A p-Laplacian Approximation for Some Mass Optimization Problems

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 118, No. 1, pp. 1–25, July 2003 ( 2003)

A p-Laplacian Approximation for Some Mass Optimization Problems G. BOUCHITTE´,1 G. BUTTAZZO,2

AND

L. DE PASCALE3

Communicated by F. Giannessi

Abstract. We show that the problem of finding the best mass distribution, in both the conductivity and elasticity cases, can be approximated by means of solutions of a p-Laplace equation as p → +S. This seems to provide a selection criterion when the optimal solutions are nonunique. Key Words. Shape optimization, p-Laplacian, mass transportation problems, Monge–Kantorovich differential equation.

1. Introduction In some recent papers (Refs. 1–2), Bouchitte´, Buttazzo, and Seppecher considered a rather general formulation for the problem of finding the optimal distribution of a given amount of material in order to obtain the most performant conductor (in the case of a scalar state function u, the temperature of the system) or elastic body (in the case of mechanical structures). The existence of an optimal solution was shown to occur in the class of nonnegative measures, under the sole assumption that the data (the heat sources in the scalar case, the forces in the case of elasticity) are bounded measures, possibly concentrated on sets of lower dimension. The precise formulation of the mass optimization problem will be recalled in Section 2. In Ref. 1 (see also Ref. 2), the problem was shown to be equivalent to a system of PDE of the Monge–Kantorovich type which, in the scalar case, is in turn equivalent to the problem of the optimal mass transportation, 1

Professor, De´partement de Mathe´matiques, Universite´ de Toulon et du Var, La Garde, France. 2 Professor, Dipartimento di Matematica, Universita` di Pisa, Pisa, Italy. 3 Assistant Professor, Dipartimento di Matematica Applicata, Universita` di Pisa, Pisa, Italy.

1 0022-3239兾03兾0700-0001兾0  2003 Plenum Publishing Corporation

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with a cost proportional to the Euclidean distance, of the form illustrated extensively in Refs. 3–4. Due to the general framework considered, when the data are singular measures, the uniqueness of an optimal solution may fail. Then, it becomes important, also from a numerical point of view, to find a criterion which, by means of an approximation procedure, selects in a canonical way the best solution among all those that solve the optimization problem. Here, we propose the approximation of the Monge–Kantorovich PDE through the p-Laplace equation as p → +S. This idea was already used in Ref. 3, in the case of data regular enough, to show that in this case the optimal mass distribution density is actually a LS function. However, as it has been shown recently in Ref. 5 (see also Ref. 6), data which are absolutely continuous with respect to the Lebesgue measure produce always optimal mass densities which are unique and Lebesgue integrable. All these results are obtained in the scalar case, thanks to the equivalence with the mass transportation problem, while nothing is known in the case of elasticity. We show that, for all the right-hand sides which are measures and in both the scalar case and the vector case, the approximation with p-power energies always produces a solution to the mass optimization problem. The numerical counterpart of these results has been developed recently by Golay and Seppecher in Ref. 7. In the final section, we give some examples and raise the question of whether or not the solution determined by approximation with p-power energies is unique.

2. Mass Optimization Problem In the following, we assume always that Ω is a bounded connected open subset of ⺢N with a smooth boundary and that Σ is a closed subset of ¯ . The notations 兩z兩 and z · z′ respectively denote the Euclidean norm and Ω the scalar product of vectors in ⺢N. The same notation will be kept when 2 z, z′ are matrices in ⺢N . We consider also a given right-hand side f, which we assume to be a bounded measure (real-valued in the scalar case and ⺢Nvalued in the case of elasticity), and a stored energy density described by a function j(z). The customary choice for the scalar case is j(z)G(1兾2)兩z兩2, whereas in the case of linear elasticity j is given by j(z)Gβ 兩zsym兩2C(α 兾2)兩tr(zsym)兩2,

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α and β being the so called Lame´ constants, and where zsym denotes the symmetric part of the matrix z. More generally, we consider a stored energy density function j of the form j(z)G(1兾p)( ρ(z))p, where ρ satisfies the conditions below: (a) (b) (c) (d)

ρ is convex; ρ is positively 1-homogeneous; ρ(z)Gρ(zsym), for all z; there exist two positive constants k1 and k2 such that k1 兩zsym兩⁄ ρ(z)⁄k2 兩zsym兩,

∀z∈⺢NBN.

From now on, unless specified differently, we refer to the case of elasticity; it is intended that similar statements hold for the scalar case as well. ¯ will be described by nonnegative measures µ Mass distributions in Ω ¯ . For such a measure, we consider the energy with support in Ω

E ( µ)Ginf

冦冮 j(Du) dµA〈 f, u〉: u∈U (Σ )冧 ,

(1)

where U (Σ ) denotes the class of smooth displacements u: ⺢N → ⺢N vanishing on Σ, and the corresponding compliance C ( µ)G−E ( µ). By standard duality argument, we can write also the compliance C ( µ) in the form

C ( µ)Ginf

冦冮 j*(σ ) dµ: σ ∈L (⺢ ; ⺢ p′ µ

N

N2 sym

冧

), −div(σµ)Gf, in ⺢N \Σ .

(2)

Remark 2.1. In (2), the condition −div(σµ)Gf is intended in the sense of distributions on the open set ⺢N \ Σ. Then, it contains implicitly interface conditions on ∂Ω\Σ; for instance, if µ is the Lebesgue measure on Ω and if the field σ is regular enough, then the normal trace σνΩ (νΩ being the exterior unit normal on ∂Ω) vanishes on ∂Ω\Σ. The mass optimization problem can now be written as the minimization problem for C ( µ) once the total mass of µ and a bound on its support region are prescribed, inf

冦C ( µ): 冮 dµ Gm, spt µ ⊂Ω¯ 冧 .

(3)

A key result in Ref. 1 (cf. Theorem 2.3) states that the infimum in (3) is related directly with the following gradient constrained optimization

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problem: IGI( f, Ω, Σ )_sup{〈 f, u〉: u∈U (Σ ), ρ(Du)⁄1 on Ω}. Indeed, we have inf (3)GI p′兾p′m1兾(pA1). Moreover, the infimum in (3) is actually a minimum provided I( f, Ω, Σ ) is finite, which (see Ref. 1, Proposition 3.1) turns out to be equivalent to the following compatibility condition: 〈 f, u〉G0,

2

∀u∈R Σ G{r(x)GAxCb: b∈⺢N, A∈⺢N skew, rG0 on Σ},

(4)

N2

where ⺢skew denotes the family of skew-symmetric NBN matrices. In the scalar case, the previous condition changes so that either Σ is nonempty or that the source term is balanced, i.e.,

冮 dfG0. In order to relax the maximization problem above, we introduce the ¯ ; ⺢N) of the set {u∈U (Σ ): ρ(Du)⁄1, class Lip1,ρ (Ω, Σ ) as the closure in C(Ω on Ω}. A characterization of Lip1,ρ (Ω, Σ ) is given in Section 3. Then, it is easy to check that, under condition (4), IGmax{〈 f, u〉: u∈Lip1, ρ (Ω, Σ )}.

(5)

The dual formulation of (5) involves measure-valued stress vector fields and, by standard convex analysis, we find (see Ref. 1) that I agrees with the quantity min

冦冮 ρ (λ ):λ ∈M (Ω, ⺢ 0

冧

), −div(λ )Gf in ⺢N \Σ ,

NBN

(6)

where ρ0 is the polar function associated with ρ,

ρ0 (z)Gsup{z · ξ : ρ(ξ )⁄1}, and where the integral is intended in the sense of convex 1-homogeneous functionals on measures. A second key result (see Ref. 1, Theorem 2.3) states the following implication:

λ solution of (6) ⇒ µ G(m兾I )ρ0 (λ ) solution of (3).

(7)

As a consequence, the solutions of the mass optimization problem (3) do not depend on the exponent p. Then, it seems natural to send p to infinity

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interpreting (5) as the limit problem associated with the strain problem (1). This procedure is the aim of Section 4. To complete the picture, we now give the system of optimality conditions which allows us to characterize the solutions of the different problems described above. In Ref. 1, it is proved that, if µ solves the mass optimization problem (3) and if u solves (5), then (6) has a solution of the form

λ Gσµ,

σ ∈Lµ1 (⺢N, ⺢NBN),

and the triplet (u, mσ 兾I, Iµ兾m) satisfies the following system, that we call the Monge–Kantorovich equation: (i)

−div(σµ)Gf,

on ⺢N \Σ,

(ii)

σ ∈∂ jµ (x, eµ (u)),

µ-a.e. on ⺢ ,

(8a) N

(iii) u∈Lip1, ρ (Ω, Σ ),

(8b) (8c)

(iv)

jµ (x, eµ (u))G1兾p,

µ-a.e. on ⺢ ,

(8d)

(v)

µ (Σ )G0,

¯. spt µ ⊂Ω

(8e)

N

Let us stress that, in (8), both u and µ are unknown. Here, the tangential strain eµ (u) and the integrand jµ are defined in a precise way which is related to some notion of tangent space to the measure µ which will be recalled in Section 3. On the other hand, up to a rescaling, the converse also holds (see Ref. 1, Theorem 3.9): if a triplet (u, σ , µ) solves the Monge–Kantorovich equation (8), then u is a solution of problem (5), the measure mµ兾I is a solution of the mass optimization problem (3), and σµ is optimal for (6) yielding that Iσ 兾m is a solution of the stress problem (2) related to optimal measure mµ兾I. Let us point out that, in the scalar isotropic case, conditions (ii) and (iv) above become simpler and the Monge–Kantorovich equation can be rewritten as follows (see Ref. 8): (i)

−div(∇µ uµ)Gf,

(ii)

u∈Lip1 (Ω, Σ ),

(iii) 兩∇µ u兩G1, (iv)

µ (Σ )G0,

on ⺢N \Σ,

(9a) (9b)

µ-a.e. on ⺢N,

(9c)

¯. spt µ ⊂Ω

(9d)

3. Some Preliminary Results In order to exploit equation (8), it is convenient to give a simpler characterization of the space Lip1,ρ (Ω, Σ ). We notice that, in the scalar case,

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when ρ(z)G兩z兩 or more generally when ρ(z)¤兩z兩, every function in Lip1,ρ (Ω, Σ ) is Lipschitz continuous; this is no more true in the case of elasticity, because of the lack of Korn inequality in W 1,S (see Ref. 9). The following space introduced in Ref. 9 will be useful: U S (Ω; ⺢N)G{u∈LS (Ω; ⺢N): e(u)∈LS (Ω, ⺢NBN)}.

(10)

Thanks to the Korn inequalities, it turns out that U S (Ω; ⺢N) is the intersection of the spaces {u∈W 1,s (Ω; ⺢N): e(u)∈LS (Ω, ⺢NBN)} for 1⁄sF+S. Theorem 3.1. We have Lip1,ρ (Ω, Σ )G{u∈U S (Ω; ⺢N): uG0 on Σ, ρ(e(u))⁄1 a.e. on Ω}. Proof. The proof will be divided in several steps. Step 1. Let u∈U S be such that uG0, on Σ, and ρ(e(u))⁄1, a.e.; then, un G(1A1兾n)u is still in U S, un G0 on Σ and ρ(e(un))⁄1A1兾n. Moreover, un → u,

uniformly.

Then, to prove the inclusion ⊆ , it is enough to prove that each function û in U S, with ûG0, on Σ, and ρ(e(û))⁄cF1, can be approximated uniformly by a sequence ûn in

D (⺢N ; ⺢N) such that

ûn G0, on Σ, and ρ(e(ûn))⁄1, in Ω. Step 2. Let û in U S with ûG0 on Σ. Fix δ H0; we now show that there exists ûδ ∈D (⺢N ; ⺢N) such that 兩ûAûδ 兩⁄ δ ,

(11)

兩兩兩ρ(e(û))兩兩SA兩兩ρ(e(ûδ ))兩兩S 兩⁄3δ ,

(12)

ûδ G0,

on Σ.

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By using Step 1, the inclusion ⊆ will then follow. Since û vanishes on Σ, we have 兩兩ρ(e(û))兩兩LS(Ω) G兩兩ρ(e(û))兩兩LS(Ω \ Σ), and therefore we may choose a suitable rH0 such that the open set Ω0 G{x∈Ω: d(x, Σ )H1兾r} satisfies 兩兩ρ(e(û))兩兩LS(Ω 0) ¤兩兩ρ(e(û))兩兩LS(Ω)Aδ .

(14)

Then, we define the open sets Ω j , A j as follows: Ω j G{x∈Ω: d(x, Σ )H1兾( jCr)}, A1 GΩ2 ,

¯ jA1 , jH1. A j GΩ jC1 \Ω

Clearly, Ω\ΣG* Ω j , j

¯ j ⊂Ω jC1 , Ω j ⊂Ω

and the sets Aj constitute a covering of Ω\Σ. Now, we consider a partition of unity φ j related to this covering and select a sequence of mollifiers g( j, with g( j compactly supported in the ball B(0, ( j), where (j is chosen decreasing in such a way that Ω0CB(0, (1)⊂A1 , A2CB(0, (2)⊂Ω0c ∩A3 , A jCB(0, ( j)⊂A jA1 ∪A j ∪A jC1 ,

for j¤3.

In addition, since by the continuity of ûφ j , g( ∗ (φ j û) converges uniformly ¯ , we can choose (j such that to φ j û on Ω 兩兩g( j ∗ (φ j û)A(φ j û)兩兩S ⁄ δ 2−j , 兩兩g( j ∗ φ jAφ j 兩兩S ⁄ δ 2−j ,

ρ(g( j ∗ (∇φ j ⊗sym û)A(∇φ j ⊗sym û))⁄ δ 2−j , where the last choice is allowed by the fact that

ρ(z)⁄k兩z兩,

for some constant k.

Moreover, we can choose (1 so small that 兩兩兩ρ(g(1 ∗ (φ 1 e(û)))兩兩LS(Ω 0)A兩兩ρ(φ 1 e(û))兩兩LS(Ω 0)兩⁄ δ .

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Now, we consider S

ûδ G ∑ g( j ∗ (φ j û). j G1

The function ûδ is in D (⺢N, ⺢N) and vanishes on Σ (because each term is zero on Σ). We have, thanks to the previous construction, S

兩ûAûδ 兩⁄ ∑ 兩g( j ∗ (φ j û)Aφ j û兩⁄ δ ; j G1

moreover, e(ûδ )G∑ g( j ∗ (∇φ j ⊗sym ûCe(û)φ j), j

and since

∑ j ∇φ j G0, we obtain S

S

e(ûδ )G ∑ (g( j ∗ (∇φ j ⊗sym u)A∇φ j ⊗sym û)C ∑ g( j ∗ (e(û)φ j). j G1

j G1

Then, since ρ is convex, subadditive, and 1-homogeneous, we obtain S

ρ(e(ûδ ))⁄ ∑ ρ(g( j ∗ (∇φ j ⊗sym û)A∇φ j ⊗sym û) j G1

S

C ∑ ρ(g( j ∗ (e(u)φ j)) j G1

S

⁄ ∑ δ 2−jCδCρ(e(û))⁄2δCρ(e(û)).

(15)

j G1

On the other hand, if we consider a point x∈Ω0 , we have ûδ (x)Gg(1 ∗ (φ 1 û)(x) and e(ûδ )(x)G(g(1 ∗ (φ 1 e(û)))(x); therefore, 兩兩ρ(e(ûδ ))兩兩LS(Ω 0) ¤兩兩ρ(e(û))兩兩LS(Ω 0)A2δ , which together with (14) and (15) concludes the second step. Step 3. It remains to show the inclusion ⊆. This is a consequence of the Korn inequalities in W 1,s for 1FsF+S. Indeed, let u∈Lip1, ρ (Ω; Σ )

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and let un be a sequence in that

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D (⺢N ; ⺢N) converging to u uniformly and such

ρ(e(un))⁄1 and un G0 on Σ. If sF+S, using the Korn inequality and the growth conditions on ρ, we have

冮 兩Du 兩 dx⁄C 冮 ( ρ(e(u )) dx⁄C 兩Ω兩. s

s

n

s

Ω

n

s

Ω

We infer that un is weakly precompact in W 1,s ; therefore, u∈W 1,s and e(un) → e(u) weakly in Ls. Since ρs is a convex continuous integrand, the following lower semicontinuity inequality holds:

冤冮 ( ρ(e(u))) 冥

s 1兾s

Ω

⁄lim inf n→S

冤冮 ( ρ(e(u ))) 冥

s 1兾s

n

⁄兩Ω兩1兾s.

Ω

As this is valid for every sH1, the conclusion follows by sending s to infinity. The proof of Theorem 3.1 is now complete. 䊐 In the scalar isotropic case, the characterization of Lip1 (Ω; Σ ) becomes simpler. See the following theorem. Theorem 3.2. In the scalar case, we have Lip1 (Ω; Σ )G{u∈W 1,S (Ω ): uG0 on Σ, 兩Du兩⁄1 a.e. on Ω}. Proof. The proof of the inclusion ⊆ is similar to the third step of the previous theorem. To prove the opposite inclusion, we consider an element u∈W 1,S (Ω ) such that uG0, on Σ, and 兩Du兩⁄1, a.e.. Let dΣ denote the distance function from Σ and define, for every (H0, the function û( Gα ( u, where α ( is the following cutoff function:

冦

0, α ( (x)_ [dΣ (x)A(]兾dΣ (x), 1A1(, Since u vanishes on Σ, we clearly have u(x)⁄dΣ (x).

if dΣ (x)⁄(, if dΣ (x)∈[(,1(], if dΣ (x)¤ 1(.

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Then, it is easy to check that û( vanishes in an (-tubular neighborhood of Σ, converges uniformly to u as ( → 0 and that the norm of the gradient of û( remains bounded by 1. Mollifying û( will provide the requested approximating sequence for u. 䊐 Here, we recall some features of the calculus of variations with respect to a measure, which are needed to give a precise meaning to equations (8) and (9). These notions have been introduced in Ref. 2 and further details can be found also in Refs. 1, 8, 10–12, where this theory has been fruitfully applied. Given µ ∈M (Ω ), we consider the space of tangent vector fields Xµ G{ϕ ∈Lµ1 : −div(ϕµ)∈M (Ω )}, where the divergence operator is intended in the sense of distributions, that is, 〈div(ϕµ),ψ 〉_−

冮 ϕ · ∇ ψ d µ,

∀ψ ∈C 0S (⺢N).

Ω

Then, we define locally the tangent space to µ for µ-a.e. x∈Ω as Tµ (x)GµAess * {ϕ (x): ϕ ∈Xµ }. The µ essential union is defined as a µ-measurable closed multifunction such that: (i) ϕ ∈Xµ ⇒ ϕ (x)∈Tµ (x), for µ-a.e. x∈Ω; (ii) between all the multifunctions with the previous property, the µ essential union is minimal with respect to the inclusion µ-a.e. We notice that, in the case

µ GH k LS, with S a k-dimensional Lipschitz manifold in ⺢N, k⁄n, then Tµ coincides µ-a.e. with the usual tangent bundle TS given by differential geometry. Once we have the notion of tangent space to µ, it is natural to define the notion of µ-tangential gradient of a smooth scalar function u as ∇µ u(x)GPTµ(x) (∇u(x)),

µ-a.e.,

where PTµ(x) denotes the orthogonal projection on Tµ (x). It turns out that the operator ∇µ is closable in Lµp and this allows us to define the Sobolev space W 1,p as the domain of the unique closed extension (see Ref. 2). For µ ¯ , every our purpose, an important issue is that, under the condition spt µ ⊂Ω element of Lip1 (Ω, Σ ) belongs to W 1,p (for every p); therefore, equation (9c) µ is meaningful.

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Similarly to what was done in the scalar case, for µ-a.e. x, we define the linear space of matrices 2

Mµ (x)_ µAess * {Φ(x): Φ∈Xµ (⺢N, ⺢N sym)},

(16)

where 2

2

2 N N 2 N Xµ (⺢N, ⺢N sym)_{Φ∈Lµ (⺢ , ⺢sym): div(Φµ)∈Lµ (Ω, ⺢ )}.

(17)

For every smooth vector field ψ on Ω, the µ-tangential strain eµ (ψ ) can be 2 defined by PMµ (e(ψ )), PMµ being the orthogonal projector from ⺢N sym onto Mµ. In particular, one can show that, if µ is the Lebesgue measure over an open subset of ⺢N, then eµ (ψ ) coincides with the usual strain tensor e(ψ ), while, if µ is the Hausdorff measure H k over a k-dimensional smooth manifold in ⺢N, then eµ (ψ )GPTµ (e(ψ ))PTµ. The operator 2

eµ : Lµp (⺢N ; ⺢N) → Lµp (⺢N, ⺢N sym), defined either on C 0S (⺢N ; ⺢N) or on the space of smooth fields compactly supported in ⺢N \Σ, turns out to be closable with respect to the norm of Lµp and the domain of its unique closed extension gives a Banach space D µp (Ω; ⺢N) of µ-admissible displacements and in the latter case the Banach space D p,0 of µ-admissible displacements vanishing on Σ. Similarly to the µ scalar case, as we have the inclusion of Lip1,ρ (Ω, Σ ) in D p,0 µ , the expression eµ (u) in equation (86) makes sense, where the integrand jµ (x, z) is defined by jµ (x, z)Ginf{j(x, zCξ ): ξ ∈(Mµ (x))⊥}.

(18)

This new integrand jµ , which depends on x through the geometry of µ, plays a crucial role in the relaxation process of the strain problem (1). Briefly, if j satisfies the assumptions (a), (c), (d) of Section 2, then the relaxation in the weak-Lµp topology of the functional defined on smooth functions by 兰Ω j(x, Du) dµ is given by

冮 j (x, ∇ u) dµ, µ

µ

N N if u∈D µp (⺢N ; ⺢N) [resp. D p,0 µ (⺢ ; ⺢ )],

Ω

+S, otherwise, if one starts with functions u vanishing on Σ. Finally, we will need some extension of the notion of convergence in sense of Young measures and some related results which can be found in Ref. 11. This kind of result can be found also in the framework of the theory of varifolds for which we refer to Ref. 13.

D p,0 µ

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Lemma 3.1. Let {µh }⊂ M , with µh % µ. Let d be a positive integer, and let fh ∈(Lµp h)d, with 兩兩 fh 兩兩(Lpµh )d ⁄M for some p∈[1, +S[. Then, the sequence {fh µh } is bounded in M d, and the weak limit of any convergent subsequence is absolutely continuous with respect to µ, with a density f∈(Lµp )d. Moreover, for every convex and lower semicontinuous function Φ: ⺢d → [0, +S[, we have lim inf h→S

冮 Φ( f ) dµ ¤ 冮 Φ( f) dµ. h

h

Definition 3.1. Let µh , µ ∈M +, with µh % µ, and let fh , f: ⺢N → ⺢d be respectively µh and µ integrable. We say that fh weakly converges to f with w( µh , µ) f, if fh µh % fµ in M d. respect to the pair ( µh , µ), and we write fh → w( µh , µ) When fh → f, it is natural to consider the sequence N νh ∈Y (⺢ , µh ; ⺢d) of Young measures associated to fh , which describe the oscillation of fh with respect to the measure µh ; that is,

冮

〈νh , ϕ〉G ϕ (x, fh (x)) dµh (x),

∀ϕ ∈C0 (⺢NB⺢d).

Since {νh } is a bounded sequence, by possibly extracting a subsequence, we may assume that it converges to some measure ν on the product space ⺢NB⺢d whose marginal on the first factor is µ, and the disintegration of ν yields

冮

〈ν, ϕ〉G 〈ϕ (x, · ), νx 〉 dµ (x),

∀ϕ ∈C0 (⺢NB⺢d).

This justifies the following definition. Definition 3.2. We say that {fh } weakly converges to the family of probabilities (νx) in the sense of Young measures with respect to ( µh , µ), Y( µh , µ) (νx), if and we write fh →

冮 ϕ(x, f (x)) dµ (x) → 冮 〈ϕ(x, · ), ν 〉 dµ (x), h

h

x

∀ϕ ∈C0 (⺢NB⺢d).

w( µh , µ) Proposition 3.1. Let fh ∈(Lµ1 h)d, with fh → (νx). Then:

(i)

For every nonnegative lower-semicontinuous function ϕ on ⺢NB⺢d, we have lim inf h→S

冮 ϕ(x, f (x)) dµ (x)¤ 冮 〈ϕ(x, · ), ν 〉 dµ (x). h

h

x

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(ii)

13

Assume in addition that {fh } is uniformly bounded in the (Lµp h)d-norm, for some pH1, and that µh →µ tightly; i.e., µh (⺢N) →µ (⺢N). Then, for any continuous function ϕ on ⺢NB⺢d satisfying a growth condition of the type 兩ϕ (x, z)兩⁄C(1C兩z兩q),

with qFp,

we have lim

h→S

冮 ϕ(x, f (x)) dµ (x)G冮 〈ϕ(x, · ), ν 〉 dµ (x). h

h

x

By applying Proposition 3.1 (ii), with

ϕ (x, z)Gz · Ψ(x) and Ψ∈D (⺢N ; ⺢d), we find that w( µh , µ) f, fh →

where f (x)G[νx ];

here,

冮

[νx ]_ ξ dνx (ξ ) denotes the first momentum of νx. In fact, the family {νx } describes the oscillations of {fh }. Claiming that νx is a Dirac mass at f (x) means that no oscillations occur; this situation motivates the following definition. Definition 3.3. Let µh , µ ∈M +, with µh % µ, and let fh , f: ⺢N → ⺢d be respectively µh and µ integrable. We say that fh strongly converges to f with s( µh , µ) f, if respect to the pair ( µh , µ), and we write fh → lim

h→S

冮 ϕ(x, f (x)) dµ G冮 ϕ(x, f (x)) dµ, h

h

∀ϕ ∈C0 (⺢NB⺢d).

In Lemma 3.2 below, we give a particular case of strong convergence which will be used in Section 4. Lemma 3.2. Let {fp : p∈⺞} be a sequence of measurable nonnegative functions such that 兰Ω 兩 fp 兩p dx⁄C. Set µp G兩 fp 兩pA2 and assume that µp % µ. Then, the following implication holds: lim

p→S

冮 兩 f 兩 dxG lim 冮 兩 f 兩 p

p

Ω

p→S

p

Ω

pA1

s( µh , µ) dx ⇒ fp → 1.

(19)

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Proof. Possibly passing to a subsequence, we may assume that Y (µp , µ) νx . fp →

For every t∈[0, 2], set

冮

h(x, t)_ 兩z兩t dνx . Since

冮 兩 f 兩 dµ ⁄C, 2

p

p

by Proposition 3.1(i) applied with ϕ (x, z)G兩z兩p, we have +SHlim inf p→S

冮 兩 f 兩 dxGlim inf 冮 兩 f 兩 dµ ¤ 冮 h(x, 2) dµ. 2

p

p

p→S

Ω

p

p

Ω

r Ω

Therefore h(x, · ) is continuous on [0, 2] for µ-a.e. x∈Ω. In addition, as Ω ¯ ; applying Proposition is bounded, the convergence of µh to µ is tight on Ω ¯ ), we have 3.1 (ii), we obtain that, for every s∈[0, 2[ and for every Ψ∈C 0 (Ω lim

p→S

冮 兩f 兩

pA2Cs

p

Ψ dxG lim

p→S

Ω

G

冮 兩 f 兩 Ψ dµ s

p

p

Ω

冮 h(x, s)Ψ(x) dµ.

(20)

r Ω

In the previous equalities, we may substitute Ψ with the characteristic function of any ball B such that µ (∂B)G0 and then, by the Ho¨ lder inequality, we obtain that, for every 0⁄t1 Ft2 F2,

冮 h(x, t ) dµG lim 冮 兩 f 兩 2

B

p→S

¤ lim

p→S

p

pA2Ct2

dx

B

冤冮

兩 fp 兩pA2Ct1 dx

B

冥

冮

G h(x, t1) dµ. B

Then, we deduce the monotonicity property h(x, t1)⁄h(x, t2),

(pA2Ct2)兾(pA2Ct1)

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which can be extended to t2 G2 using the continuity of h(x, · ) on [0, 2]. This implies that 1Gh(x, 0)⁄h(x, 1)⁄h(x, 2). Now, thanks to (20), the equality in the right-hand member of (19) implies that

冮 h(x, 1) dµ G冮 h(x, 2) dµ; r Ω

r Ω

hence, there exists a suitable f∈Lµ2 which satisfies, for every t∈[1, 2], ¯. µ a.e. x∈Ω

h(x, t)Gf (x), In particular, we have that 1⁄f (x),

µ-a.e.

On the other hand, by the Jensen inequality and since f(x) is the barycenter of νx ,

冮

f (x)Gh(x, 2)G 兩z兩2ûx (dz)¤( f (x))2. Therefore, f (x)G1,

µ-a.e.,

yielding that the previous Jensen inequality becomes an equality. Thus, νx is the Dirac mass at f(x). 䊐

4. Approximation Result Our approximation procedure for the mass optimization problem focuses on problem (5) introduced in Section 2. From now on, the function ρ is fixed so that it fulfills conditions (a), (b), (c), (d) of Section 2. For every pHN, we consider the p-power version of the strain problem (1), i.e.,

冦

α p _inf (1兾p)

冮 ρ(e(u)) dxA〈 f,u〉: p

冧

u∈W 1,p (Ω, ⺢N), uG0 on Σ ,

Ω

(21)

expecting that, for p tending to infinity, we will have α p →α , where

α _−I( f, Ω, Σ )Ginf{〈Af, u〉: u∈Lip1, ρ (Ω, Σ)}.

(22)

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Through this section, we will assume that the compatibility condition (4) holds ensuring that α is finite; this assumption is needed to have the existence of an optimal mass distribution and then it is not restrictive for our purpose. Under (4), it is easy to check also, using the Korn inequality in W 1,p (see Lemma 4.2 below) and embedding in the space of continuous functions for pHN, that (21) has at least one minimizer up. Moreover, by standard convex analysis methods, taking into account the growth conditions, we can express α p in terms of the dual stress problem,

冦

α p G−min (1兾p′)

冮 (ρ (σ )) 0

p′

冧

dx: σ ∈Lp′ (Ω, ⺢NBN), −divσ Gf, in Ω\Σ .

Ω

(23) Before stating the main result, we complete the picture with the following lemma. Lemma 4.1. Let up be a minimizer of (21). Then, any solution of (23) can be written in the form σ p Gξ p ( ρ(e(up)))pA2, where ξ p is an element of ∂ρ(e(up)). Moreover, the following relationship holds:

冮 ρ(e(u )) dxG〈 f, u 〉G[ p兾(1Ap)]α . p

p

p

(24)

p

Ω

Proof. To prove the first part of this lemma, we just notice that any solution σ p of (23) satisfies the equality (1兾p′)

冮 ρ (σ ) 0

p′

p

dxC(1兾p)

Ω

冮 ρ(e(u )) dxG〈 f, u 〉G冮 σ p

p

p

Ω

p

· e(up) dx,

Ω

from which, by the Young inequality, we derive that

ρ(e(up))ρ0 (σ p)Gσ p · e(up),

ρ0 (σ p)Gρ(e(up))pA1.

(25)

Now, if we write

σ p Gξ p ρ(e(up))pA2, we deduce from (25) that

ρ0 (ξ p)Gρ(e(up)),

ρ0 (ξ p)ρ(e(up))Gξ p · e(up).

Therefore,

ξ p ∈∂ρ(e(up)). In addition, we deduce from (26) that

σ p · e(up)Gξ p · e(up)ρ(e(up))pA2 Gρ(e(up))p.

(26)

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17

Then, since −div σ p Gf holds on Ω\Σ, we obtain after integrating by parts 〈 f, up 〉G

冮 ρ(e(u )) dx. p

p

Ω

The second statement now follows, since up is a solution of (21) and we have

α p G(1兾p)

冮 ρ(e(u )) dxA〈 f, u 〉. p

p

p

䊐

Ω

The aim of this section is to prove the following result. Theorem 4.1. Assume (4) and let {(up , σ p)} be a sequence of minimizers for (21) and (23). Set

µp _ ρ(e(up))pA2,

σ p G ξ p µp .

¯ ), and suitable rigid displaceThen, there exist u∈Lip1,ρ (Ω, Σ ), µ ∈ M (Ω ments rp ∈ R Σ, such that, possibly extracting a subsequence, which we still denote by up , (i) upArp → u, uniformly; ¯ ); (ii) µp % µ, in M (Ω ¯ , ⺢N); (iii) ξ p µp % ξµ, in M (Ω (iv) u is a minimizer of (5) and (u,ξ , µ) is a solution of (8). Moreover, if the set {ρ0 ⁄1} is strictly convex, then the strong convergence s( µp , µ) ξ p → ξ holds in the sense of Definition 3.3. Remark 4.1. In the scalar case of conductivity, Theorem 4.1 can be reformulated as follows: assume that ΣG∅ or that 兰 fG0, and let up be the unique minimizer (up to an additive constant) of −∆ p up Gf,

on Ω\Σ,

up G0,

on Σ,

∂up 兾∂ûG0,

on ∂Ω\Σ.

Then, by possibly extracting a subsequence, we have up → u,

uniformly,

兩∇up 兩pA2 % µ,

¯ ), in M (Ω

兩∇up 兩pA2∇up % ∇µ u,

¯ , ⺢N), in M (Ω

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where (u, µ) solves (9). In addition, there is the strong convergence s( µp , µ) ∇up → ∇µ u.

The proof will be obtained after several lemmas and propositions. Lemma 4.2 (Korn Inequality). Let q∈(1, +S); then, there exists a constant Cq such that, for every û∈W 1,q (Ω, ⺢N), inf{兩兩ûAr兩兩W 1,q(Ω, ⺢N), r∈R Σ}⁄Cq 兩兩e(û)兩兩Lq(Ω, ⺢N 2).

(27)

Proof. The conclusion follows by contradiction considering a sequence {ûh } such that dist(ûh , R Σ)G1,

兩兩e(ûh)兩兩Lq → 0,

and applying the classical Korn inequality in W 1,q (Ω, ⺢N).

䊐

Remark 4.2. The fact that the Korn inequality fails if qG+S or qG1 implies that the optimal constant Cq in (27) tends to infinity as q → S or as q → 1. In the scalar case, (27) is valid replacing e(û) by ∇û and R Σ by the null space if Σ ≠ ∅ or by constant functions if ΣG∅ and Ω is connected. It is straightforward from the Poincare´ inequality that the constant Cq is uniformly bounded in this case. Proposition 4.1 (Γ-Convergence). Let us consider the following func¯ , ⺢N): tionals on C 0 (Ω Fp (û)G(1兾p)

冮 ( ρ(Dû)) dx, p

if û∈W 1,p (Ω, ⺢N), ûG0 on Σ,

Ω

Fp (û)G+S,

otherwise,

FS (û)G0,

if û∈Lip1,ρ (Ω, Σ ),

FS (û)GS,

otherwise.

Then, Fp Γ-converges to FS with respect to the uniform convergence of continuous functions. Proof. As the Γ-lim sup inequality is obvious, we prove only the Γlim inf inequality. Let ûp → û uniformly in Ω and assume without loss of generality that {Fp (ûp)}⁄C. Noticing that, for every tH0, the function s > (tsA1)兾s is monotone increasing on (0, +S), we infer that q⁄p ⇒ Fq (w)⁄Fp (w)C(1兾qA1兾p)兩Ω兩,

w∈W 1,p (Ω, ⺢N).

(28)

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19

Therefore, for every qH1, we have sup Fq (ûp)F+S. p¤q

Then, from the growth conditions and Lemma 4.2, we deduce that ûp belongs to W 1,q for p large enough and that ûp % û in W 1,q. Thanks to the semicontinuity of the convex functional w>

冮 (ρ(e(w))) dx, q

Ω

we obtain

冢冮

(ρ(e(û)))q dx

Ω

冣

1兾q

⁄兩Ω兩1兾qC 1兾q,

∀qH1.

(29)

Passing to the limit in (29) as q → S, we obtain e(û)∈LS and ρ(e(û))⁄1. Since clearly ûG0 on Σ, the conclusion follows from Theorem 3.1.

䊐

Proposition 4.2. Boundedness. Let {up } be a sequence of minimizers for (21). Then, there exists a constant C such that

冮 (ρ(e(u ))) dx⁄C. p

p

(30)

Ω

Proof. By (24), it is enough to prove that α p has a uniform lower bound. We know that α q is finite for every qHN and that, by (28), we have

α q ⁄ α pC(1兾qA1兾p)兩Ω兩. The conclusion then follows.

䊐

Corollary 4.1. For each fixed q∈[1,S[, the family {e(up)} is bounded 2 in Lq (Ω, ⺢N ) and there exist rigid motions {rp } in R Σ such that the family ¯ , ⺢N). {upArp } is bounded in W 1,q (Ω, ⺢N) and then it is precompact in C(Ω Proof. It follows from Proposition 4.2 and from Lemma 4.2.

䊐

Proof of Theorem 4.1. By (30), (26), and the growth condition on ρ, we have that sup p

冮 (1C兩ξ 兩 ) dµ F+S. 2

p

Ω

p

(31)

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Taking into account Corollary 4.1 and Lemma 3.1, we can then assume that, up to subsequences,

µp % µ,

in

M (Ω ),

(32)

ξ p µp % ξµ,

in

M (Ω, ⺢NBN),

(33)

upArp → u,

uniformly on Ω,

(34)

where ξ , rp are suitable elements in Lµ2 (Ω, ⺢NBN) and in R Σ respectively. We have to show that the triplet (u,ξ , µ) is a solution of equation (8) or equivalently (see Section 2) that: (i) (ii)

u is a solution of (5); ρ0 (ξ )G1, µ-a.e., and λ _ ξµ is a solution of (6).

Indeed, under (ii), we deduce from (7) that

µ˜ _mµ兾I (Gmρ0 (λ )兾I ) is optimal for (3) and λ can be written as

λ Gσµ˜ , (i)

with σ GIξ 兾m.

It is a straightforward consequence of (34) and of the Γ-convergence result of Proposition 4.1. In addition, we have the convergence of minima, i.e.,

α p →α , (ii)

α G−I( f, Ω, Σ ).

(35)

By (23) and by using the inequality s 兾p′¤sA(1兾p), sH0, we obtain p′

−α p G(1兾p′)

冮 ( ρ (σ )) 0

p′

p

Ω

dx¤

冮 ρ (σ ) dxA1兾p. 0

p

Ω

Thus, passing to the limit as p → S, with the help of (35), (33), and by using Lemma 3.1, we deduce that Aα ¤lim inf p

0

p

Ω

Glim inf p

¤

冮 ρ (σ ) dx 冮 ρ ( ξ ) dµ 0

p

冮 ρ (ξ ) d µ . 0

r Ω

p

Ω

(36)

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21

Clearly, λ Gξµ satisfies −div λ Gf,

in Ω\Σ,

and so problem (6) whose infimum is Aα GI admits λ as a solution. In turn, the inequalities in (36) become equalities and we obtain −α G lim

冮 ρ (ξ ) d µ G 冮 ρ (ξ ) d µ . 0

p→S

0

p

(37)

p

Ω

r Ω

Let us now prove that ρ0 (ξ )G1 holds µ-a.e.. This is a consequence of Lemma 3.2, which we apply to the sequence fp _ ρ0 (ξ p). Indeed, by (26), we have fp Gρ0 (ξ p)Gρ(e(up)), hence

µp Gf ppA2

and

冮 兩 f 兩 dxF+S p

p

Ω

by (30). Moreover, by (24) and (37), we have lim

p→S

冮 兩 f 兩 dµ G lim 冮 (ρ (ξ )) dµ 2

0

p

p

Ω

G lim

p→S

G lim

p→S

2

p

p→S

p

Ω

冮 ρ (ξ ) d µ 0

p

p

Ω

冮 兩 f 兩 dµ . p

p

(38)

Ω

We derive from (19) that ρ0 (ξ p) converges (strongly) to 1. Applying the assertion (i) of Proposition 3.1, with ϕ (x, z)Gθ (x)ρ0 (z), where θ is a nonnegative test function and applying the Jensen inequality, we infer that ρ0 (ξ )⁄1, µ-a.e. holds. The claim (ii) now follows, since by (37) we have

冮

¯ )G ρ0 (ξ ) dµ. −α Gµ (Ω To conclude the proof, apply again assertion (i) of Proposition 3.1 with ϕ (x, z)G( ρ0)2 (z). Then, the equalities in (38) imply that ( ρ0)2 satisfies the Jensen equality with respect to the family of Young measures {νx } generated by {ξ p }. If the set {ρ0 ⁄1} is strictly convex, the strict convexity of ( ρ0)2 yields that νx Gδ ξ (x) , that is the strong convergence of {ξ p }. 䊐

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5. Examples The main feature of Theorem 4.1 is the convergence of the measures that we denoted by µp to a solution µ of the shape optimization problem (up to a normalization factor). We remark that, in many cases, the mass optimization problem that we are considering has many solutions (see for example Ref. 1). An important issue discovered in Ref. 1 is that, in the scalar case, such optimal solutions can be written explicitly in terms of optimal transport plans in the following way: we define dΩ, Σ as follows: dΩ, Σ (x, y)Gmin{dΩ (x, y), dΩ (x, Σ )CdΩ ( y, Σ )}, where dΩ denotes the geodesic distance in Ω. Let γ be an optimal transport plan for the following problem: min

PΣ (f +,f −)

冮

dΩ, Σ (x, y) dγ (x, y).

(39)

ΩBΩ

For each pair (x, y)∈spt(γ ), let [x, y] denote the geodesic line joining x and ¯ . For every Borel set A⊂Ω, if we define y in Ω

µ (A)_

冮

H 1 (A∩[x, y]) dγ (x, y),

(40)

r BΩ r Ω

then µ is an optimal mass distribution (see Ref. 1, Theorem 4.6) and for each solution u of (5) the triplet (u, Du, µ) is a solution of the scalar version of (8). Moreover in Ref. 6 it was proved (under the assumption Ω convex and ΣG∅) that, for all the optimal mass distribution µ, there exists a suitable optimal plan γ such that µ is the optimal measure associated to γ through formula (40). Still in the same paper (Ref. 6), it is also proved that, if in addition f∈L1, then there is only one optimal mass distribution µ. As we will see, in many situations, guessing an optimal plan is much simpler than guessing the optimal mass distributions. Here, we would like to discuss some cases in which it is possible to prove that the whole sequence µp converges to a particular solution of the limit problem. Example 5.1. Consider the one-dimensional case, NG1,

Ω_[−1, 1],

fGdx L (−1, 0)Adx L (0, 1).

In this case, the p-Laplacian equation is reduced to −(兩up ′兩pA2up ′)′Gsign x,

on ]A1, 1[,

(41)

whose unique solution is given by up G[ p兾( pA1)] sign x[1A(1A兩x兩)p兾(pA1)].

(42)

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23

Then clearly, the unique limit of 兩up ′兩pA2 dx is given by µ G(1A兩x兩) dx, which is also the unique solution of the mass optimization problem. Example 5.2. Consider ΩG⺢2 and fG{δ (−1,1)Cδ (1, −1)Aδ (1,1)Aδ (−1, −1)}. In this case, formula (40) implies that all the solutions µ of the limit problem are supported on the boundary of the unitary square of ⺢2 and can be written as follows. We denote by A the segment [(−1,1), (1,1)], by B the segment [(1,1), (1, −1)], by C the segment [(1, −1), (−1, −1)] and finally by D the segment [(−1, −1), (−1,1)]. Then, given two numbers α and β in the interval [0, 1], we can write an optimal measure µ as follows:

µ Gα H 1 L AC(1Aα )H 1 L DCβ H 1 L BC(1Aβ )H 1 L C.

(43)

However, the symmetry of the data implies that the solutions of the approximating problems up are invariant with respect to the action of π rotations and of symmetry with respect to the lines yGx and yG−x. It then follows that the solution µ selected by the p-Laplacian is invariant with respect to the same symmetries and therefore is unique and given by

µ G(1兾2)H 1 L AC(1兾2)H 1 L DC(1兾2)H 1 L BC(1兾2)H 1 L C. (44) Example 5.3. Here, we consider the Dirichlet problem ΩGB(0, 1),

NG2,

ΣG∂Ω,

fGδ (0,0).

Here, the transport plans are of the form

γ Gδ 0,0 ⊗ α , α being a probability measure on Σ. Since the distance from (0, 0) is constant on Σ, they are all optimal and then according to formula (40) produce the optimal mass distribution µ given by µ (A)G

冮H

1

(Sx ∩A) dα (x),

(45)

Σ

where Sx denotes the segment [0, x]. Making the probability measure α vary in formula (45) we are led to an infinite family of solutions of the mass optimization problem. However, the invariance of the data with respect to the action of the group of the rotation of ⺢N gives the invariance of (up , µp) with respect to the action of the same group. It follows that, among

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all the solution given by formula (45), the sequence µp selects the one given by the uniform probability α G(1兾2π )H 1 L Σ. Remark 5.1. In the same setting of the previous example, we could consider a Dirichlet condition assigned only on a subset Γ of the boundary ∂Ω. In this case, the approximating solutions up and µp are not radial anymore; indeed, the radial solutions do not satisfy the Neumann homogeneous conditions (required by the minimization problem) on the remaining part of the boundary ∂Ω\Γ. This very simple variant of Example 5.3 particularizes very well the open problem of understanding if the whole sequence µp converges to some µ (which is of course an optimal mass distribution), and then of finding a selection criterium in order to identify this particular solution µ.

References 1. BOUCHITTE´ , G., and BUTTAZZO, G., Characterization of Optimal Shapes and Masses through the Monge–Kantoroûich Equation, Journal of the European Mathematical Society, Vol. 3, pp. 139–168, 2001. 2. BOUCHITTE´ , G., BUTTAZZO, G., and SEPPECHER, P., Energy with Respect to a Measure and Applications to Low-Dimensional Structures, Calculus of Variations and Partial Differential Equations, Vol. 5, pp. 37–54, 1997. 3. EVANS, L. C., and GANGBO, W., Differential Equations Methods for the Monge– Kantoroûich Mass Transfer Problem, Memoires of the American Mathematical Society, Providence, Rhode Island, Vol. 137, 1999. 4. EVANS, L. C., Partial Differential Equations and Monge–Kantoroûich Mass Transfer, Current Developments in Mathematics 1997, International Press, Boston, Massachusetts, pp. 65–126, 1999. 5. FELDMAN, M., and MCCANN, R. J., Monge’s Transport Problem on a Riemannian Manifold, Transactions of the American Mathematical Society, Vol. 354, pp. 1667–1697, 2002. 6. AMBROSIO, L., Lecture Notes on Optimal Transport Problems, Lectures given in Madeira, Portugal, 2–9 July 2000. Preprint available on http:兾兾cvgmt.sns.it兾 people兾ambrosio兾. 7. GOLAY, F., and SEPPECHER, P., Locking Materials and the Topology of Optimal Shapes, European Journal of Mechanics兾Solids, Vol. 20A, pp. 631–644, 2001. 8. BOUCHITTE´ , G., BUTTAZZO, G., and SEPPECHER, P., Shape Optimization Solutions ûia Monge–Kantoroûich Equation, Comptes Rendus de l’ Academie de Sciences de Paris, Vol. 324, pp. 1185–1191, 1997. 9. DEMENGEL, F., De´ placements a´ De´ formations Borne´ es et Champs de Contrainte Mesures, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Vol. 12, pp. 243–318, 1985.

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10. BOUCHITTE´ , G., BUTTAZZO, G., and FRAGALA´ , I., Mean Curûature of a Measure and Related Variational Problems, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Vol. 25, pp. 179–196, 1997. 11. BOUCHITTE´ , G., BUTTAZZO, G., and FRAGALA´ , I., Conûergence of Soboleû Spaces on Varying Manifolds, Journal of Geometric Analysis, Vol. 11, pp. 399– 422, 2001. 12. FRAGALA´ , I., and MANTEGAZZA, C., On Some Notion of Tangent Space to a Measure, Proceedings of the Royal Society of Edinburgh, Mathematics, Vol. 129A, pp. 331–342, 1999. 13. HUTCHINSON, J. E., Second Fundamental Form for Varifolds and the Existence of Surfaces Minimizing Curûature, Indiana University Mathematics Journal, Vol. 35, pp. 45–71, 1986. 14. ALLAIRE, G., and KOHN, R. V., Optimal Design for Minimum Weight and Compliance in Plane Stress Using Extremal Microstructures, European Journal of Mechanics兾Solids, Vol. 12A, pp. 839–878, 1993.