 0 and a velocity field V such that

for some constant c > 0 independent o f t , the map T given by (4.4)-(4.6) satisfies conditions (Tl), (T2), and

for some constant c > 0 independent oft. Moreover, condition (T3) is satisfied and there exists r' > 0 such that

for some constant c independent o f t . (ii) Given r > 0 and T : [0, r] x R N —> R N satisfying conditions (4.20) and T(0, •) = I, there exists T' > 0 such that the velocity field V(i] = f ' ( t ) o T^"1 satisfies conditions (V) and (4.19) in [0,-r'j. Proof. We prove the theorem for fc = 0. The general case is obtained by induction over k. (i) By assumption on V', the conditions (V) given by (4.2) are satisfied and by Theorem 4.1 the corresponding family T satisfies conditions (Tl) to (T3). Conditions (4.20) on f . For any x and s < t

By assumption on V and Gronwall's inequality

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Chapter 7. Transformations versus Flows of Velocities

for another constant c independent of t. Moreover,

for some other constant c' by the second condition (Tl). Again by Gronwall's inequality, there exists another constant c such that

Moreover, /'(£) = V(t) o Tt and

Finally,

and c(/'(t)) < c for some new constant c independent of t. Therefore,

Conditions (4.21) on g. Since conditions (Tl) and (T2) are satisfied there exists r' > 0 such that conditions (T3) are satisfied by Theorem 4.1 (hi). Moreover, from conditions (4.20)

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307

Choose a new r" = minJT', l/(2c)}. Then for 0 < t < r", c(g(t)) < let. Now

For t in [0,r"], ct < 1/2, and

The conditions (4.20) on / are satisfied for k = 0. For /e = 1 we start from the equation

and use the fact that DT^1 = [DTt]~l o T^1 in connection with the identity

(ii) From conditions (4.20) on /, the transformation T satisfies conditions (Tl). To check condition (T2) we consider two cases: k > 1 and k — 0. For k > 1 the function t \-> Df(t] = DTt - I : [0,r] -> C' fc - 1 (RN,R N ) 7V is continuous. Hence t i—>• detDTt : [0,r] —> R is continuous and detDTo — 1. So there exists r' > 0 such that Tt is invertible for all t in [0,r'] and (T2) is satisfied in [0,r']. In the case /c = 0 consider for any Y the map h(X) =Y — f(t}(X). For any X\ and X2, |/i(X2) - /i(Xi) < c(/(t)) |^2 - Xi|. But by assumption / € ^([O,^;^ 0 '^^)) and c(/(0)) = 0 since /(O) = 0. Hence there exists r' > 0 such that c(/(t)) < 1/2 for all t in [0, T'} and /i is a contraction. So for all Y in R N there exists a unique X such that

Tt is bijective, and condition (T2) is satisfied in [0,r']. By Theorem 4.1 (iii) from (Tl) and (T2), there exists another r' > 0 for which conditions (T3) on g and (V) on V(t) = /'(£) oXp 1 are also satisfied. Moreover, we have seen in the proof of part (i) that conditions (4.21) on g follow from (T2) and (4.20). Using conditions (4.20) and (4.21),

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Chapter 7. Transformations versus Flows of Velocities

and c(V(i)} < c'. Also

Therefore, since both g and /' are continuous,

for some constant c independent of t. This proves the result for k = 0. As in part (i), for k = 1 we use the identity

and proceed in the sane way. The general case is obtained by induction over k. We now turn to the case of velocities in CQ (R N , R N ). As in Chapter 2, it will be convenient to use the notation CQ for the space <7o(R N ,R N ), C fc (R N ) for the space C f c (RN,R N ) and C M (RN) for the space C M (R*,R N ). Theorem 4.4. Let k > I be an integer. (i) Given T > 0 and a velocity field V such that

the map T given by (4.4)-(4.6) satisfies conditions (Tl), (T2), and

Moreover, conditions (T3) is satisfied and there exists r' > 0 such that

(ii) Given r > 0 and T : [0, r] x RN —> RN satisfying conditions (4.24) and T(0, •) = /, there exists T' > 0 such that the velocity field V(t) = f ' ( t ) o T^1 satisfies conditions (V) and (4.23) on [0,r']. Proof. As in the proof of Theorem 4.3 we only prove the theorem for k = 1. The general case is obtained by induction on fc, the various identities on /, g, /', and V, and the techniques of Theorem 8.1 and Lemmas 8.2 and 8.3 in Chapter 2. (i) By the embedding Cg(R N ,R^) C C 1 (R N ,R N ) C C°' 1 (R N ,R N ), it follows from (4.23) that V G CQO^jC^R^R 1 *)) and condition (4.19) of Theorem 4.3 are satisfied. Therefore, conditions (4.20) and (4.21) of Theorem 4.3 are also satisfied in some interval [0, r'], T' > 0.

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309

Conditions (4.24) on f . It remains to show that f(t) and /'(£) belong to the subspace Co(R N ,R N ) of C(R N ,R N ) and to prove the appropriate properties for Df(t) and D f ' ( t ) . Recall from the proof of the previous theorems that there exists c > 0 such that

By assumption on V(0), for E > 0 there exists a compact set K such that

and there exists £, 0 < 6 < 1, such that

Proceeding in this fashion from the interval [0, 6] to the next interval [<5, 26} using the inequality

the uniform continuity of V

and the fact that V(6) G Co, that is, that there exists a compact set K(6) such that

we get /(t) G C0, 5 < t < 26, and hence / G C([0,r];C 0 ). For /'(£) we make use of the identity /'(£) = V(t) o Tt. Again by assumption for any £ > 0 there exists a compact set K(t) such that |l/(t)(x)| < e on C/C(t). Thus by choosing the compact K't = T t ~ l ( K ( t ) ) , f'(t)(x)\ < £ on C^, and /' e C([0,r];C 0 ). In order to complete the proof, it remains to establish the same properties for Df(t) and D f ' ( i ) . The matrix Df(t) is solution of the equations

From the proof of Theorem 8.1 in Chapter 2 for each t the elements of the matrix

belong to Co since DV(t] and /(t) do. By assumption, V € (7([0,r];Co) and I/ and all its derivatives daV are uniformly continuous in [0,r] x R N . Therefore, for each e > 0 there exists 6 > 0 such that

310

Chapter 7. Transformations versus Flows of Velocities

Pick 0 < 6' < 6 such that

For each x, Df(t}(x) is the unique solution of the linear matrix equation (4.26). To show that D f ( t ) 6 (Co]N we first show that Df(t)(x) is uniformly continuous for x in R N . For any x and y

But / e C([0,r];Co) is uniformly continuous in ( t , x ) : for each s > 0 there exists 6, 0 < 6 < £/(2cr), such that

Substituting in the previous inequality for each e > 0, there exists 6 > 0 such that

Hence Df(t) is uniformly continuous in R N . Furthermore, from the equation (4.26) we have the following inequality:

since V G C([0, r];Cg). By Gronwall's inequality,

for some other constant c independent oft. Thence Df(t) E C(R N ,R N ) j V . Finally to show that Df(t) vanishes at infinity we start from the integral form of (4.26):

4. Unconstrained Families of Domains

311

By the same technique as before for /(t), it follows that the elements of Df(t) belong to CQ since both /(s) and DV(r) do. Finally for the continuity with respect to t

Again, by Gronwall's inequality, there exists another constant c such that

Therefore, Df G C([0,r]; (C0)N) and / G C([0,r];Cj). For Df we repeat the proof for /' using the identity

to get

Conditions (4.25) on g. From the remark at the beginning of part (i) of the proof, the conclusions of Theorem 4.3 are true for g, and it remains to check the remaining properties for g and Dg using the identities

By the proof of Theorem 8.1 in Chapter 2, g(t) G CQ since Df(t) and g(t) do. Therefore, g(t) G CQ. The continuity follows by the same argument as for /' and peCaO.rhCo 1 ). (ii) By assumption from conditions (4.24), conditions (Tl) are satisfied. For (T2) observe that for k > 1 the function t i-> Df(t) = DTt - I : [0,r] -» C' fc - 1 (RN,R N ) 7V is continuous. Hence t ^ det DTt : [0,r] -> R is continuous and detDTo = 1. So there exists r' > 0 such that Tt is invertible for all t in [0,r'] and (T2) is satisfied in [0,r']. Furthermore, from the proof of part (i) conditions (T3) and (4.25) on g are also satisfied in some interval [0,r'], r' > 0. Therefore, the velocity field

satisfies the conditions (V) specified by (4.2) in [0,r']. By the proof of Theorem 8.1 in Chapter 2, V(t] e CQ since f ' ( t ) and g(t) belong to CQ. By assumption, / G C 1 ([0,r];Cg)- Hence /' and all its derivatives 5a/', a\ < fc, are uniformly continuous on [0, r] x R N ; that is, given e > 0, there exists 6 > 0 such that

312

Chapter 7. Transformations versus Flows of Velocities

Similarly g € (7([0, r'];CJo) and there exists 0 < 6' < 6 such that

Therefore, for \t - s\ < 6'

and since 6' < 6

We then proceed to the first derivative of V,

and by uniform continuity of the right-hand side V 6 C([0,r'];Co). By induction on fc, we finally get V € C([0, T'];Cj). The proof of the last theorem is based on the fact that the vector functions involved are uniformly continuous. The fact that they vanish at infinity is not an essential element of the proof. Therefore, the theorem is valid with C f c (R N ,R N ) in place of C$(R N ,R N ). Theorem 4.5. Let k > I be an integer. (i) Given r > 0 and a velocity field V such that

the map T given by (4.4)-(4.6) satisfies conditions (Tl), (T2), and

Moreover, conditions (T3) are satisfied and there exists r' > 0 such that

(ii) Given r > 0 and T : [0, r] x RN —>• RN satisfying conditions (4.29) and T(0, •) = /, there exists r' > 0 such that the velocity field V(t) = f ' ( t ] o T^1 satisfies conditions (V) and (4.28) on [0,r'].

5

Constrained Families of Domains

We now turn to the case where the family of admissible domains fJ is constrained to lie in a fixed larger subset D of RN or its closure. For instance, D can be an open set or a closed submanifold of R N .

5. Constrained Families of Domains

5.1

313

Equivalence between Velocities and Transformations

Given a nonempty subset D of R N , consider a family of transformations

with the following properties:

where under assumption (T2/?), T~l is denned from the inverse of Tt as

Those three properties are the analogue for D of the same three properties obtained for R N . In fact, Theorem 4.1 extends from RN to D by adding one assumption to (V). Specifically we shall consider for r > 0 velocities

such that

where Tp(x) is the Bouligand contingent cone to D at the point x in D

and B is the unit disk in RN (cf. Aubin and Cellina [1, p. 176]). This definition is equivalent to

(cf. Aubin and Frankowska [1, pp. 121-122; 17; 21]). Note that when D is bounded inR N ,

314

Chapter 7. Transformations versus Flows of Velocities

When D is equal to R N , Tp(x) = RN for all x and condition (V2p) can be dropped. When D is equal to the boundary dA of a set A of class C1'1 in R N , dA is a C1'1submanifold of RN and

that is, at each point of dA, the velocity field is tangent to dA: it belongs to the tangent linear space of dA. The next theorem is a generalization of Theorem 4.1 from RN to an arbitrary set D which shows the equivalence between velocity and transformation viewpoints.

Theorem 5.1. (i) Let T > 0 and V be a family of velocity fields satisfying conditions (Vl£>) and (V2/)) and consider the family of transformations

where x ( - , X ) is the solution of

Then the family of transformations T satisfies conditions (Tl/p) to (T3£>). (ii) Conversely, given a family of transformations T satisfying conditions (Tl/j) to (T3.c>), the family of velocity fields

satisfies conditions (Vlp) and (V2z?). //, in addition, T(0, •) = I, then T(-, X) is the solution of (5.9) for that V. (iii) Given a real number r > 0 and a map T : [0, T] x D —> D satisfying assumptions (Tl£)) and (T2^)) and T(0, •) = /, then there exists r' > 0 such that the conclusions of part (ii) hold on [0, T'} . Remark 5.1. Assumption (V2rj} is a double viability condition. Nagumo's [1] usual viability condition

is a necessary and sufficient condition for a viable solution to (5.9):

(cf. Aubin and Cellina [1, pp. 174 and 180]). Condition (V2D):

5. Constrained Families of Domains

315

is a strict viability condition which says that Tt maps D into D and

In particular, it maps interior points onto interior points and boundary points onto boundary points (cf. Dugundji [1, pp. 87-88]).

Remark 5.2. Condition (V2£>) is a generalization to an arbitrary set D of the following condition used by Zolesio [12] in 1979: For all x in 3D,

Proof of Theorem 5.1. (i) Existence and uniqueness of viable solutions to (5.9). Apply Nagumo's [1] theorem to the augmented system on fO,r]:

that is,

where x(t) = (z 0 (t),z(t)) e R N+1 , V(x) = ( l , V ( x ) ) , and

It is easy to check that systems (5.15) and (5.16) are equivalent on [0,r] and that x(t) = (t,x(t)). The new velocity field on V C R +1 is continuous at each point x £ D by the first assumption (Vlrj} since

and for 0 < XQ , yo < r

In addition,

316

Chapter 7. Transformations versus Flows of Velocities

Moreover, V(D] is bounded and R^+1 is finite-dimensional. By using the version of Nagumo's theorem given in Aubin and Cellina [1, Thm. 3, part (b), pp. 182-183], there exists a viable solution x to (5.16) for all t > 0. In particular,

which is necessarily of the form x(t) = (£, x(i)}. Hence there exists a viable solution x,x(t} G D on [0, r], to (5.9). The uniqueness now follows from the Lipschitz condition (VID}- The Lipschitzian continuity (Tip) can be established by a standard argument. Condition (T2/?). Associate with X in D the function

Then

For each x G D, the differential equation

has a unique viable solution in C1([0, £];R N ):

since by assumption (F2/))

The proof is the same as above. The solutions of (5.19) define a Lipschitzian mapping such that

Now in view of (5.18) and (5.19)

To obtain the other identity, consider the function

where y ( - , x ) is the solution of equation (5.19). By definition,

5. Constrained Families of Domains

317

and necessarily

Condition (TS^)- The uniform Lipschitz continuity in (T3£>) follows from (5.21) and (T2^), and we need only to show that

Given t in [0,r], choose an arbitrary sequence {tn}, tn —-> t. Then for each x e D there exists X £ D such that

from (Tl£>). But

By the uniform Lipschitz continuity of T^1

and the last term converges to zero as tn goes to t. (ii) The first condition (Vl^) is satisfied since for each x £ D and t, s in [0, r]

The second condition (Vl£>) follows from (T1D) and (T3.o) and the following inequality: for all x and y in D

To check condition (V2£>), recall the definition (5.6) of Bouligand contingent cone:

Chapter 7. Transformations versus Flows of Velocities

318

where B is the unit disk in R N . We first show that

By (T2/)), Tt is bijective. So it is equivalent to show that

For simplicity we use the notation

By the definition of Tp(x(t)), we must prove that

such that x'(t] G u + eB and x(t) + hu 6 D. Choose 5,0 < 6 < a, such that

Then fix i',0< t' -t <6,

Therefore choose u = [x(t') - x(t}]/(t' - t). Now

and choose h = t' — t since 0 < t' — t < 6 < a:

by assumption on T. This proves (5.22). The second part of (V2£>) is

which is equivalent to proving that

or with the simplified notation

5. Constrained Families of Domains

319

We proceed exactly as in the proof of (5.22) except that we choose t' such that 0 < t - t' < 6, h = t - t', and u = -[x(t] - x(t'}]/(t - t'}. Then

and

and we get (5.23). (iii) From (T1D) and (T2 D )

For r'

and the second condition (T3) is satisfied on [0,r']. The first one follows by the same argument as in part (i). Therefore, the conclusions of part (ii) are true on [0, r']. This completes the proof of the theorem. D

5.2 Transformation of Condition (V2D) into a Linear Constraint Condition (V2^) is equivalent to

since T^(x] = TD(X}. If TD(X) were convex, then the above intersection would be a closed linear subspace of R N . This is true when D is convex. In that case TD(X) — CD(X}-, where CD(X] is Clarke tangent cone and

is a closed linear subspace of R N . This means that (V2#) reduces to

It turns out that for continuous vector fields V(t, •), the equivalence of (V2£>) and (5.26) extends to arbitrary domains D. This equivalence generally fails for discontinuous vector fields. Other equivalences might be possible between TO and some intermediary convex cone between CD and TD, but there is no evidence so far of that fact. For smooth bounded open domains Q, the two cones coincide and the condition reduces to V • n = 0, n the normal to <9fJ, and V(t,x) belongs to the tangent space to d$l in each point of d£l.

320

Chapter 7. Transformations versus Flows of Velocities

Theorem 5.2. (i) Given a velocity field V satisfying (VID], condition (V2rj) is equivalent to

where CD(X) is the (closed convex) Clarke tangent cone to D at x,

and —^ denotes the convergence in D. D (ii) LD(X) is a closed linear subspace o/R N . Proof, (i) The equivalence of (V2£>) and (V2c) is a direct consequence of the following lemma. Lemma 5.1. Given a vector field W € C(D; R N ), the following two conditions are equivalent:

(ii) The set Lrj(x) is closed as the intersection of two closed sets. To show that it is linear, we show that for all a 6 R and V e Lp(x), aV G LD(X), and for all V and W in LD(X), V + W G LD(X). Since ±CD(X) are cones,

By convexity of ±Cx>(z)

This completes the proof of the theorem. Proof of Lemma 5.1. Assume that (5.29) is verified. By definition CD(X) C TD(X) and (5.29) =>• (5.28). Conversely, either x is an isolated point and Tp(x) = {0} = CD(X), or there are points x ^ y G D such that y —^ x. In the latter case we know that

(cf., for instance, Aubin and Frankowska [1, Thm. 4.1.10, section 4.1.5, p. 130]). Since W is continuous in D and (5.28) is satisfied, then for each x G D

and (5.28) implies (5.29).

6. Continuity of Shape Functions

321

Remark 5.3.

Lemma 5.1 essentially says that for continuous vector fields we can relax the condition of Nagumo's [1] theorem from (V2£>) involving the Bouligand contingent cone to (V2cO involving the smaller Clarke convex tangent cone. In dimension N = 3, LD(X] is {0}, a line, a plane, or the whole space.

Notation 5.1.

In what follows, it will be convenient to introduce the following spaces and subspaces:

and for an arbitrary domain D in RN

For any integers k > 0 and m > 0 and any compact subset K of RN define the following subspaces of £:

where ~Dk(K: R N ) is the space of fc-times continuously differentiable transformations of RN with compact support in K. In all cases V^' C CK- As usual T>°°(K, R N ) will de written T>(K,RN).

6

Continuity of Shape Functions

In this section we give a characterization of the continuity of a shape function

defined on a family A in ^(R1^) (cf. Definition 2.1) with values in a Banach space B with respect to the Courant metric in terms of its continuity along the flows generated by a family of velocity fields using the equivalence Theorems 4.3, 4.4, and 4.5. Checking the continuity along flows is usually easier and more natural. We specifically consider the continuity of shape functions with respect to the Courant metric associated with the quotient spaces of transformations FQ /<5(£t) of section 8 and JF fc (RN)/£(Q) and Fk>l(lV*)/g(n) of sectionjU in Chapter 2 corresponding to the families of velocity fields C$(R N ,R N ), C fc (RN,R N ), and C M (R*,R N )6.1

Courant Metrics and Flows of Velocities

We start with the space Cj(R N ) = Cj(R N ,R N ) used in Micheletti [1].

322

Chapter 7. Transformations versus Flows of Velocities

Theorem 6.1. Let k >l be an integer, B a Banach space, and ft a nonempty open subset o/R N . Consider a shape function J : N$i([I}} —»• B defined in a neighborhood Nfi([I}) of [I] in FQ/G(£L}. Then J is continuous at ft for the Courant metric if and only if

for all families of velocity fields {V(t) : 0 < t < T} satisfying the condition

Proof. It is sufficient to prove the theorem for a real-valued function J. The Banach space case is readily obtained by considering the new real-valued function j(T) =

\j(T(ft)) - j(ft)\.

(i) If J is (5-continuous at Q, then for all e > 0 there exists 6 > 0 such that

Condition (6.3) on V coincides with condition (4.23) of Theorem 4.4, which implies conditions (4.24) and (4.25):

But by definition of the metric 6,

and we get the convergence (6.2) of the function J(Tt(ft)) to J(ft) as t goes to zero for all V satisfying (6.3). (ii) Conversely, it is sufficient to prove that for any sequence {[Tn]} such that <5([Tn], [/]) goes to zero, there exists a subsequence such that

Indeed let

By definition of the liminf, there is a subsequence, still indexed by n, such that i — liminf n _,oo J(Tn(ft)). But since there exists a subsequence {Tnk} of {Tn} such that J ( T n k ( f t ) } —> J(ft), then necessarily i = J(ft). The same reasoning applies to the limsup and hence the whole sequence J(Tn(ft)) converges to J(fi) and we have the continuity of J at ft. We prove that we can construct a velocity V associated with a subsequence of {Tn} verifying conditions (4.23) of Theorem 4.4 and hence conditions (6.3). By Corollary 1 of Theorem 8.2 in Chapter 2 and the same technique as in the proofs

6. Continuity of Shape Functions

323

of Lemma 8.7 and Theorem 8.3 in Chapter 2, associate with a sequence {Tn} such that <5([Tn], [/])—» 0 a subsequence, still denoted {Tn}, such that

For n > I set tn = f2rn and observe that tn — tn+\ = — 2~(n+1\ Define the following Cl-interpolation in (0,1/2]: for t in [t n+ i,t n ]

where p G P3[0,1] is the polynomial of order 3 on [0,1] such that p(0) = 1 and p(l) = 0 and p^(0) = 0 = p (1) (l)Conditions on f. By definition for all t, 0 < t < 1/2, /(t) = Tt -1 e Cj(R N ). Moreover, for 0 < t < 1/2

f (t) = 8T/dt(t,.) e C 0 fe (R N ) and /(•)(*) = T(-,X) - 7 e ^((0,1/2]; R N ). By definition, /(O) = 0. For each 0 < t < 1/2 there exists n > N such that tn+i
Define at t = 0, f ' ( t ] = 0. By the same technique there exists a constant c > 0, and for each 0 < t < 1/2 there exists n > N such that tn+\
So for each X the functions t ^ f(t)(X) and t H-> T t ( X ) belong to C^QO, 1/2]; R N ). By uniform Ck-continuity of the Tn's and the continuity with respect to t for each X, it follows that / e C^QO, l/2];Cj(R N )) and the condition (4.24) of Theorem 4.4 is satisfied. Hence the corresponding velocity V satisfies conditions (4.23). Finally V satisfies conditions (6.3) and by (6.2) for all e > 0 there exists 5 > 0 such that

324

Chapter 7. Transformations versus Flows of Velocities

In particular there exists N > 0 such that for all n > N, tn < 5, and

and this proves the 5-continuity for the subsequence {Tn}. The case of the Courant metric associated with the space C fc (R N ) = CA:(RN, RN is a corollary to Theorem 6.1. Theorem 6.2. Let k > 1 be an integer, B a Banach space, and f2 a nonempty open subset o/R N . Consider a shape function J : Afo([/]) —>• B defined in a neighborhood Afo([/]) of [I] in ^ r/c (R N )/^(n). Then J is continuous at ft for the Courant metric if and only if

for all families of velocity fields {V(t) : 0 < t < T} satisfying the condition

The proof of the theorem for the Courant metric topology associated with the space C M (RN) = C M (RN, R N ) is similar to the proof of Theorem 6.1 with obvious changes. Theorem 6.3. Let k > 0 be an integer, 0 a nonempty open subset o/R N , and B a Banach space. Consider a shape function J : Afo([/]) —» B defined in a neighborhood Nfi([I}) of [I] in J? rfc ' 1 (R N )/^(0). Then J is continuous at fi for the Courant metric if and only if

for all families {V(t) : 0 < t < T} of velocity fields in C fc)1 (R N ,R N ) satisfying the conditions

for some constant c independent of t. Proof. As in the proof of Theorem 6.1, it is sufficient to prove the theorem for a real-valued function J. (i) If J is 5-continuous at Q, then for all £ > 0 there exists 6 > 0 such that

Under condition (6.7), from Theorem 4.3

6. Continuity of Shape Functions

325

But by definition of the metric 6 and we get the convergence (6.6) of the function J(T t (Q)) to J(0) as t goes to zero for all V satisfying (6.7). (ii) Conversely, as in the proof of Theorem 6.1, it is sufficient to prove that given any sequence {[Tn]} such that <5([Tn], [/]) —> 0 there exists a subsequence such that By Theorem 9.1 (i) and the same technique as in the proofs of Lemma 8.7 and Theorem 8.3 of Chapter 2, associate with a sequence {Tn} such that 5([Tn], [/]) —» 0 a subsequence, still denoted {Tn}, such that For n > 1 set tn = 2~ n and observe that tn—tn+i = — 2~( n+1 ). Define the following (^-interpolation in (0,1/2]: for t in [t n+ i,t n ],

where p G -P3[0,1] is the polynomial of order 3 on [0,1] such that p(0) = 1 and p(l) = 0 and pW(Q) = 0 = p (1) (l)Conditions on f. By definition for all t, 0 < t < 1/2, f ( t ) =Tt-Ie Ck'l(BF). Moreover, for 0 < t < 1/2

/'(t) = dT/dt e C^HR^) and /(-)PO - T(-,X) -I £ Cl((Q, 1/2];RN). By definition, /(O) = 0. For each 0 < t < 1/2 there exists n > N such that tn+i
Define at t — 0 f ' ( t ] ~ 0. By the same technique there exists a constant c > 0, and for each 0 < t < 1/2 there exists n > N such that tn+i < t < tn and

326

Chapter 7. Transformations versus Flows of Velocities

and for each X the functions 11-> f(t)(X) and 11-> r t (X) belong to C^QO,1/2]; R N ). By uniform Ck-continuity of the Tn's and the continuity with respect to t for each X, it follows that / E ^([O, l/2];C fe (RF)). Moreover, it can be shown that

for some c' > 0. The result is straightforward for k — 0 and then the general case follows by induction on k. As a result / e C([0, l/2];C fc)1 (R N )) and the condition (4.20) of Theorem 4.3 is satisfied. Hence the corresponding velocity V satisfies conditions (4.19). Finally the velocity field V satisfies conditions (6.7), and by (6.6) for all e > 0 there exists 6 > 0 such that

In particular there exists N > 0 such that for all n > N, tn < 6 and

and we have the ^-continuity for the subsequence {Tn}.

Remark 6.1. The conclusions of Theorems 6.1, 6.2, and 6.3 are generic. They also have their counterpart in the constrained case. For instance, a generalization of Theorem 6.1 in the constrained case has been announced by Boisgerault and Gomez [1] for an open subset D of RN of class C2. The difficulty lies in the second part of the theorem, which requires a special construction to make sure that the family of transformations {Tt ' 0 < t < T} constructed from the sequence {Tn} are homeomorphisms of D.

6.2

Shape Continuity and Velocity Method

These results point to a natural notion of directional shape continuity associated with the velocity method which is very much in the spirit of geometry. This intimate relationship between the Courant metric on quotient spaces of transformations and the velocity method is a nice mathematical result and a powerful tool for the analysis of shape problems. It will be particularly useful for the differentiability of shape functions in Chapter 8.

Definition 6.1.

Let / be a shape function with values in a Banach space B. (i) / is shape continuous at O in the direction V (resp., 0) if

where Tt is specified by the solutions of

6. Continuity of Shape Functions

327

where {V(t) : 0 < t < T} is a family of velocity fields which satisfies condition (V) given by (4.2) for some r > 0 (resp., Tt is specified by

where 9 e Lip (R N ,R N ) is a uniform Lipschitzian transformation of R N :

(ii) / is said to be shape continuous at 0 with respect to a class V of families of velocity fields if it is shape continuous at 0 in the direction V for all V in the class V satisfying conditions (V) given by (4.2). (iii) / is said to be shape continuous at 0 with respect to a family 0 of transformations in Lip (R N ,R N ) if it is shape continuous at 0 in the direction 9 for all 0 e 6.

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Chapter 8

jijiape Derivatives and calculus, and Tangential Differential Calculus

1

Introduction

Chapter 7 has studied the equivalence between the two points of view of transformations and velocities. Equivalent characterizations of shape continuity by the Courant metric and continuity along flows of velocities have been established. Since the velocity approach readily extends to the constrained case, and in particular to submanifolds of R N , this chapter adopts the velocity framework to study shape semiderivatives. In section 2 we give a self-contained review of semiderivatives and derivatives in vector spaces in order to prepare the ground for shape derivatives. Section 3 gives the definitions and the main properties of first-order semiderivatives and derivatives of shape functions. Section 3.2 specializes the definitions and results of section 3 to the generic shape spaces endowed with the Courant metric. Finally, a canonical definition of the shape gradient is given along with the main structure theorem in section 3.3. Before going to second-order derivatives, the main elements of the shape calculus are introduced in section 4 and the basic formulae for domain and boundary integrals are given in sections 4.1 and 4.2. Their application is illustrated in a series of examples in section 4.3. The final expressions of the gradients in the examples of section 4 always lead to a domain and a boundary expression. The boundary expression contains fundamental properties of the gradients and the natural way to untangle some of the resulting terms is to use the tangential calculus. Section 5 gives the main elements of that calculus for a C2-submanifold of R N of codimension 1 including Stokes's and Green's formulae in section 5.5 and the relationship between tangential and covariant derivatives in section 5.6. This is applied to the derivative of the integral of the square of the normal derivative of section 4.3.3. Section 6 extends definitions and structure theorems to second-order derivatives. In order to develop a better feeling for the abstract definitions the secondorder derivative of the domain integral is computed in section 6.1 using the combined strengths of the shape and tangential calculi. A basic formula for the second-order 329

330 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus semiderivative of the domain integral is given in section 6.2. This is completed with structure theorems in the rionautonomous case in section 6.3 and in the autonomous case in section 6.4. The shape Hessian is decomposed into a symmetrical term plus the gradient acting on the first half of the Lie bracket in section 6.5. This symmetrical part is itself decomposed into a symmetrical part that only depends on the normal component of the velocity field and a symmetrical term made up of the gradient acting on a generic group of terms that occurs in all examples considered in section 6.

2

Review of Differentiation in Banach Spaces

In this section we review some elements of derivatives and semiderivatives in vector spaces. In that context we shall start with the weaker notion of Gateaux semiderivative and emphasize the key role played by the Hadamard semiderivative in the semiderivative of the composition of two functions, since in most applications the extension of the chain rule is a central ingredient of a good differential calculus.

2.1

Definitions of Semiderivatives and Derivatives

Definition 2.1. Let / be a real-valued function defined in a neighborhood of a point re of a topological vector space E. (i) We say that / has a Gateaux semiderivative at a point x 6 U in the direction v G E if the following limit exists:

when it exists it will be denoted by df(x; v). (ii) We say that / has a Hadamard semiderivative at x e U in the direction v G E if the following limit exists:

when it exists it will be denoted by dfjf(x;v). The above definitions extend from a real-valued function to a function / into another topological vector space F. It is clear that df(x; v} exists and is equal to d n f ( x ; v) whenever the semiderivative dnf(x;v) exists, but the converse is not true without additional assumptions, as can be seen from the following example.

2. Review of Differentiation in Banach Spaces

331

Example 2.1. Consider the function / : R2 —» R:

It is readily seen that / has a Gateaux semiderivative at (0,0) in all directions v in R and that

which is trivially linear and continuous with respect to v. However, if for e > 0 we choose the directions

then

and djjf(Q, 0; 1,0) does not exist. We shall also need the following notions of full derivatives. Definition 2.2. Let / be a real-valued function denned in a neighborhood of a point x of a normed vector space E. (i) / has a Gateaux derivative at x if Vv 6 E, df(x;v) exists and v i—> df(x; v) : E —> R is linear and continuous.

(2-5)

Whenever it exists the linear map (2.5) will be denoted V/ : E —» E'. (ii) / has a Frechet derivative at x if it has a Gateaux derivative at x and

where (-,-}E denotes the duality pairing between E' and E. The above definitions extend from a real-valued function to a function / into another normed vector space F. The Gateaux semiderivative is generally neither linear nor continuous with respect to the direction, even in finite dimension.

332 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

Example 2.2.

Consider the following function / : R2 —> R:

It is Gateaux semidifferentiable at (0,0) in all directions v = (vi,v%) in R2, and

is neither linear nor continuous with respect to v. As for the Gateaux semiderivative the Hadamard semiderivative is generally neither linear nor continuous with respect to the direction.

Example 2.3.

The Hadamard semiderivative of the norm f(x} = \X\E at x = 0 is given by

It is continuous but not linear in v. When / has a Gateaux semiderivative in all directions v at a point x that is continuous with respect to v, it has a Hadamard semiderivative in all directions

but again it is not necessarily linear in v. 2.2

Locally Lipschitz Functions

The following theorem gives a sufficient condition for the equivalence of Gateaux and Hadamard semiderivatives (resp., Gateaux and Frechet derivatives). Theorem 2.1. Let E be a normed vector space. Given a function f : U —> R which is uniformly Lipschitzian in a neighborhood U of x in E, that is,

(i) If df(x;v) exists, then dnf(x',v} exists and df(x;v) = dfjf(x;v}. Moreover, if df(x; v) exists for all v £ E, then

(ii) If f has a Gateaux derivative at x, then it has a Frechet derivative at x and they are equal.

2. Review of Differentiation in Banach Spaces

333

Proof, (i) There exist s > 0 and a neighborhood W of v in E such that

Then

and

So as e —» 0 and w —> v, d#/(ar, t>) = d/(x; v). (ii) This part of the proof is obtained from Lemma 3.1 in Chapter 4 extended to a normed vector space. D

2.3

Chain Rule for Semiderivatives

The main difference between the two semiderivatives is that the composition of two functions that are semidifferentiable in the sense of Hadamard is semidifferentiable in the sense of Hadamard. Theorem 2.2. Let E and F be two topological vector spaces and let h be the composition of two mappings f and g:

in a neighborhood U of a point x in E, where

Assume that (i) g has a Gateaux (resp., Hadamard) semiderivative at x in the direction v, and (ii) d H f ( g ( x } \ d g ( x ] v } }

exists.

Then

Proof, (a) For 5 > 0 small enough let

334 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

By assumption, m(e) —> 0 as e —> 0. By the definition of dh(x; v) we want to find the limit of the differential quotient

which can be rewritten as

where So by the definition of du /,

(b) When g is Hadamard semidifferentiable we replace m(e) and d(e) by

and

and proceed as in part (a). In general we cannot improve the semiderivative of h by improving the semiderivative of g when / is not Hadamard semidifferentiable.

Example 2.4.

Consider the composition of the function / : R —> R in Example 2.1 and the map

The map / is Gateaux, but not Hadamard semidifferentiable, and the map g is infinitely differentiable. However, the composition

is not even Gateaux semidifferentiable at 0. We reiterate that the Hadamard semidifferentiability is a key property for the "chain rule." In general, the composition h — f o g of two maps will fail to have a semiderivative unless / has a Hadamard semiderivative—even if the map g : E —> F is Frechet differentiable at the point x.

2. Review of Differentiation in Banach Spaces

2.4

335

Semiderivatives of Convex Functions

Finally, the class of functions that are Hadamard semidifferentiable is not restrictive since it contains the classical continuously differeritiable functions and the convex continuous functions. Theorem 2.3. Let f : U C E —> R be a convex function defined in a convex neighborhood U of a point x of a topological vector space E. (i) There exists a neighborhood V of x such that

(ii) /// is continuous at x, then there exists a neighborhood W of x such that

Proof. Part (ii) is a consequence of part (i) and the fact that a continuous convex function at x is locally Lipschitzian in a neighborhood of x (cf. Ekeland and Temam [1, pp. 11-12]) by direct application of Theorem 2.1. We give the proof of part (i) for completeness. Let 0 e ]0,1]. Notice that for fixed x and v,

For this fixed a, we show that

This follows from the identity

and the convexity of /,

This can be rewritten

and yields (2.17). Define

336 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

Now we show that ^ is a monotone increasing function of 0 > 0. For all Oi and 6-2, 0 < #i < 6>2 < 00:

which implies that <^(#i) < ^(^2)- As (p is a monotone increasing function which is bounded below, its limit exists as 9 goes to 0. By definition it is equal to df(x; v).

2.5

Conditions for Predict Differentiability

It turns out that for finite-dimensional spaces E there is a complete analogy between the Gateaux derivative and the linearity and continuity of the Gateaux semiderivative on one hand and the Frechet derivative and the linearity and continuity of the Hadamard semiderivative on the other hand. Theorem 2.4. If a function f has a Frechet derivative atx, then f has a Hadamard semiderivative at x for all v E E and the map

is linear and continuous. The converse is true when E is finite dimensional. Proof. (=£>) Consider the quotient

as w —•> v and t \ 0. Hence h(t, w) = tw —•> 0 as w —» v and t \ 0. Then

where

and 0 if h — 0. By assumption, Q(h(t, w)) —>• 0 since h(t, w) = tw —> 0, v = 0, and (Vf(x},w)E —* (V/(x),f) J 5; as w —>• v by continuity. Therefore,

and dnf(x\v] exists and dfjf(x;v} = (Vf(x),v)E is linear and continuous with respect to v. (•£=) Define the element V/(x) of E' as

2. Review of Differentiation in Banach Spaces

337

Denote by Q the limsup of Q(h] as \h\ —» 0, which can possibly be infinite. Notice that for h ^ 0

Since E is finite dimensional, there exists a sequence {hn} and v E E such that

By letting tn = hn\,

Therefore, Q = 0 and / is Frechet differentiable at x. We complete this section with a classical sufficient condition for the Frechet differentiability. Theorem 2.5. Given a normed vector space E and a map f : E —>• R, assume that there exists a neighborhood V(x] of x E E such that (i) for all y E V(x], f has a Gateaux derivative V/(y), and (ii) the map V/ : V(x) —> E' is continuous at x. Then f has a Frechet derivative at x. Proof. There exists p > 0 such that the open ball B(x, p) is contained in V(x). For h £ E, Q < \h < p, consider the quotient

There exists a, 0 < a < 1, such that

This shows that / is Frechet differentiate at x.

338 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

2.6

Hadamard Semiderivative and Velocity Method

In section 2 of Chapter 7 we have drawn an analogy between Gateaux and Hadamard semiderivatives on one hand and shape semiderivatives obtained by a perturbation of the identity and the velocity method on the other hand. The next theorem relates the Hadamard semiderivative and the semiderivative obtained by the velocity method for real functions denned on R N . Theorem 2.6. Let f : NX -^ R be a real function defined in a neighborhood NX of a point X in R N . Then f is Hadamard semidifferentiable at (X, v) if and only if there exists r > 0 such that for all velocity fields V : [0, T] —> R satisfying the assumptions (a) (b)

(c) the limit

exists and for all V satisfying (a) and (b) and V(0) = v,

where Tt(V}(X]

= x(i) is the solution of the differential equation

Proof. (=£>) Let V be a vector field satisfying conditions (a), (b), and (c). Define

It is continuous on ]0, r] and

Therefore

2. Review of Differentiation in Banach Spaces

339

So

since / is Hadamard semidifferentiable at (X, v). The limit only depends on V(0) = v. (<£=) Conversely, saying that the dfjf(x;v) exists is equivalent to saying that for the sequence tn = 2~ n and any sequence wn —> v the limit

exists and only depends on v. We now show that we can associate with such a sequence {wn} a velocity field V satisfying (a) to (c) such that V(tn,X} = wn and Ttn(V)(X) = X + tnwn. Then from properties (2.18) and (2.19) we conclude that d n f ( x ] v ] exists. Set to = 1, tn = 2~ n , xn = X + tnwn and observe that tn - tn+i = -2~( n+1 ). Define for m > 1 the following CTO-interpolation in (0,1]: for t in [t n + i,t n ]

where j>, QQ, q\ G P 2m+1 [0,1] are polynomials of order 2m + 1 on [0,1] such that p(0) = 0 and p(l] = 1 and p^(0) = 0 = pW(l), 1 < I < m, q0(Q) = 0 = q Q ( l ) , q'0(Q) = 1, ^(1) = 0 and ^>(0) = 0 = 9^(1), 2 < t < m, 9l(0) = 0 = 9l (l), ^(0) = 0, q { ( l ) = I and q[l}(0) = 0 = q [ e ) ( l ) , 2 < i < m. By definition T ( - , X ) G C m ((0,1]; R N ). Moreover, for 1 < ^ < m,

We now show that T satisfies conditions (a) and (b) which are equivalent to condition (4.2) in Chapter 7 on V. First observe that Tt(X] - X and dT/dt are both independent of X and (Tt - I)(X) 6 C m ((0,1]; R N ). Define

340 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

Hence T and dT/dt clearly satisfy condition (Tl) in (4.8) in Chapter 7. It satisfies condition (T2) since Tt~l(Y) = Y-f(t), and a fortiori T~l satisfies condition (T3). Therefore, the velocity field

satisfies condition (V) given by (4.2) of Chapter 7. Thus properties (a) and (b) are satisfied. It remains to show that df/dt(t) —> v as t —*• 0. It is easy to check that by construction1 —p'(r) +
The left-hand side now clearly converges since the three polynomials are bounded by a constant independent of n. This is sufficient to prove the theorem.

Remark 2.1.

The equivalence Theorem 2.6 shows how to extend the Hadamard semiderivative from linear spaces to submanifolds of R N .

3

First-Order Semiderivatives and Shape Gradient

Recall for the unconstrained case conditions (V) given by (4.2) in Chapter 7:

Similarly, in the constrained case for a closed subset D of R1N recall the conditions (Vl£>) and (V2£>) given by (5.5) in Chapter 7 in their following equivalent form (VD):

^y interpolation of the function g(r) = r — 1, g(r) = g(0)p(r) + g'(0)qo(T) + g'(l)qi(r) = —P(T) +
3. First-Order Semiderivatives and Shape Gradient

341

where LD(X] = CD(X) fl { — C D ( X } } is the linear tangent space to D at the point x E 3D. Introduce the following linear subspace of Lip (Z), R N ):

Under the action of a velocity field V satisfying conditions (3.2), a domain ft in D is transformed into a new domain,

also contained in D.

3.1

Definitions of Semiderivatives and Derivatives

We give the definitions in the general constrained case. Recall from Definition 2.1 of section 2 in Chapter 7 that a function J(fl) defined on a family of subsets ft of RN (resp., D} is called a shape function if for any transformation T of R N (resp., D such that T(D) = D} T(Q) = fl implies that J(T(fi)) = J(ft).

Definition 3.1.

Let 0 be a topological vector subspace of Lip L (D,R N ) and J a real-valued shape function. (i) Given a velocity field V satisfying conditions (3.2), J is said to have a Eulerian semiderivative at ft in the direction V if the following limit exists and is finite:

(ii) For 9 E Lip L (D,R N ) and the autonomous velocity field

we shall use the notation dJ(ft;9]

or simply dJ(fi;0).

(iii) Given 9 E 9, J is said to have a Hadamard semiderivative at fl in the direction 9 with respect to 0 if for all V satisfying conditions (3.1), V(t) E 0, and V(Q) = 9, dJ($l\ V) exists and only depends on V(0) — 9. In that case the semiderivative will be denoted du J(0; 9} and necessarily

(iv) J is said to be differentiate at 0 in 0' if it has a Eulerian semiderivative at fl in all directions 9 E 0 and the map

is linear and continuous. The map (3.8) is denoted G(fT) and referred to as the gradient of J in the topological dual 0' of 0.

342 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus The definition of a Eulerian semiderivative is quite general. For instance, it readily applies to shape functions defined on closed submanifolds D of R N . It includes cases where dJ(£l; V) is dependent not only on F(0) but also on V(t) in a neighborhood of t — 0. We shall see that this will not occur under some continuity assumption on the map V H-> dJ(fi; V). When dJ(£l; V] only depends on V(0), the analysis can be specialized to autonomous vector fields V, and the semiderivative can be related to the gradient of J. If J has a Hadamard semiderivative at f2 in the direction 0, it has a Eulerian semiderivative at Jl in the direction 6 and

Example 3.1. For any measurable subset $1 of R N , consider the volume function

For O with finite volume and V in C([0, r]; CQ (R N , R N )), consider the transformations T t (fJ) of ft:

for t small since det(DTo) = det J = 1. This yields the Eulerian semiderivative

By definition it only depends on V(0) and hence J has a Hadamard semiderivative I

and even a gradient <2(fZ) for 6 = Cg(R N , R N ). The following simple continuity condition can also be used to obtain the Hadamard semidifferentiability. Theorem 3.1. Let Q be a Banach subspace o/Lip I / (D,R N ) ; J a real-valued shape function, and f2 a subset of R N . (i) Given 9 e 6, i/ and «/ the map

is continuous for the subspace of V 's such that V(0) = 0, then J is Hadamard semidifferentiable at ft in the direction 9 with respect to 6 and

3. First-Order Semiderivatives and Shape Gradient

343

(ii) If for all V in C([0,r];6), dJ($l\V] exists and the map

is continuous, then J is Hadamard semidifferentiable V(0) with respect to 0 and

at 0 in the direction

Proof. Given V in C([0,r];0) such that V(0) — 0, construct the sequence

Note that for each n > 1, dJ(Q; Vn) = dJ(Q; V] since for t < r/n, Tt(Vn] =Tt(V). By continuity of V, {V^} converges in C([0,r]; 6) to the autonomous field V(t) = V(Q):

Hence

Hence by continuity of the map (3.15)

But since dJ(O; V") is only dependent on V^(0), for all W E O such that W(0] = V(0) we have the identity dJ(fJ; W) = dJ(fi; V'). Therefore, J has a Hadamard semiderivative at 0 in the direction V(0) and

This concludes the proof of the theorem. For simplicity, the last theorem was only given for a Banach space 0. This includes the spaces Cj(R N ), C f c (RN), C0'1^), and £ f c (R N ,R N ) considered at the end of Chapter 2. However, its conclusions are not limited to Banach spaces. Other constructions can be used as illustrated below in the unconstrained case. Consider velocity fields in

344 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus where for m > 0,

and lim denotes the inductive limit set with respect to K endowed with its natural inductive limit topology. For autonomous fields, this construction reduces to

For k > 1, Vk = £> fc (R N ,R N ). In all cases conditions (3.1) are satisfied. Theorem 3.2. Let J be a real-valued shape function, SI a subset o/R N , and k > 0 an integer. (i) Given 9 e Vk, assume that

and that the map

is continuous for all V's such that V(0) = 9. Then J is Hadamard semidifferentiable in O in the direction 0 with respect to Vk and

(ii) Assume that for all V G V°' fe , dJ(£l; V) exists and that the map

is continuous. Then J is Hadamard semidifferentiable V(Q) with respect to Vk and

in SI in the direction

Proof. It is sufficient to prove the theorem for any compact subset K of RN and hence only for velocities in V^ = C([0, r]; V^), where V^ is a Banach space contained in (7o(R N ,R N ). So the theorem follows from Theorem 3.1.

3. First-Order Semiderivatives and Shape Gradient

3.2

345

Perturbations of the Identity and Predict Derivative

In the unconstrained case (D = R N ), we have introduced in section 2 of this chapter and section 4 of Chapter 7 a notion of directional semiderivative associated with first- and second-order perturbations of the identity. Even if this approach does not naturally extend to the constrained case, it is interesting to compare the definitions and results with those associated with the velocity method of Definition 3.1. Go back to the generic metric spaces of Micheletti in section 9 of Chapter 2 and their Courant metric. They all use a Banach subspace 6 of transformations from D to R N , 6 C Lip L (D,R N ), and the space

Associate with F £ .F(O)

where the infima are taken over all finite factorizations in ^(O) of the form

for /j, gi in 0. Extend this definition to all pairs F and G in ^"(0):

Define for some open or closed subset £1 of D the subgroup

and the Courant metric on JF(0)/£?(17),

Assume that ^"(0) is complete. Here we consider the unconstrained case (D = R N ) and assume that there exists a ball Be of radius e > 0 in 0 and a constant c > 0 such that

This is true in all cases considered in Chapter 2. Hence the maps

are well defined and continuous in / = 0 since

346 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus For a shape function J, the map

is well denned since J is invariant on Q(£t] and the map

is continuous in / = 0 if J is continuous in 0 for the Courant metric on .F(0)/(y(J7). The following definitions are now the standard definitions of section 2 applied to the function Jn(/) defined on the ball B£ in the topological vector space 0. They parallel the ones of Definition 3.1.

Definition 3.2.

Let J be a real-valued shape function and 0 a topological vector subspace of Lip (RN, R N ). For / e 0 such that [/ + / ] € .F(0), denote [/ + /](fi) by fy. (i) Jfi is said to have a Gateaux semiderivative at / in the direction 0 € G if the following limit exists and is finite:

(ii) Jn is said to be Gateaux differentiable at / if it has a Gateaux semiderivative at / in all directions (9 E 0 and the map

is linear and continuous. The map (3.31) is denoted VJ^(/) and referred to as the gradient of J^ in the topological dual 0' of 0. (iii) If, in addition, 0 is a normed vector space, we say that J is Frechet tiable at / if J is Gateaux differentiable at / and

differen-

The semiderivatives of J and JQ are related. Theorem 3.3. Let J be a real-valued shape function. (i) Assume that J^ has a Gateaux semiderivative at f in the direction 0 E 0; then J has a Eulerian semiderivative at O/ in the direction V/ and

3. First-Order Semiderivatives and Shape Gradient

347

(ii) // J has a Hadamard semiderivative at Q/ in the direction 9 o [I + f}~~1, then Jfi has a Gateaux semiderivative at f in the direction 9 and

Conversely, if JQ has a Gateaux semiderivative at f in the direction 0o [/ + /], then J has a Hadamard semiderivative at Q/ in the direction 0. If either dJsi(f\Q} or d#J(Q/;$) is linear and continuous with respect to all 9 in Q, so is the other and

Proof. By definition,

for the family of transformations

and from Theorem 4.1 in section 4.1 of Chapter 7, T/ corresponds to the velocity field

Identity (3.34) now follows from the fact that J has a Hadamard semiderivative at O/. The other properties readily follow from the definitions. We have standard sufficient conditions for the Frechet differentiability from Theorem 2.5. Theorem 3.4. Let J be a real-valued shape function. Let 0 be a subset o/R N and 6 be C 0 fc+1 (R N ) 7 C fc+1 (R^), (^(RN), /5 fc+1 (R N ,R N ), k > 0. //J n is Gateaux differentiate for all f in B£ and the map

is continuous in f = 0, then J$i is Frechet differentiate

in f = 0.

348 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

3.3

Shape Gradient and Structure Theorem

In view of the previous discussion we now specialize to autonomous vector fields V to further study the properties and the structure of dJ(£l;V}. For simplicity we also specialize to the unconstrained case in R N . The constrained case yields similar result but is technically more involved (cf. Delfour and Zolesio [14] for D open in R N ). The choice of a shape gradient depends on the choice of the topological vector subspace 6 of Lip (R N , R N ). We choose to work in the classical framework of the Theory of Distributions (cf. Schwartz [3]) with 8 = £>(R N ,R N ), the space of all infinitely differentiable transformations 0 of RN with compact support. For these velocity fields V, conditions (3.1) are satisfied.

Definition 3.3.

Let J be a real-valued shape function. Let £1 be a subset of R N . (i) The function J is said to be shape differentiable Oforall0in£>(RN,RN).

at 0 if it is differentiable at

(ii) The map (3.8) defines a vector distribution (7(0) in £>(R N ,R N )', which will be referred to as the shape gradient of J at fi. (iii) When, for some finite k > 0, (7(Q) is continuous for the £> fc (R N , RN)-topology, we say that the shape gradient G(0) is of order k. The next theorem gives additional properties of shape differentiable functions.

Notation 3.1.

Associate with a subset A of R N and an integer k > 0 the set

where LA(x) = (-CU(z)} flCUOc) and CA(X) is given by (5.27) in Chapter 7. Theorem 3.5 (structure theorem). Let J be a real-valued shape function. Assume that J has a shape gradient G(£l] for some subset J7 of RN with boundary r. (i) The support of the shape gradient G($l) is contained in F. (ii) If 0 is open or closed in RN and the shape gradient is of order k for some k > 0, then there exists [G(fi)] in (Dk/L^}r such that for all V in Vk =f D f e (R N ,R N ),

where q^ : T>k —> T>k /L^ is the canonical quotient surjection. Moreover,

where (qiY denotes the transpose of the linear map q^.

3. First-Order Semiderivatives and Shape Gradient

349

Proof, (i) Any V in T> such that V — 0 on F satisfies assumptions (Vl^) and (Via) in (5.5) (with_0 in place of D) and V G L^. Then by Theorem 5.1 (i) of Chapter 7, Ts : Cl —> fj is a homeomorphism and (fj) s = Ts(Cl) = & and (int f£) s = Ts(int 0) = int f2. Thus when 0 is closed (0 = fj) or open (f2 = intfi),

(ii) It is sufficient to prove that dJ(Q;V) = 0 for all V in L^. The other statements follow by standard arguments and the fact that L^ is a closed linear subspace of T>k. From part (i) we know that the result is true for all V in L^3 and hence by a density argument for all V in LQ.

Remark 3.1. When the boundary F of £1 is compact and J is shape differentiate at £7, the distribution G(0) is of finite order. Once this is known, the conclusions of Theorem 3.5 (ii) apply with k equal to the order of G(fl). Hence (7(Q) will belong to a Hilbert space #~ S (R N ) for some s > 0. The quotient space is very much related to a trace on the boundary F, and when the boundary F is sufficiently smooth we can indeed make that identification. Corollary 1. Assume that the assumptions of Theorem 3.5 are satisfied for an open domain 17, that the order of G(f2) is k > 0, and that the boundary F of $1 is Ck+1. Then for all x in F, LQ(X) is an (N — 1)-dimensional hyperplane to 0 at x and there exists a unique outward unit normal n(x) which belongs to C f c (F;R N ). As a result, the kernel of the map

coincides with L^, where 7r : £> f c (R N ,R N ) -> C f c (F,R N ) is the trace of V on F. Moreover, the map p^(V}

is a well-defined isomorphism. In particular, there exists a scalar distribution g(T) in R N with support in F such that g(T) e C rfc (F) / and for all V in £> f c (R N ,R N )

and

Wheng(Y] 6 Ll(T)

where 7r is the trace operator on F.

350 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

Proof. The surjectivity of (3.39) is a consequence of the fact that F is compact and that for a Ck+1 boundary, k > 0, it is always possible to construct an extension N of the unit normal n on F which belongs P fc (R N , R N ) with support in a neighborhood of F. Then for any v in C fc (F), there exists also an extension v in £> fc (R N ) with support in a neighborhood of F, and the vector V = vN belongs to £> fc (R N ,R N ) and coincides with vn on F.

Remark 3.2. In 1907, Hadamard [1] used displacements along the normal to the boundary F of a C^-domain (as in section 3.1 of Chapter 7) to compute the derivative of the first eigenvalue of the clamped plate. Theorem 3.5 and its corollary are generalizations to arbitrary shape functions of that property to open or closed domains with an arbitrary boundary. The structure theorem for shape functions on open domains with a (7fc+1-boundary is due to Zolesio [12] in 1979 and not to Hadamard even if the formula (3.42) is often called the Hadamard formula.

Example 3.2.

For any measurable subset SI of R N , consider the volume shape function (3.9) of Example 3.1:

For O with finite volume and V in P 1 (R N , R N ), we have seen in Example 3.1 that

and this is formula (3.40) with k — I . For an open domain Q, with a Cl compact boundary F,

which is also continuous with respect to V in D°(R N , R N ). Here the smoothness of the boundary decreases the order of the distribution G($l}. This raises the question of the characterization of the family of all subsets fi of R N for which the map

can be continuously extended to P°(RN, R N ). But this is the family of locally finite perimeter sets: sets Jl whose characteristic function belongs to -BVbc(RN).

4

Elements of Shape Calculus

In this section we recall a number of basic formulae from Zolesio [7, 8] for the derivative of domain and boundary integrals. The reader is also referred to the companion book of Sokolowski and Zolesio [9] for the computation of shape derivatives associated with a wide range of partial differential equations.

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351

A more modern treatment of the derivative of boundary integrals is given that emphasizes a recent and apparently new approach to the tangential calculus on a C2 submanifold of RN of codimension one. This development took place in the context of the theory of shells, where a simple and self-contained intrinsic differential calculus has been developed by using the oriented distance function of Chapter 5. It completely avoids the use of local maps and bases and Christoffel's symbols. In this section a new, considerably simplified proof of the formula for the derivative of boundary integrals is presented by using the oriented distance function.

4.1

Basic Formula for Domain Integrals

The simplest examples of domain functions are given by volume integrals over a bounded open domain f2 in R N . They use a basic formula in connection with the family of transformations {Tt : 0 < t < T}. Assume that condition (3.1) is satisfied by the velocity field {V(t} : 0 < t < T}. Further assume that V G C°([0,r];C' 1 1 oc (R N ,R N )) and that r > 0 is such that the Jacobian Jt is strictly positive:

where DTt(X) is the Jacobian matrix of the transformation Tt — Tt(V) associated with the velocity vector field V. Given a function (p in W lo ' c (R N ), consider for 0 < t < r the volume integral

where 0^(V) = Tt(V)(£i). By the change of variable formula,

and the following formulae and results are easy to check. Theorem 4.1. Let (p be a function in W lo ' c (R N ). Assume that the vector field V = {V(t} : 0 < t < r} satisfies condition (V}. (i) For each t G [0, r] the map

and its inverse are both locally Lipschitzian and

(ii) I f V G COQO^C^R^RN)), then the map

352 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

is well defined and for each t

Hence the function

(iii) IfVe C°([0,T};Ctoc(RN,RN)),

is differentiate

then the map

and

Hence the map t i—>• Jt belongs to C1([0,r]; C^ C (R N )). Indeed it is easy to check that

and (4.4) follows directly by definition of J t ( X ) . From (4.2), (4.3), and (4.4)

If fi has a Lipschitzian boundary, then by Stokes's theorem

Theorem 4.2. Assume that there exists r > 0 such that the velocity field V(t) satisfies conditions (V) and V e C°([0, r];C' 1 Q C (R N ,R N )). Given a function tp e C(0, r; Wj0'c (R N )) PI C rl (0,r;L 1 1 oc (R N )) and a bounded measurable domain 17 with boundary F, the semiderivative of the function

4. Elements of Shape Calculus

353

at t = 0 is given by

where ip(Q)(x) =
4.2

Basic Formula for Boundary Integrals

Given ift in //j^ c (R N ), consider for some bounded open Lipschitzian domain fi in RN the shape function

This integral is invariant with respect to homeomorphisms which map 17 onto itself (and hence F onto itself). Given the velocity field V and t > 0, consider the expression

Using the change of variable Tt(V) and the material introduced in (3.13) of section 3.2 and (5.12) of section 5 in Chapter 2, this integral can be brought back from r t to T:

where the density ujt is given as

n is the outward normal field on F, and M(DTt) is the cofactor matrix of DTt, that is,

It can easily be checked from (4.10) and (4.11) that t H-> ujt is differentiate in C°(F) and that the limit

354 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

in the C°(F)-norm is linear and continuous with respect to V(0) in the CjQC(RN, R N ) Prechet topology. Hence

From Corollary 1 to the structure theorem (Theorem 3.5), dJ(Q; V) depends only on the normal component vn of the velocity field V(0) on F:

through Hadamard's formula (3.40). In view of this property any other velocity field with the same smoothness and normal component on F will yield the same limit. Given k > 0 consider the tubular neighborhood

of F in RN for the oriented distance function b = b& associated with £). Assuming that F is compact and of class C2, there exists h > 0 such that b e C2 (62/1 CO). Let (f> £ £>(RN) be such that (p = 1 in S/i(F) and
Clearly, the normal component of W(0) on F coincides with vn. Moreover, in ^(F)

since V6|r = n, D2bVb — 0, and H = A6 is the additive curvature, that is, the sum of the N — 1 curvatures of F or AT — 1 times the mean curvature H. Finally,

We have proved the following result.

4. Elements of Shape Calculus

355

Theorem 4.3. Let F be the boundary of a bounded open subset 0 o/R N of class C2 andifj an element oj'(^(M; tf£c(RN)). Assume that V e C°([0,r]; C^R*, R N )). Consider the function

Then the derivative of Jy(t) tyzt/i respect to t in t = 0 is given by the expression

where iJ>f(Q)(x) =f d^/dt(Q,x). Note that, as in the case of the volume integral, we have two formulae. Hence the following identity:

4.3

Examples of Shape Derivative

4.3.1

Volume of ft and Area of T

Consider the volume shape function

This shape function is used as a constraint on the domain in several examples of shape optimization problems. We get

and, if F is Lipschitzian,

A sufficient condition on the field F(0) to preserve the volume is div V(Q) = 0 in f2 and, if F is Lipschitzian, V(0) • n = 0 on F.

356 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus Consider the (shape) area function

Assuming that F is of class C2 we get from (4.17)

where H = A6 is the additive curvature. The condition for the surface of F to be preserved is that V(0) • n be orthogonal (in I/ 2 (F)) to H. 4.3.2

H1(fi)-Norm

Given 0 and ip in H^OC(R^)^ consider the shape function

By using the change of variable Tt(V), ftt = Slt(V) = Tt(V)(Sl) and

where A(V] is the following matrix associated with the field V:

and J(t) = det(DTt). Expression (4.23) is easily obtained from the identity

If V G C°([0,T];C£ C (R N ;R N )), k > 1, TandT" 1 belong to ^([0,^; Cf oc (R N ,R N )) and A(V) to CHtO^C^R^R*2)). Then if 0,^ belongs to #i2oc(RN) we get t^Kf>oTt which is differentiable in /^(R1*), with d^/dt o r t | t=0 = V^ • F(0), which belongs to ^f[QC(RN). Finally, we obtain

where A'(V) is the derivative in the C^^R^^^^^-norm

4. Elements of Shape Calculus

357

and £:(V(0)) is the symmetrized Jacobian matrix (the strain tensor associated to the field V(0) in elasticity)

We finally obtain the volume expression

When F is a Cfl-submanifold, 0, ifi G J£/1^)C(RN), we obtain the simpler boundary expression by directly using formula (4.17):

This simple example nicely illustrates the notion of density gradient g. Prom expression (4.28) it was obvious that the mapping V i—» dJ($l\ V) was well defined, linear, and continuous on C 1 ([0,r]; Cj^ c (R N ;R N )). By the structure theorem and Hadamard's formula, we knew that dJ could be written in the form

But from (4.29) we know that g, which is an element of P 1 (F) / , is an element of Wl/2'l(T] given by

The direct calculation of g from expression (4.24) would have been very fastidious. 4.3.3

Normal Derivative

Let F be of class C2 and 0 e H^OC(R^) be given. Consider the following shape function:

By the change of variable formula we get, with ftt = Tt(V)(£t) and Tt = Tt(V)(T),

where nt o Tt is the transported normal field nt from Tt onto F. The derivative can be obtained by using formula (4.17) of Theorem 4.3 and one of the above two expressions. However, the first expression first requires the construction of an extension Nt of the normal nt in a neighborhood of F. In both cases the following result will be useful.

358 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

Theorem 4.4. Let k > I be. an integer. Given a velocity field V(t] satisfying condition (V] such that V E C([0,r]; C£ C (R N ,R N )), then

where n and rit are the respective outward normals to 0 and f^ on T and Tt and M(DTt) is the cofactor's matrix of DTt. Proof. Go back to Definition 3.1 in section 3.1 of Chapter 2. We have shown in that section that for a domain J7 of class Ck the unit outward normal at any point y G r x = F H U(x) is given by expression (3.9)

where hx is the local diffeomorphism specified by (3.3):

For i1t — r^(V) and xt = Tt(x), choose the following new neighborhood and local diffeomorphism

On Tt(V) n Ut the normal in given by the same expression but with ht in place of hx:

However,

Therefore, by using the previous expressions for n and nt,

The other expression for nt o Tt in terms of the cofactor matrix M(DTt) of DTt readily follows from the identity M(DTt] = J(t) *(DTt)~l. Recalling the expression for ut,

4. Elements of Shape Calculus our boundary integral becomes

Using expression (4.26) of A'(V) and expression (4.12) of u/ we get

This formula can be somewhat simplified by using identity (4.18) with

We obtain

359

360 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus and hence

This formula can be more readily obtained from the first expression (4.31) and the extension

of the normal nt. To compute N', decompose Nt as follows:

So finally

and

The first part of the integral (4.36) explicitly depends on the normal component of V(0). Yet we know from the structure theorem that for this function, the shape derivative only depends on the normal component of V(Q). To make this explicit, it is necessary to introduce some elements of tangential calculus.

5

Elements of Tangential Calculus

In this section some basic elements of the differential calculus on a Cr2-submanifold of codimension 1 are introduced. The approach avoids local bases and coordinates by using the intrinsic tangential derivatives. A more systematic treatment combined

5. Elements of Tangential Calculus

361

with the use of the distance function has been successfully used in the theory of thin and asymptotic shells with a C1'1-midsurface. Let fi be an open domain of class C2 in RN with compact boundary T. Therefore, there exists h > 0 such that b = b^ € C2(S'2h(^}}- The projection of a point x onto F is given by and the orthogonal projection operator of a vector onto the tangent plane T p ( x )F is given by Notice that, as a transformation of Tp(x^T,

is the identity transformation on Tp^T. In fact P(x] coincides with the first fundamental form. Similarly D2b can be considered as a transformation of Tp(x) since D2b(x)n(x} — D 2 6(x)V6(x) = 0. We shall show in section 5.6 that

coincides with the second fundamental form of F. Similarly D2b(x}'2 coincides with the third fundamental form. Finally,

5.1

Intrinsic Definition of the Tangential Gradient

The classical way to define the tangential gradient of a scalar function / : F —> R is through an appropriately smooth extension F of / in a neighborhood of F using the fact that the resulting expression on F is independent of the choice of the extension F. In this section an equivalent direct intrinsic definition is given in terms of the extension / op of the function /. This is the basis of a simple differential calculus on F which uses the Euclidean differential calculus in the ambient neighborhood of F.

This is the orthogonal projection P(x)VF(x) of VF(x) onto the tangent plane TXT to F at x. To get something intrinsic, g(F] must be independent of the choice of F. It is sufficient to show that g(F) = 0 for / = 0. But F = f = 0 on F and the tangential component of VF is 0 on F, and

362 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

Definition 5.1 (general extension). Assume that F is compact and that there exists h > 0 such that fo € C 2 (52^(r)). Given an extension F 6 C1(52/l(F)) of / G C fl (r), the tangential gradient of / in a point of F is denned as

The notation is quite natural. The subscript F of Vp/ indicates that the gradient is with respect to the variable x in the submanifold F. Theorem 5.1. Assume that F is compact and that there exists h > 0 such that bn e C2(52/l(F)) and that f 6 C^F). Then

Proof, (i) By definition

Moreover,

(ii) F — f op is a C1-extension of / and

But by definition of Vr/

Recall, from Theorem 4.3(i) in Chapter 5, that n o p = V6 so that

Again

5. Elements ofTangential Calculus

363

from part (i) and In view of part (ii) of the theorem, f o p plays the role of a canonical extension of a map / : F —> R to a neighborhood $2/1 (F) of F, and its gradient is tangent to the level sets of 6. This suggests the use of the following definition of tangential gradient which will be the clue to the tangential differential calculus.

Definition 5.2 (canonical extension).

Under the assumptions of Definition 5.1 on 6^, associate with / e C*1(F)

Remark 5.1.

This definition naturally extends to nonempty sets A such that d2A belongs to C*1'1 (5*2/1 (^4)) 5 since the projection onto .A,

is C 0>1 . They are the sets of positive reach introduced by Federer. They include convex sets and submanifolds of codimension larger than or equal to 1. Theorem 5.2. Under the assumption of Definition 5.1 on b^ for f € C1(F), (i) V6 • V(/ o p) = 0 in Sh(T) and n • V r / = 0 on T.

(ii) VF|r - |f n = (PVF}\r = V r / and V(/ op) = [/ - bD2b] V r / op m F. Proof, (i) Consider

But and

(ii) By definition However, since Fop = fopouT and

By restricting to F

364 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

5.2

First-Order Derivatives

The tangential Jacobian matrix of a vector function v E C 1 (r) M , M > 1, is defined in the same way as the gradient

If % = (vi,... ,VM), then where Vr^ is a column vector. From the previous theorems about the tangential gradient we can recover the definition from an extension V 6 C1(5<2/l(r))M of v:

and

Also, and we have for the extension

Note that For a vector function V £ C1(F)jV define the tangential divergence as

and it is easy to show that

The tangential linear strain tensor of linear elasticity is given by

5. Elements of Tangential Calculus

365

The tangential Jacobian matrix of the normal n is especially interesting since, from Theorem 4.3 (i) in Chapter 5, n o p = V6 = Vb o p. As a result,

The tangential vectorial divergence of a matrix or tensor function A is defined as

5.3

Second-Order Derivatives

Assume that Q is of class C3. The simplest second-order derivative is the LaplaceBeltrami operator of a function / e C 2 (F), which is denned as

Recall from Theorem 5.2 the identity

Then

and by taking restrictions to F,

The tangential Hessian matrix of second-order derivatives is defined as

Here the curvatures of the submanifold begin to appear. The Hessian matrix is not symmetrical and does not coincide with the restriction of the Hessian matrix of the canonical extension. Specifically,

366 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus Of course, since D2(f op] is symmetrical we also have

As a final result we have the following identity:

since by definition *£)£/ = *(Dpf). So the Hessian and its transpose differ by terms that contain first-order derivatives, as is well known in differential geometry. Note that D2bVpf = — *Dr(Vr/)ft = —*D^fn and that we also can write

5.4

A Few Useful Formulae and the Chain Rule

Associate with F G C^(S^h(T)) and V G C1(S2h(T))N

where vp and vn are the respective tangential part and the normal component of v. In view of the previous definitions the following identities are easy to check:

Decomposing v into its tangential part and its normal component,

Given / G C fl (F) and g G C'1(F;r), consider the canonical extensions / op G C (S2h(T)} and g op G C1(5(2/i(r); R N ) and the gradient of the composition l

and for a vector-valued function v G C1(F; R N ),

5. Elements of Tangential Calculus

5.5

367

The Stokes and Green Formulae

One interesting application of the shape calculus in connection with the tangential calculus is the tangential Stokes formula. Given v G Cl(T)N, consider Stokes's formula in RN for the vector function v o p:

Given an autonomous velocity field V, differentiate both sides of Stokes's formula with respect to t:

where Nt is the extension (4.37) of nt- This gives the new identity

Now choose the velocity field V — V&V, where ip G T>(S<2h(T)) is chosen in such a way that ip = I on 5/l(F). Using expression (4.38) for TV',

Finally we get the tangential Stokes formula with H = A6:

For a function / 6 C l ( T ] and a vector v G Cl(T)N, the above formula also yields the tangential Green's formula,

5.6

Relation between Tangential and Covariant Derivatives

One of the simplest examples of a tangential derivative is when fi is the half-space

368 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

for some orthonormal basis {ei,..., ejv} in RN and F is the boundary of H+ denoted

If PH denotes the projection onto 77, thenp//(£) = PH((,'•, CAT) = C' — (Ci> • • • > Ov-i) £ #, and for any function (p G C1 (ff) the tangential gradient

coincides with the gradient of (p in H:

With the notation of section 3.1 of Chapter 2, consider a set 0 locally of class C2 in R N . Its boundary F = <9Q is an (N — l)-dimensional submanifold of RN of class C2. At each point x G F, there is a C2-diffeomorphism hx from the open unit ball B onto a neighborhood U(x) of a: such that

where B+ = H+ n 5 and -B0 = Hr\B and #+ and /f are as defined above for some appropriate orthonormal basis. For a function / e Cl(T), f o $ € C1^) with $ =f /ix|So : 5o -» F. Observing that f ° 3> °PH — f °P° & °PH, we have

The covariant basis associated with $ is defined as

where from identity (3.9) of section 3.1 in Chapter 2 we know that a^ o <1>~1 is the inward unit normal to 0, that is,

The covariant partial derivatives of / are denned as

369

5. Elements of Tangential Calculus But from the previous observations

For a vector function v e Cl(T; R N )

Again,

and

The second fundamental form of F is defined from the inward normal a AT,

where from (5.28) a^v = —n o <£, and from (5.10) D^n — D 2 6|r,

Recall that since |V6| = 1, D2bVb — 0 and V6 is an eigenvector for the eigenvalue 0. As a result the eigenvalues of D2b at a point of F are the eigenvalues of the second fundamental form bap (the N — I principal curvatures) plus {0}. In particular,

where H is the mean curvature of F and H is the additive curvature (cf. section 3.3 of Chapter 2).

370 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

5.7

Back to the Example of Section 4.3.3

Coming back to formula (4.36) for c?J(f2; V^(O)) of the boundary integral of the square of the normal derivative in section 4.3.3, the tangential calculus is now used on the term

From identity (5.18) with v = V(0)|r and identity (5.23),

Therefore, by using the tangential Stokes formula (5.26),

Substituting into expression (4.36), we finally get the explicit formula in term of V(Q) • n as predicted by the structure theorem

6

Second-Order Semiderivative and Shape Hessian

The object of this section is to study second-order derivatives and semiderivatives by the velocity method for smooth and nonsmooth domains and discuss its relationship with the method of perturbations of the identity in the unconstrained case. The analysis of this section is motivated by first computing the second-order derivative of a domain integral by using the combined strengths of the shape and tangential calculi in section 6.1. A basic formula for the second-order semiderivative of the domain integral is given in section 6.2 which also reveals the general structure

6. Second-Order Semiderivative and Shape Hessian

371

of this derivative. Structure theorems for the second-order Eulerian semiderivative d2J(Q;V;W) of a function J(0) for two vector fields V and W are given in sections 6.3 and 6.4. A first theorem shows that under some natural continuity assumptions,

where V'(0) is the time-partial derivative dtV(t,x) at t = 0. As in the study of first-order Eulerian semiderivatives, this first theorem reduces the study of secondorder Eulerian semiderivatives to the autonomous case. So we then specialize to fields in D f c (D,R N ) and give the equivalent of Hadamard's structure theorem for the term d2 j(fi; V(0); W(Q)). This bilinear term is decomposed into a symmetrical term plus the gradient acting on the first half of the Lie bracket [V(0), W(0)] in section 6.5. The symmetrical part is itself decomposed into a symmetrical part that depends only on the normal component of the velocity fields and a symmetrical term made up of the gradient acting on a generic group of terms, which occurs in all examples considered in this section.

6.1

Second-Order Derivative of the Domain Integral

Given / G Cf oc (R N ) and a domain f2 of class C2, consider the function

where

for some pair of autonomous velocity fields V and W satisfying condition (3.1) and additional smoothness conditions as necessary. The objective is to compute

From formula (4.7) in Theorem 4.2 of the previous section, we already know that

where TV,, = NS(W) is the extension (4.37) of the normal ns. So we can readily use formula (4.17) from Theorem 4.3:

372 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

where N'(W) is the derivative of the extension Ns = NS(W) given by expression (4.38). This yields

It remains to untangle the following term in the first part of the integral:

Using the notation v = V\r and w = W\r,

Finally,

and by using the tangential Stokes formula (5.26)

6. Second-Order Semiderivative and Shape Hessian

373

Finally, we get two equivalent expressions

Since the above expressions involved the composition TS(W) o Tt(V), it is expected that the condition for the symmetry of expression (6.3) will involve the Lie bracket [V, W] = DVW - DWV. Indeed, by using identity (5.23)

and substituting in the first expression, we get a symmetrical term plus the first half of the Lie bracket

Thus

from which either / [V, W] • n — 0 on F or div (/ [V, W]) — 0 on £1 can be used as sufficient conditions. Example 6.1. Let 0 = { ( x , y ) : x2 + y2 < 1} be the unit ball in R2 with boundary F = {(x,y] : x2 + y2 = 1). Choose

6.2

Basic Formula for Domain Integrals

We have proved the following result.

374 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

Theorem 6.1. Let f G C 2 ([0,r] x [0,r];#£ c (R N )) and T be the boundary of a bounded open subset ft of RN of class C2. Assume that V belongs to C°([0, T]; Cf oc (R N ,R N )) andW to C0([0,T}-Cfoc(RN,RN)}. Consider the function

Then the partial derivative of Jy,w(t, s) with respect to t in t = 0 is given by

and the second-order mixed derivative of Jy,w(t, s) in ( t , s } = (0,0),

is given by the expression

The last term in the second integral can be expressed in terms of v = V(0)\p and w = W(0)|r as follows:

6.3

Nonautonomous Case

The framework introduced in sections 4 and 5 of Chapter 7 has reduced the computation of the Eulerian semiderivative of J(O) to the computation of the derivative of the function

6. Second-Order Semiderivative and Shape Hessian

375

for a velocity field V G C([0, T]; C£C(RN; R N )). In t > 0

since Ts+t(V] = Ts(Vt) o Tt(V), where Vt(s) d= K(* + s) and 1^(0) = V(t). This suggests the following definition.

Definition 6.1.

Let J be a real-valued shape function. Let V and W satisfy condition (V) and assume that for all t e [0,r], dJ(Slt(W)\ Vt) exists for Slt(W) = Tt(W)(ty. The function J is said to have a second-order Eulerian semiderivative at £1 in the directions (V, W) if the following limit exists:

When it exists, it is denoted d2 J(Q; F; W). If, for all t, J has a Hadamard semiderivative at Q$(W), recall that

and the above definition reduces to

Remark 6.1. This last definition is compatible with the second-order expansion of j(t] with respect to t around t = 0:

where

The next theorem is the analogue of Theorem 3.2 and provides the canonical structure of the second-order Eulerian semiderivative (cf. (3.17) to (3.19) in section 3.1 for the definitions of V m'f and )/). Theorem 6.2. Let J be a real-valued shape function, 0 a subset o/R N , and m > 0 and I > 0 two integers. Assume that

376 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus (ii) VW 6 Vm^, Vt € [0, r], J has a shape gradient of order t and is Hadamard semidifferentiable at fit(W); (iii) W e Ve, the map

is continuous. Then for all V in Vm+1>e and all W in Vm*,

where

Proof. The differential quotient (6.13) can be split into the sum of two terms

In view of (i) and (iii), for all U in V^,

by the same argument as in the proof of Theorem 3.2 for the gradient. Hence the first term converges to

For the second term recall that V belongs to y m +M an(} observe that the vector field

belongs to Vm^ and that F(0) = V'(0). Thus by linearity of dJ($l; V], the second term in (6.19) can be written as

But for any V in Vm+2'1, V belongs to v"m+M. Then by assumption (i),

6. Second-Order Semiderivative and Shape Hessian

377

which implies that

Now by assumption (ii), the map U *-+ dJ(£l;U) is linear and continuous on £>^(R N ,R N ), and the map

is linear and continuous (hence uniformly continuous) for the topology V m+1 '^ for all V in the dense subspace V m + 2 ' £ . Hence it uniquely and continuously extends to all elements of V m+l>e. This completes the proof of the theorem. This important theorem gives the canonical structure of the second-order Eulerian semiderivative: a first term that depends on V(0) and W(0) and a second term that is equal to dJ(tl; V'(0)). When V is autonomous the second term disappears and the semiderivative coincides with d 2 j(0; V\ W(0)) which can be separately studied for autonomous vector fields in V1. We conclude this section with the explicit computation of the second-order Eulerian semiderivative for a shape function J(0) with respect to two velocity fields V and W satisfying the conditions of Theorem 6.2 and such that the shape gradient at any t is of the form

for some function g(t) £ C(Tt(W)). Further assume that the family of functions g(t) has an extension Q G C^QO,?-]; Cf oc (7V(F); R N )) to an open neighborhood N(T) of T such that U {Tt(W) : 0 < t < r} C N(T). Therefore using the extension Nt(W) of the normal nt on Ft(W), it amounts to differentiating the expression

Apply the first formula (4.17) of Theorem 4.3 to get

where Q'w(0) only depends on W. But the last three terms have already been computed in several forms. They constitute expression (6.2) in section 6.1, which yields (6.3) and (6.4) with / = Q(0), V = WO), and W = W(0). This yields with

378 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus the notation v =

Remark 6.2. When V is autonomous the term in V'(0) disappears. In that case the first half of the Lie bracket can be eliminated by restarting the computation with V(t) = VoTj~1(W) in place of V since V(0) = V and

Remark 6.3. Except for the terms that contain the first half of the Lie bracket DVW and V'(0), the only term that might not be symmetrical in the first expression (6.21) is the one in Q'W(Q). In fact, according to the second expression and our theorem, Q'W(Q) = Q'WfQ\ only depends on W(0). Now choose autonomous velocity fields V and W. Furthermore, assume that W is of the form W = wr ° p. Since W • n = 0 on F, dJ(ttt(W);V(ty) = dJ(Sl;V(Q)) and necessarily d2J(tt;V;wr op] = 0. Therefore, d 2 J(f2; V; W) only depends on wn and hence Q'w(ty = Q'Wn(Q) and the integral

only depends on wn and vn. The above expressions give valuable information on the structure of the secondorder Eulerian derivative. Other expressions can also be obtained. For instance, if j ( t ) is transformed into the volume integral

we get from formula (4.6) in Theorem 4.2 the equivalent volume expression

which can obviously be transformed into a boundary expression.

6. Second-Order Semiderivative and Shape Hessian

379

6.4 Autonomous Case Definition 6.2. Let J be a real-valued shape function. Let 0 be a subset of R N . (i) The function J(Q) is said to be twice shape differentiable

at fi if

and the map

is bilinear and continuous. We denote by h the map (6.23). (ii) Denote by H(tt) the vector distribution in (D(R N ,R N ) £>(R N ,R N ))' associated with h:

where V (g> W is the tensor product of V and W denned as

and Vi(x) (resp., Wj(y)} is the ith (resp., jth) component of the vector V (resp., W) (cf. Schwartz's [1] kernel theorem and Gelfand and Vilenkin [I]). H(£V) will be called the shape Hessian of J at Q. (iii) When there exists a finite integer i > 0 such that H(£l] is continuous for the D £ (R N , R N ) ® £> £ (R N , RN)-topology, we say that H(Sl) is of order L In what follows, the compact notation Z>£ will be used in place of P^(R N , R N ). Theorem 6.3. Lei J be a real-valued shape function and 0 a subset of R N with boundary T. Assume that J is twice shape differentiable. (i) The vector distribution H(£l) has support in T x F. (ii) // J7 is an open or closed domain in RN and H(£l) is of order i > 0, then there exists a continuous bilinear form

such that for all [V] in T>e/Dlr and [W] m T>e/L^,

where go '• 'D1 —» £^/-Dp and q^ : T>^ —> T>1 /L^ are the canonical quotient surjections and

380 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

Proof, (i) It is sufficient to prove the following two properties: (a)VV, W € T> such that W = 0 in a neighborhood of F, d2 J(fi; V\ W) = 0. (b) W, W £ X> such that F = 0 in a neighborhood of F, d2 J(fi; F; W) = 0. In case (a) the proof is similar to the one in Theorem 3.5 for the gradient and we prove the stronger result that for W such that W — 0 on F,

In case (b) V — 0 in a neighborhood N of T and in C-ftT, the complement of the compact support K of V. So U = £>K is a neighborhood of T where V — 0. By construction U fl K — 0 and there exists a bounded neighborhood U of K such that U fl F = 0. Since ZY is compact and F is closed, the minimum distance d from U to F is finite and nonzero. Let

where For all X in F

and by condition (3.1) on W,

and it can easily be shown that for t < 1/c

Thus But W is continuous with compact support. Therefore,

and there exist r > 0 such that

By definition and the previous inequalities

6. Second-Order Semiderivative and Shape Hessian

381

for all 5 in [0, r] and all X € T. This implies that

By construction, V = 0 in N(T) since the distance from K to F is greater than or equal to d. Therefore, and as in the proof of Theorem 3.5, d/(Q s (W); V) = 0 and necessarily d2 J(0; V; W} = 0. (ii) We have already established in (i) that the bilinear form

is zero for all V 6 V and W € T> such that W = 0 on F and also zero for all W £ T> and V 6 X> for which V = 0 in a neighborhood of F. By density all this is still true in £>^, and now by the same argument as in the proof of Theorem 3.5 for all V inZ*, is well defined, linear, and continuous. For the first component it is necessary to show that for all W in

the bilinear form h(V, W) = 0. We first prove the result for the subspace

Then by density and continuity the result holds for the T>1 (R N , RN)-closure A of A. Finally we prove that A = Df. For any V in A, there exist Vi £ £>(17;RN) and V-2 G £>(CO;R N ) such that V = Vi+V2. Moreover,

are compact subsets of the open sets J7 and Cfi, respectively. Hence V\ — 0 (resp., Vf, £>f C D £ (fi; R N ) 0 P£(CO; R N ). Now A c Df,

and

By construction each V in A is of the form V — Vi + V2 for

382 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

Hence and V 6 X^(R N ,R N ). Moreover,

This proves that A C Dr and thence A = Dp. To complete the proof notice that by continuity of V >—> h(V, W), for all W in T>1 the map

is well defined, linear, and continuous. Finally, the map

is well defined, bilinear, and continuous. The next and last result is the extension of the structure theorem, Theorem 3.5, to second-order Eulerian semiderivatives. We need the result established in the corollary to Theorem 3.5. For a domain 17 with a boundary F which is Ce+1, t > 0, the map

is a well-defined isomorphism. This will be used for the ^-component. For the VF-component we need the following lemma. Lemma 6.1. Assume that the boundary T o f f t i s C^+l, l>0. Then the map

is a well-defined isomorphism, where

is the canonical surjection and Dr is given by (6.28). Proof. The proof follows by standard arguments. Theorem 6.4. Let J be a real-valued shape function. Assume that the conditions of Theorem 6.3 (ii) are satisfied and that the boundary F of the open domain 17 is Ci+l for t > 0. (i) The map

is bilinear and continuous, and for all V and W in Z^(R N ,R N )

where P(V] is a linear combination of derivatives of V up to order L

6. Second-Order Semiderivative and Shape Hessian

383

(ii) This induces a vector distribution h(T <8> F) on C£(F, R N )
In the regular case the reader is also referred to the early papers of Zolesio [24, p. 434] and Bucur and Zolesio [12].

Remark 6.4. Finally, under the assumptions of Theorems 6.3 and 6.4

for all V in V*™+M and W in Vm^.

Example 6.2. Go back to the example of the domain integral in section 6.1,

ForV eT>l(R^,HN)

Using the domain expression with V in £> 2 (R N ,R N ) and W in ^(R^R111),

we readily get

and if F is C1,

is continuous for pairs

384 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus 6.5

Decomposition of d2J(ft; V(o), W(o))

One important observation in the explicit computation of the second-order Eulerian semiderivative of the domain integral (6.1) in section 6.1 was the lack of symmetry and the appearance of the first half of the Lie bracket in (6.4). The same phenomenon was observed in the final form of the basic formula (6.10) for domain integrals (6.6) in Theorem 6.1, and also in section 6.3 for the derivative of the shape gradient (6.21) when it can be represented in integral form (6.20). In this section perturbations of the identity will be used to show that d2«7(fi; V(0), W(0)) can be further decomposed into a symmetric term plus the gradient applied to the velocity DV(Q)W(0): Furthermore, the symmetrical term can be obtained by the velocity method

Theorem 6.5. Let J be a real-valued shape function and 0 a Banach subspace of Lip(R N ;R N ). (i) Given f , 9, and £ in Be, assume that there exists r > 0 such that Then

for £lf = [I + /](0) and the velocity fields

(ii) If f, 0 belong to Vi+l and £ to Vi, and V and W^ satisfy the conditions of Theorem 6.2, then

(iii) // V and W satisfy the conditions of Theorem 6.2 and V belongs to V m + 1 >^+ 1 ; then

6. Second-Order Semiderivative and Shape Hessian

385

and

Proof, (i) Assume that oPJn(/;$;£) exists and consider the differential quotient

From Theorem 3.3 (ii),

Define

From Theorem 4.1 in Chapter 7, Tt = Tt(W{) for the velocity field

Therefore

since q(t) converges as t goes to 0. The converse is obvious. (ii) is now a direct consequence of Theorem 6.2 and the fact that

(iii) From Theorem 6.2 and part (ii) with / = 0, 0 = V(0) and £ = W(0). If D2Jft(f) exists in a neighborhood of / = 0 and if it is continuous at / = 0, then D2 Jn(0) is symmetrical and this completes the decomposition of the shape gradient into a symmetrical operator and the gradient applied to the first half of the Lie bracket [V(0), W(fy]

386 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus

But this is not the end of the story. From the computation of the Hessian of the domain integral (6.4) in section 6.1,

the result of the computation (6.10) in section 6.2,

and expression (6.21) of the derivative of (6.20) in section 6.3,

it is readily seen that the symmetrical term (D2J^(0) F(0), W(0)} further decomposes into a symmetrical term which only depends on the normal components vn and wn of V(Q) and VF(0) and another symmetrical term, which is boxed in the above expressions. This last term is the same in all expressions and only depends on the trace of G(0) (here of / and Q(0)) on F and the group of terms

involving vr, wr, and tangential derivatives of vn and wn on It would be interesting to further investigate this structure. In the context of a domain optimization problem, if the shape gradient is zero, both the term (6.48) and the first half of the Lie bracket will be multiplied by zero. Thus they won't contribute to the Hessian which will reduce to the symmetrical part that depends only on vn and wn; that is, for autonomous velocity fields V and W,

and the term

6. Second-Order Semiderivative and Shape Hessian

387

is symmetrical. For earlier results on the structure of the second-order shape derivative the reader is referred to Zolesio [24, p. 434] and Bucur and Zolesio [12]. Of course, the task of establishing similar results in the constrained case (D ^ R N ) remains to be done.

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j Chapter 9

ShapeGradients under a

State Equation Constraint

1

Introduction

When a shape functional depends on the solution of a boundary value problem defined on the underlying domain, it is said to be constrained. This special type of constraint is to be distinguished from constraints on the geometry such as the volume, perimeter, curvatures, etc. Using the generic terminology of control theory the solution of the boundary value problem will be called the state and its corresponding equation or inequality the state equation or inequality. The domains will be identified with the controls, and the constraints on the domains with the control constraints. Additional constraints on the solution of the state equation are usually called state constraints. Such problems have received a considerable amount of attention over the last half-century, and a rich and abundant literature can be found in mechanics, control theory, and optimization. Treating the state equation as an equality constraint and using Lagrange multipliers naturally yields a dual variable which is solution of the adjoint equation of the linearized state equation. This dual variable is called the adjoint state. Of course, under appropriate differentiability assumptions with respect to the state and the control variables, necessary conditions can be obtained within the classical framework of the calculus of variations. Probably one of the most influential contributions of the last half of the 20th century to optimal control theory was made by Pontryagin et al. [I] in the 1960s, when they showed that the differentiability with respect to the control could be relaxed and replaced by the pointwise maximization with respect to the control variable of the Hamiltonian constructed from the objective function and the coupled state and adjoint state equations, the so-called maximum principle. One of the important technical advantages of the control theory approach is to avoid the differentiation of the state with respect to the control. This is not only true for the characterization of optimal controls, but also to obtain explicit expressions of the derivative of a state equation constrained objective function with respect to the control. Using a Lagrangian or Hamiltonian formulation, the derivative of 389

390

Chapter 9. Shape Gradients under a State Equation Constraint

the constrained objective function with respect to the control is typically equal to the "partial" derivative of the Lagrangian or the Hamiltonian with respect to the control, where the adjoint variable is a solution of an appropriate adjoint state equation that is coupled with the state equation. This chapter will concentrate on two generic examples often encountered in shape optimization. The first one is associated with the so-called compliance problems, where the shape functional is equal to the minimum of a domain-dependent energy functional. The special feature of such functionals is that the adjoint state coincides with the state. This obviously leads to considerable simplifications in the analysis. In that case it will be shown that theorems on the differentiability of the minimum of a functional with respect to a real parameter readily give explicit expressions of the Eulerian semiderivative even when the minimizer is not unique. The second one will deal with a shape functional which can be expressed as the saddle point of some appropriate Lagrangian. As in the first example, theorems on the differentiability of the saddle point of a functional with respect to a real parameter readily give explicit expressions of the Eulerian semiderivative even when the solution of the saddle point equations is not unique. Avoiding the differentiation of the state equation with respect to the domain is particularly advantageous in shape problems. Here the state variable (the solution of the boundary value problem) lives in a function space (Banach, Hilbert, Sobolev, etc.) which depends on the control (underlying variable domain)! Thus the notion of derivative of the state with respect to the domain is more delicate. In this chapter two techniques will be presented to get around this difficulty: function space parametrization and function space embedding. Function space parametrization consists in transporting the functions on variable domains back onto the initial domain, where they can be compared. It is related to the notion of material derivative in continuum and structural mechanics. This will be illustrated by computing the volume and boundary integral expressions of the Eulerian semiderivative of a simple shape functional for the homogeneous Dirichlet and Neumann boundary value problems by the theorems on the differentiation of a minimum and a saddle point. Function space embedding consists in constructing extensions of the domaindependent boundary value problems to a larger fixed domain. For homogeneous Dirichlet boundary conditions, it is sufficient to consider extensions by zero to R N . The nonhomogeneous case requires the introduction of a multiplier to take into account the boundary condition. This technique will be illustrated by computing the boundary integral expression of the Eulerian semiderivative of a simple shape functional for the nonhomogeneous Dirichlet boundary value problem by the theorems on the differentiation of a saddle point. In addition to the generic example, the theorem on the differentiation of an infimum with respect to a parameter will be applied to the example of the buckling of columns considered in section 4 of Chapter 3. In section 3 an explicit expression of the semiderivative of Euler's buckling load with respect to the cross-sectional area will be given. A necessary and sufficient condition will also be given to characterize the maximum Euler's buckling load with respect to a family of cross-sectional areas. The theory is further illustrated in section 4 by providing the semiderivative

2. Min Formulation

391

of the first eigenvalue of several boundary problems over a bounded open domain: Laplace equation, bi-Laplace equation, linear elasticity. In general the first eigenvalue is not simple over an arbitrary bounded open domain and the eigenvalue is not differentiable; yet the main theorem provides explicit domain expressions for a bounded open domain and boundary expressions of the semiderivatives for a sufficiently smooth bounded open domain.

2 2.1

Min Formulation An Illustrative Example and a Shape Variational Principle

Let £1 be a bounded open domain in R N with a smooth boundary F. Let y = y(Q) be the solution of the Dirichlet problem

where / is a fixed function in H 1 ( H N } . The solution of (2.1) is the minimizing element in HQ (f2) of the energy functional

Introduce the shape function

We want to show that

This example is the prototype of a free boundary problem1 which can be obtained from the following shape variational principle: It yields the extra boundary condition2

Equations (2.1) and (2.6) characterize a free boundary problem. It is the simplest example of a large family of problems where the shape function is an extremal of a natural internal energy (cf. section 6 of Chapter 3). They correspond to the so-called "compliance problems" in elasticity theory. They also occur in fracture theory and image segmentation. The first-order variation of this shape functional yields the extra boundary condition, which is characteristic of a free boundary problem. Usually with other constraints, such as a volume constraint on fi and/or an area constraint on P. 2 With a volume equality constraint we would get that \dy/dn\2 is equal to a constant on F.

392

2.2

Chapter 9. Shape Gradients under a State Equation Constraint

Function Space Parametrization

To compute the first-order derivative of J(f£) we perturb the bounded open domain 0 by a velocity field V, which generates the family of transformations {Tt : 0 < t < r} of RN and the family of domains {ftt - Tt(tt) : 0 < t < T}. At t,

and the minimizing element yt = y(&t) is the solution of the Dirichlet problem

where Tt is the boundary of fif. We want to compute the derivative

of the function

We need a theorem that would give the derivative of a Min with respect to a parameter i > 0 at t = 0. The difficulty here is that the function space Hl($lt} depends on the parameter t. To get around this and obtain a Min with respect to a function space that is independent of t, introduce the following parametrization:

Notice that since Tt is a homeomorphism, it transforms the open domain fi into the open domain Qt and sends the boundary F of fi onto the boundary Tt of fit. In particular, when V is sufficiently smooth for all in Hg(f2), (p o Tp1 E .//o(f^), and conversely, for all ^ in #0(^)5 V ; o ^t £ ^o(^)- This parametrization does not affect the value of the minimum J(f2 t ) but changes the functional E:

This parametrization is typical in shape analysis. It amounts to introducing the new energy functional

Our objective in the next section is to compute the limit (2.9) for

This will be done. Before closing, it is interesting to characterize the minimizing element y1 in HQ(££) of

2. Min Formulation

393

which is the solution of the following variational equation: to find y* 6 HQ(^I) such that for all
Compare this expression with the characterization of the minimizing element yt of E(n t, (p) on #o(^t): to find yt ^ HO (tit) such that for all

It is easy to verify that

So yt is the solution yt of (2.8) transported back onto the fixed domain £7 by the change of variable induced by Tt. In view of this expression (2.15) can be rewritten on the fixed domain Q

where for t in [0, T] small

With this change of variable yt is now characterized by the variational equation

2.3

Differentiability of a Minimum with Respect to a Parameter

Consider a functional

for some T > 0 and some set X. For each t in [0, r] define

The objective is to characterize the limit

394

Chapter 9. Shape Gradients under a State Equation Constraint

when X ( t ) is not empty for 0 < t < r. When X(t) = {x*} is a singleton, 0 < t < r, and the derivative

of x is known, then it is easy to obtain dg(Q) under appropriate differentiability of the functional G with respect to t and x. When x is not readily available or when the sets X ( t ) are not singletons, this direct approach fails or becomes very intricate. In this section we present a theorem that gives an explicit expression for dg(Q), the derivative of the Min of the functional G with respect to t at t = 0. Its originality is that the differentiability of x* is replaced by a continuity assumption on the setvalued function and the existence of the partial derivative of the functional G with respect to the parameter t. In other words this technique does not require a priori knowledge of the derivative x of the minimizing elements x* with respect to t. Theorem 2.1. Let X be an arbitrary set, r > 0, and G: [0,T] x X —>• R a welldefined functional. Assume that the following conditions are satisfied: (HI) For all t G [0,r], X(t) + 0; (H2) for all x in (Jte[o T] X ( t ) , dtG(t,x) exists everywhere in [0, r]; (H3) there exists a topology TX on X such that for any sequence {tn} C ]0, T], tn —» ^o = 0, 3xo G -X"(0), 3 a subsequence {tnk} of {tn}, and for each k > I , 3xnk G X(tnk) such that (i) xnk —> x° in the TX-topology (ii) and

(H4) for all x in X(Q), the map 11—>• dtG(t,x) is upper semicontinuous at t = 0. Then there exists x° G -^(0) such that

Remark 2.1. In the literature condition (H3)(i) is known as sequential semicontinuity for setvalued functions. When X(Q) is a singleton {#0} we readily get d
2. Min Formulation

395

Proof of Theorem 2.1. (i) We first establish upper and lower bounds to the differential quotient

Choose arbitrary XQ in X(0) and Xt in X(t). Then, by definition,

Add up the above two inequalities to obtain

By assumption (H2), there exist Ot, 0 < 9t < 1, and at, 0 < at < 1, such that

and by dividing by t > 0

(ii) Define

There exists a sequence {tn : 0 < tn < T}, tn —> 0, such that

By assumption (H3), 3x° G -X"(0), 3 a subsequence {tnk} of |tn}, and for each k > 1, 3xnfc G ^(t n f c ) such that xnk -^ x° in Tx and

So, from the first part of the estimate (2.30) for t = t n f c ,

and

396

Chapter 9. Shape Gradients under a State Equation Constraint

Therefore, and

Prom the second part of (2.30) and assumption (H4) we also obtain

and necessarily

In particular, from (2.3) and (2.31)

and x° is a minimizing point of <9(G(0, •).

2.4

Application of the Theorem

Our example has a unique minimizing point y* for t > 0 small. Here X — JYg(O), X(i) = {y*}, and it is sufficient to establish the continuity of the map t i—>• y1 at t = 0 for an appropriate topology on HQ(&}. We now check assumptions (HI) to (H4). Assume that V belongs to C°([0, T]; 2 £> (R N ,R N )) and that / e JH"1(RN). Choose r > 0 small enough such that

and that there exist constants 0 < a < (3 such that

Since the bilinear form associated with (2.23) is coercive, there exists a unique solution y1 to (2.23) and

So assumption (HI) is satisfied. To check (H2) use expression (2.19) and compute

where

2. Min Formulation

397

By assumptions on V and /, dtE(t, (p) exists everywhere in [0, r] for all (p in HQ(£I) and assumption (H2) is satisfied. To check assumption (H3)(i) we first show that {?/*} is bounded in HQ(^,). Prom (2.33)

By using the norm and the continuous injection of #o(0) into I/ 2 (Q), So from (2.38)

But Jt —» 1 as t —> 0 and / o Tt —»• / in L 2 (fJ) by the following lemma. Lemma 2.1. Asswrae that V e C°([0,r];D 1 (R N ,R N )) satisfies assumption (V) o/ Chapter 7 and that f 6 L 2 (R N ). Then

So by (2.39) y* is bounded:

The next step is to prove the continuity by subtracting (2.23) at t > 0 from (2.23) at t = 0:

Subtract and set (p = yt — y:

398

Chapter 9. Shape Gradients under a State Equation Constraint

and

But A(t) - I -> 0 and Jt f o Tt -» / in L 2 (ft), and finally y* -> y in //J(fi). So assumption (H3)(i) is satisfied for HQ (Q)-strong. For assumption (H3)(ii) we have

and for (p in #<J(n), / in Hl(RN), and V in C°([0,r];P 2 (R N ,R N ))

As
an(

^ ^ ^ 0, the first term converges to zero since

The second term is continuous with respect to

y in ffo(f2). So assumption (H3)(ii) is satisfied. Finally, (H4) is satisfied since

is continuous in [0,r]. So all the assumptions of Theorem 2.1 are satisfied and for V in C°([0,r];£> 2 (R N ,R N )) and / in ^(R N ),

For V autonomous, expression (2.43) is continuous with respect to the space 'D 1 (R N ,R N ), and the shape gradient is of order 1. We know by the structure theorem, Theorem 3.5 of Chapter 8, that for fJ open and a C2-boundary F,

The characterization of
2. Min Formulation

399

Proof of Lemma 2.1. (i) By density of ^(R1*) in L 2 (R N ) for all e > 0, there exists /£ such that

Hence

The last term is less than e and the middle term can be rewritten after a change of variable A function f£ in 'D1(RN) has a compact support and is uniformly Lipschitz continuous, that is,

Thus for all X in R N

Now

Since V G C°([0, r]; P 1 (R N , R N )) is also uniformly Lipschitz continuous by assumption (V)

and for all t i n [0, r]

It is now easy to verify that there exists a constant c > 0 such that

Finally, in view of (2.44) and (2.45) the term

400

Chapter 9. Shape Gradients under a State Equation Constraint

So for t small enough the right-hand side of (2.4) is less than 3e and this completes the proof of the first part of (2.41). (ii) The second part of (2.41) can be obtained by the following change of variable:

and the fact that /3~1 < J^1 < a~1. This completes the proof of the lemma. 2.5

Domain and Boundary Integral Expressions of the Shape Gradient

Expression (2.43) for the shape gradient is a volume (or domain) integral, and it is easy to check that the map

is linear and continuous (cf. section 3.3 in Chapter 8). So by Corollary 1 to the structure theorem, Theorem 3.5 in Chapter 8, we know that for a domain $7 with a C2-boundary F there exists a scalar distribution g(T) in Z>(r)' such that

The next objective is to further characterize the boundary expression. Recall that we have assumed that / 6 ^T1(RN). So for a C2-boundary F the solution y = y(0i) of the problem

belongs to H2(fy. For velocity fields V in C°([0,r] ; r» 1 (R N ,R N )) satisfying assumption (V) the transported solution yi in /f 2 (Q) is the solution of the system

Knowing that for all t in [0, T], y* € H2(£l) flHQ (0), we can repeat the computation of dtE(t, (p) for (p in H2(ty (~]HQ($}) instead of HQ(0). With this extra smoothness we can use the formula

for a sufficiently smooth function F: [0, r] x RN —» R. Prior to applying this formula to expression (2.15) in section 2.2, notice that for (p in JY 2 (0),

2. Min Formulation

401

but it generally does not belong to HQ($I). Then

Substitute (p — y in (2.51):

But y e H2(tt) n#d(^) and

and

But and finally

This is the boundary expression that is continuous for V(0) in the space D°(R N , R N ). It has been obtained via a parametrization of the function space appearing in the Min formulation. Thus

as predicted in section 2.1. Remark 2.3. An original application of the computations made for the generic example can be found in Delfour, Payre, and Zolesio [3]. In that paper the derivative of the energy function with respect to the nodes in a P1 finite element approximation is obtained from the volume expression of the shape semiderivative of the continuous problem. This technique also applies to a broad class of boundary value problems and to mixed hybrid finite element approximations. As observed by several authors, the boundary integral expression is not suitable since the finite element solution does not have the appropriate smoothness under which the boundary integral formula is obtained. This technique is used to obtain a triangularization which minimizes

402

Chapter 9. Shape Gradients under a State Equation Constraint

the approximation error of the solution. Other formulae of the same type have been obtained for mixed finite element approximations in Delfour, Mghzali, and Zolesio [1] and several other boundary value problems. The reader is also referred to Delfour, Payre, and Zolesio [1, 2, 4, 5] for application of those techniques to thermal problems such as diffusers and space radiators. The last paper combines the above techniques with the systematic handling of parametrized geometries

3

Buckling of Columns

Auchmuty's dual principle was used for the functional associated with the optimal design of a column against buckling in section 4 of Chapter 3. In this section we first use this construction to compute the directional semiderivative with respect to the cross-sectional area and then use it to give a necessary and sufficient condition that characterizes the maximizers. Indeed, recall the identity

and notice that A is a maximizer of A over A if and only if A is a maximizer of ^ over A. We have seen in Theorem 4.3 of section 4 of Chapter 3 that the concave upper semicontinuous functional

has maximizers over the weakly compact convex set A. Therefore, if p,(A] has a directional semiderivative dn(A\B], a maximizer is completely characterized by

This directional semiderivative always exists for concave continuous functions, and Theorem 2.1 can be used to get an explicit expression of dfj,(A; B). Given A e A, B £ L°°(0,1), and t > 0, let

In view of the fact that A > AQ > 0, there exists r > Q such that for all 0 < t < r

Define

3. Buckling of Columns

403

Theorem 3.1. Let B be an arbitrary function in L°°(0,1) and A an element of A. (i) The directional semiderivative at A in the direction B is given by the following expression:

and since fj, is concave and continuous, dfj^(A\ B) also exists. (ii) The maximizing elements of ^(A) or \(A) in A are completely characterized by the variational inequality

or equivalentiy

If we replace E(A] by

then the maximizing elements of fJ>(A) or \(A) in A are completely characterized by the variational inequality

Proof, (i) From Theorem 4.2 in Chapter 3, X ( t ) ^ 0 and assumption (Hi) is satisfied. The partial derivative of L(t, v) with respect to t is given by the expression

which is independent of t and exists for all B 6 I/°°(0,1). Hence assumptions (H2) and (H4) are trivially satisfied. Prom (4.7) in Theorem 4.1 of Chapter 3

404

Chapter 9. Shape Gradients under a State Equation Constraint

But

Therefore for any sequence tn \ #o(0,1) such that

0, there exists a subsequence {tnk}, H, and u G

Going to the limit in (3.8)

we get the following variational equation for u G #o (0, 1):

To show that u £ X(Q) = E(A), let u0 be any element of E(A). By definition,

since ||i4'l|L2 is bounded. Since ||w' fc IU 2 ~^ II W / I|L 2 5 by going to the limit in (3.10), we get

4. Eigenvalue Problems

405

Therefore, by definition of the minimum, u 6 E(A) = X ( 0 ] and assumption (H3)(i) is satisfied. To check the second part of (H3) we first show that {u^} converges not only in #o(0, l)-weak, but also in -£/o(0, l)-strong. Then assumption (H4) directly follows from that property:

The strong convergence follows from the following chain of inequalities:

since A = \(A). This complete the proof of part (i). (ii) The first characterization now follows directly from the fact that p, is concave and continuous and A is closed and convex. The second characterization uses the following identity, which comes from (3.8) for the eigenvectors and their normalization (4.11) in Theorem 4.2 of Chapter 3 as elements of E(A}: for all v e E(A]

4

Eigenvalue Problems

The first eigenvalue A(Q) of a linear boundary value problem defined on a bounded open subset fi of RN is a classical example of shape functional which comes in the form of an infimum. It is generally not differentiate since the eigenvalue can be repeated, but Theorem 2.1 of section 2.3 can be used to compute its Eulerian semiderivative. In this section we give the volume expression of the Eulerian semiderivative for the Laplacian and the bi-Laplacian with a homogeneous Dirichlet boundary condition for a general bounded open domain and its boundary expression when the domain is sufficiently smooth. In the case of the Laplacian over a smooth domain, the first eigenvalue is simple and differentiable. As in section 3 it is technically advantageous to work with Auchmuty's [1] dual principle rather than with the Rayleigh quotient. It is also technically advantageous to embed the problem of the computation of the Eulerian semiderivative for the domain fi into a larger and sufficiently smooth hold-all D, which will contain all the perturbations Flt = Tt(V}(ti) of

406

Chapter 9. Shape Gradients under a State Equation Constraint

ft for t sufficiently small. The smoothness conditions that would normally occur on ft will then occur on D. Once the expression for the semiderivative is obtained, D can be thrown away, leaving a final expression that does not require any smoothness assumption on ft. 4.1

Transport of Hj(O) by ^'^-Transformations of RN

Recall Theorem 2.1 of section 2 in Chapter 6 on the embedding of #o(ft) into HQ(D}. As a consequence, the homogeneous Dirichlet boundary value problem in ft: find y € f/o(fi) such that

is completely equivalent to the variational problem, find Y € £/o(ft;D) such that

and y|n = y. Let k > 1 and let T be a transformation of RN such that

Associate with T the map By assumption on T, the norms ||
is again a linear bijection such that both 7^ and T^1 are uniformly Lipschitzian. Given a bounded open subset D of R N , further assume that T is such that

As a result Let ft be an open subset of RN such that ft C D. Then T(ft) c D, T~l(fy C D, and H$($!,) and HQ(T($I)) can be identified with the subspaces #g(ft;.D) and H£(T(to);D) ofH£(D). Moreover,

4. Eigenvalue Problems

407

and the following diagram commutes:

4.2

Laplacian and Bi-Laplacian

For an arbitrary bounded open subset £1 of RN the bounded open hold-all D can be arbitrarily chosen such that 0 C D. For instance, D can be a large enough open ball. This will make it possible to throw on D any smoothness assumption that would normally occur on Q and work with an arbitrary 0. In this set-up, D is used as an intermediate step and can be thrown away once the semiderivative has been computed. Given a velocity field

the flow mapping Tt = Tt(V) maps D onto itself and 3D onto itself and is equal to the identity on R N \D. So it transports HQ(D] onto itself. Theorem 4.1. For k = 1,2 and 0 < t < r,

Hence for £lt = T t (fi),

Proof. The second statement follows from the fact that Tt and its inverse T^1 are both Lipschitz continuous and that, by the previous considerations or Lemma 7.3 in Chapter 6, they transport sets of zero capacity onto sets of zero capacity: Tt(D\Q) —

D\r t (n) = D\n t .

n

As Q is compact and 0 C D, there exists TV > 0 such that for all t, 0 < t < TV, fit(V) C D. In both cases the first eigenvalue is given by the Rayleigh quotient:

where

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Chapter 9. Shape Gradients under a State Equation Constraint

In view of the previous constructions ^(f^) can be replaced by #o (f^; D):

Note that if (p ^ 0 is a minimizer, then for any real number a ^ 0, oup is also a minimizer. Theorem 4.2. Given k = 1,2, let SI be a bounded open subset o/R N . Assume that

for some bounded open domain D in RN such that f2 C D. There exists at least one nonzero solution (p G HQ (^it'i D) to the minimization problem (4.3), A(fi t (V)) > A(£>) > 0, and

The solutions are completely characterized by the following variational equation: there exists (p e 11^(0.^ D} such that

or equivalently

Proof. Same technique as in the proof of Theorem 2.2 in Chapter 6. The minimization problem can be rewritten on the unit sphere in L2(D) by normalization:

The minimizing elements of (4.6) cannot be zero since the injection of HQ(D) into L2(D] is compact (cf. Theorem 2.1 in Chapter 6).

Remark 4.1. One of the consequences of the use of the embedding of HQ (0) into HQ (D) is that the characterization of the first eigenvalue on O only requires that 0 be a bounded open subset of R N . It is not necessary to assume that i7 is Lipschitzian in order to use Rellich's theorem, since D can be chosen sufficiently large (17 C D} and smooth in order to contain all variations f^ C D as t goes to zero. This technique will be exploited in the computation of the Eulerian semiderivative of A(0).

4. Eigenvalue Problems

409

Auchmuty's [1] dual variational principle for this eigenvalue problem can be chosen as

By using the embedding of H0(Q f ) into HQ(D), this problem can be rewritten as

Theorem 4.3. Given k = 1, 2, let H be a bounded open subset o/R N . Assume that

for some bounded open domain D in RN such that Jl C D. Then for 0 < t < T,

and the set of minimizers of (4.7) is given by

Proof. From the previous theorem the set E(ftt} is not empty, and for any