A dual variational approach to a class of nonlocal semilinear Tricomi problems

A DUAL VARIATIONAL APPROACH TO A CLASS OF NONLOCAL SEMILINEAR TRICOMI PROBLEMS Daniela LUPO Dipartimento di Matematica, Politecnico di Milano Piazza Leonardo da Vinci, 32 20133 Milano, Italy e-mail: [email protected] Kevin R. PAYNE Department of Mathematics and Computer Science University of Miami Coral Gables, FL 33124-4250 e-mail: [email protected] Abstract The existence of at least one nontrivial solution to a class of semilinear Tricomi problems is established via an application of the dual variational method which captures the solution as the preimage of a minimum of a suitable dual action functional. The boundary conditions are homogeneous Dirichlet conditions on a suitable part of the boundary, as dictated by uniqueness theorems for the linear problem. While there are good compactness properties for the inverse operator for the linear problem, there is a manifest asymmetry in the linear part due to the form of the boundary conditions. The linear part is symmetrized by introducing suitable reflection operators on symmetric domains, which then results in a nonlocal character of the nonlinearity.

1

Introduction and statement of the main result

In this work, we are interested in the use of variational methods to establish the existence of solutions in a suitably generalized sense to the following nonlocal semilinear Tricomi problem

1



T u ≡ −yuxx − uyy = R (sign(u)|u|p ) u=0

in Ω on AC ∪ σ,

(N ST )

where p ∈ R with 0 < p < 1, T ≡ −y∂x2 − ∂y2 is the Tricomi operator on R2 , R is the reflection operator on L2 (Ω) induced by composition with the map Φ : R2 → R2 defined by Φ(x, y) = (−x, y), and Ω is a bounded region in R2 that is symmetric with respect to the y-axis and has a piecewise smooth boundary ∂Ω of the classical Tricomi form. That is, ∂Ω consists of a smooth arc σ in the elliptic region y > 0, with endpoints on the x-axis at A = (−x0 , 0) and B = (x0 , 0), and two characteristic arcs AC and BC for the Tricomi operator in the hyperbolic region y < 0 issuing from A and B and meeting at the point C on the y-axis. One knows that 2 2 AC : (x + x0 ) − (−y)3/2 = 0 and BC : (x − x0 ) + (−y)3/2 = 0. 3 3 The form of the problem (N ST ) arises from various considerations. In a general sense, we are interested in the use of variational methods for boundary value problems involving mixed (elliptic-hyperbolic) type partial differential equations. Our motivation is twofold. On the one hand, there are interesting physical problems such as transonic potential flow past profiles which are modeled by nonlinear mixed type boundary value problems which admit variational characterizations (cf. section 4 of [6]). On the other hand, global variational treatments of mixed type problems would improve the basic understanding as to why solutions to such problems should exist at all. Even for linear problems, the techniques employed often involve pasting together solutions found independently in elliptic and hyperbolic regions; with notable exceptions such as works based upon the positive symmetric systems technique of Friedrichs (cf. [12], [16]). Variational tools would provide another approach which is independent of type, with some added ability to interpret the results. The results presented here can be thought of a first step towards understanding the obstructions to variational formulations for mixed type problems and proposing a possible method for their resolution and represent the first variational treatment of a nonlinear Tricomi problem. More precisely we consider a mixed type problem whose linear part is most well understood, the seminal linear Tricomi problem [22]  T u = f in Ω (LT ) u = 0 on AC ∪ σ , and proceed to add to it the mildest kind of nonlinear structure, namely a semilinear term. The problem (LT ) has long been connected to the problem of transonic nozzle flow through the pioneering work of Frankl’ [11]. The term Tricomi problem refers to the placement of the boundary data on only the portion AC ∪ σ of the 2

boundary; such a boundary condition is chosen because the presence of a hyperbolic region will overdetermine the problem for classical solutions if one attempts to place data on the entire boundary (cf. [3] , for a maximum principle argument). Under some restrictions on σ, there is a wealth of results on the linear Tricomi problem, including the existence of unique strong solutions in Hilbert spaces well adapted to this boundary condition (cf. the paragraph following Theorem 1.2 for remarks about other boundary conditions). However, it should be noted that despite some 70 years of study, basic information remains unknown; for example, to our knowledge, the only established spectral result is the existence of one positive eigenvalue for the Tricomi operator supplemented with the Tricomi boundary condition (cf. [13] ). The absence of spectral theory results has important consequences for variational approaches as will be described in the next paragraph. The main difficulty in using variational methods for semilinear Tricomi problems is a manifest asymmetry in the operator T that results from placing the boundary conditions on only a portion of the boundary (cf. the discussion following Proposition 2.4). In cleanest terms, T does not map a reasonable Hilbert space into its dual, but rather into the dual of the adjoint problem, in which vanishing data is placed on BC ∪ σ. Our approach involves symmetrizing the linear operator T by first assuming that Ω is symmetric and then by composing T with the reflection R, which induces an isometric isomorphism between the adjoint boundary spaces. In this way, RT does map a natural Hilbert space into its dual, and hence (N ST ) will admit a variational structure. One could consider a direct variational approach to the symmetrized problem, but this remains problematic since crucial information on the linear Tricomi operator remains unavailable for the study of the direct functional. For example, the aforementioned lack of spectral information on the Tricomi operator makes it difficult to obtain either much geometrical information on or needed compactness properties for the direct functional, which is strongly indefinite. The use of dual variational methods allows us to solve the compactness properties problem by taking advantage of the compactness of the inverse of the linear operator. In fact, the linear operator RT does possess a priori estimates with the loss of one derivative and hence it admits an inverse (RT )−1 which is compact on L2 (Ω). The sublinear growth of the nonlinearity f allows us to overcome the geometrical difficulties, and the choice of the class of nonlinearities in (N ST ) is intended as an explicit example. Sufficient knowledge of the spectrum of the linear operator would also allow one to treat other kinds of nonlinearities as well as to obtain multiplicity results. Such a program will be the subject of a forthcoming paper (cf. [15] ). On the other hand, it should be noted that the necessity of obtaining the continuity of the Nemistki operator associated to the nonlinearity constrains one, given the present knowledge of the linear operator, to consider nonlinearities with at most an asymptotically linear growth. In order to treat superlinear growth cases, one would need an appropriate Lp theory for the linear Tricomi operator, which is not present in the literature. With this introduction, we now state the main result. We denote by CΓ∞ (Ω) the set of all smooth functions on Ω such that u ≡ 0 on Γ, by WΓ1 the closure 3

of CΓ∞ (Ω) with respect to the W 1,2 (Ω) norm, and by WΓ−1 the dual of WΓ1 . The 1 space WAC∪σ is a Hilbert space in which one can find solutions u to T u = f for all f ∈ L2 (Ω) in a strong sense (recalled in section 2). We will find generalized solutions u to (N ST ) in the following sense. 1 Definition 1.1. One says that u ∈ WAC∪σ is a generalized solution of (N ST ) if and only if

T u = R (sign(u)|u|p ) in L2 (Ω),

(1.1)

∞ and there exists a sequence {uj } ⊂ CAC∪σ (Ω) such that

1 lim kuj − ukWAC∪σ = 0 and

j→∞

lim kT uj − R (sign(uj )|uj |p ) kW −1

j→∞

= 0. (1.2)

BC∪σ

We remark that in the condition (1.1) above, the operator T is actually TAC , an 1 extension by continuity of T to WAC∪σ as described in section 2. Our main result is the following theorem, where we note that u ≡ 0 is always a solution of (N ST ). Theorem 1.2 Let Ω be an admissible Tricomi domain in the sense of Definition 2.1 that is symmetric with respect to the y axis and p ∈ (0, 1). Then the problem (N ST ) admits at least one nontrivial generalized solution in the sense of Definition 1.1. We conclude with a few additional remarks about the problem (N ST ). First, one could equally well consider boundary conditions of Guderley-Morawetz type, in which data is placed on ∂Ω\Γ where Γ is a characteristic gap (cf. Theorems 1,2 of [17] ) , where the overdetermining nature of the hyperbolicity continues to be respected. On the other hand, a boundary condition of Dirichlet type on the entire boundary, which is natural for flows past profiles, involves the a priori treatment of singularities which must be present and hence the behavior of this linear problem is much less well understood (cf. [17], [18] .. Second, one could try to follow the path of variational methods for nonpotential operators (cf. [10] ), but such an approach has not been carried out for a nonlinear mixed type problem, and, moreover, it is not clear what kind of nonlinear problem is actually amenable to this approach. Third, there are non variational treatments to semilinear Tricomi problems (cf. [19], [4] and references therein). Finally, while we cannot say that the presence of the nonlocal effect in (N ST ) results directly from physical reasoning for example, there are reasons to believe that the problem is sound. Not only does it possess a variational structure, but it is possible that the corresponding problem without the reflection possesses only the trivial solution. This is, in fact, the case for sublinear increasing nonlinearities that are C 1 as follows from the uniqueness theorems of [20] (our nonlinearities are not Lipschitz near zero). These considerations are analogous with the problem

4

of finding periodic orbits of Hamiltonian systems, for example. The presence of the symplectic matrix J in the system z˙ = J∇H(z) ensures a variational structure as well as the existence of nontrivial periodic solutions, which fail to be true otherwise. At the very least, (N ST ) gives a new situation in which the dual variational method applies and the resolution proceeds along familiar lines once one has sufficient control on the linear problem.

2

The linear results

In this section, we consider the linear Tricomi problem:  T u = f in Ω u = 0 on AC ∪ σ,

(LT )

where T = −y∂x2 − ∂y2 is the Tricomi operator and Ω is a Tricomi domain. Such a domain Ω is an open, bounded, and simply connected subset of R2 with boundary ∂Ω = σ ∪ AC ∪ BC, where AC and BC are the characteristics of negative and positive slopes respectively issuing from the points A = (x1 , 0) and B = (x2 , 0), with x1 < x2 , which meet at the point C in the hyperbolic region {y < 0}. The curve σ is a piecewise C 2 simple arc that joins A to B in the elliptic region {y > 0}. The main tools are the classical (a, b, c)-integral method of Friedrichs and spaces of positive and negative norms in the sense of Leray and Lax, as developed by Berezanskii [5] and Didenko [8]. ∞ ∞ (Ω) and CBC∪σ (Ω) the subspaces of C ∞ (Ω) whose eleWe denote by CAC∪σ 1 1 ments vanish on AC ∪ σ and BC ∪ σ respectively and by WAC∪σ and WBC∪σ their 1,2 closures with respect the the W (Ω) norm. The theory of spaces with negative −1 −1 1 1 norms [14] shows that the dual spaces WAC∪σ and WBC∪σ of WAC∪σ and WBC∪σ 2 can be characterized as norm closures of L (Ω) with respect to a norm such as kwkW −1

AC∪σ

=

sup 1 06=ϕ∈WAC∪σ

|(w, ϕ)L2 | , 1 kϕkWAC∪σ

(2.1)

1 where k · kWAC∪σ = k · kW 1,2 (Ω) and (·, ·)L2 is the standard inner product on L2 (Ω). −1 The norm on WBC∪σ is defined in an analogous way and one obtains rigged triples of Hilbert spaces with inclusion chains such as

−1 1 WAC∪σ ⊂ L2 (Ω) ⊂ WAC∪σ

−1 1 and WBC∪σ ⊂ L2 (Ω) ⊂ WBC∪σ .

One has generalized Cauchy-Schwarz (GCS) inequalities such as |hw, ϕiAC | ≤ kwkW −1

AC∪σ

1 kϕkWAC∪σ ,

−1 1 w ∈ WAC∪σ , ϕ ∈ WAC∪σ ,

(2.2)

where h·, ·iAC is the duality pairing which can be computed by limj→∞ (wj , ϕ)L2 −1 for some sequence {wj } ⊂ L2 (Ω) converging to w in the WAC∪σ norm. 5

For a generic Tricomi domain, an application of the divergence theorem yields the fundamental identity R (T u, v)L2 = Ω (yux vx + uy vy ) dxdy (2.3) ∞ ∞ = (u, T v)L2 , u ∈ CAC∪σ (Ω), v ∈ CBC∪σ (Ω), where the simple vanishing of u and v on requisite pieces of the boundary annihilates the boundary integrals; those integrals along the characteristics involve integrands which are proportional to directional derivatives along the characteristics. Applying the GCS inequality and the definitions of the norms to the identity (2.3) shows that there exist positive constants C1 , C2 such that kTAC ukW −1

BC∪σ

1 ≤ C1 kukWAC∪σ ,

1 u ∈ WAC∪σ

(2.4)

1 v ∈ WBC∪σ ,

(2.5)

and kTBC vkW −1

AC∪σ

1 ≤ C2 kvkWBC∪σ ,

where TAC and TBC are the unique continuous extensions of T relative to the ∞ ∞ (Ω) respectively. dense subspaces CAC∪σ (Ω) and CBC∪σ In order to obtain solvability results for the problem (LT ) and its adjoint problem (LT )∗ , in which the formal transpose T 0 of T is again T and the boundary conditions are placed on BC ∪ σ, one exploits the fact that a priori estimates with the loss of one derivative are often possible to establish. We encode this principle into the following definition. Definition 2.1. A Tricomi domain Ω will be called admissible if there exist positive constants C3 and C4 such that kukL2 ≤ C3 kTAC ukW −1

,

1 u ∈ WAC∪σ

(2.6)

kvkL2 ≤ C4 kTBC vkW −1

,

1 v ∈ WBC∪σ .

(2.7)

BC∪σ

and AC∪σ

−1 −1 Such L2 − WAC∪σ and L2 − WBC∪σ a priori estimates were first obtained by Didenko [8] using a mild variant of the classical (a, b, c)-integral method in which one seeks to estimate from below the quadratic integral (Du, T u)L2 in u, where Du = aux + buy + c is a first order differential operator whose coefficients (a, b, c) are to be chosen in such a way as to make the quadratic integral non-negative. Didenko’s variation is to estimate instead (u, T Du)L2 which yields estimates (2.6)− (2.7) at one unit of regularity lower that the classical method can give directly. The following proposition gives a class of admissible Tricomi domains suggested by [8], whose proof will be outlined in the appendix.

6

Proposition 2.2. Let Ω be a Tricomi domain with A = (−x0 , 0) and B = (x0 , 0) Assume that Ω and its boundary ∂Ω = σ ∪ AC ∪ BC satisfy the following technical conditions. (i) |x| ≤ x0 on Ω (ii) The arc σ is given by a C 2 graph: y = g(x) for − x0 ≤ x ≤ x0 with g(−x0 ) = 0 = g(x0 ) (iii) There exists a positive constant h with h < (3x0 /2)−1/3 such that −h < g 0 (x) < h for − x0 < x < x0 . Then Ω is admissible in the sense of Definition (2.1). The technical condition (i) is redundant, but is used throughout the proof and hence listed in the statement. The quantity (3x0 /2)−1/3 in hypothesis (iii) is just (−yC )1/2 , where the point C = (0, yC ). The a priori estimates (2.6) − (2.7) plus standard functional analysis yield the following result on solvability [8]. Proposition 2.3. Let Ω be an admissible Tricomi domain. Then for every f ∈ 1 L2 (Ω) there exists a unique strong solution u ∈ WAC∪σ to the problem (LT ) in the ∞ following sense: there exists a sequence {uj } ⊂ CAC∪σ (Ω) such that 1 lim kuj − ukWAC∪σ = 0 and

j→∞

lim kT uj − f kW −1

j→∞

= 0.

BC∪σ

An analogous statement holds for the adjoint problem (LT )∗ . We mention only the main points in the argument. The estimate (2.6) plus 1 the injectivity of the inclusion WAC∪σ ,→ L2 (Ω) gives the uniqueness. The estimate (2.7) sets up a Riesz representation theorem argument to show the existence of a 1 weak solution u ∈ WAC∪σ in the sense that hf, viBC = hTBC v, uiBC ,

1 v ∈ WBC∪σ ,

∗ where TBC turns out to be TAC . Then a continuity argument exploiting the density ∞ 1 of CAC∪σ (Ω) in WAC∪σ is used to show that the weak solution must be a strong solution.

This solvability result, together with the estimates (2.6)−(2.7) and the closed −1 graph theorem, allows one to define a continuous left inverse TAC which takes 1 values in a proper subspace of WAC∪σ . More precisely, one considers the restriction TAC |W : W → L2 (Ω),

7

(2.8)

where W is the subspace 1 W = {u ∈ WAC∪σ : TAC ∈ L2 (Ω)}.

(2.9)

For u to belong to W it is necessary and sufficient for there to exist a sequence ∞ {uj } ⊂ CAC∪σ (Ω) and a function f ∈ L2 (Ω) such that 1 = 0 and lim kuj − ukWAC∪σ

j→∞

lim kT uj − f kW −1

j→∞

= 0.

(2.10)

BC∪σ

1 1 The subspace W is dense in WAC∪σ as W contains the dense subspace WAC∪σ ∩ 2,2 1 ∞ W (Ω) of WAC∪σ . Moreover, C0 (Ω) ⊂ W since

TAC ([[u]]) = [T (u)],

u ∈ C0∞ (Ω),

1 where [·] and [[·]] are L2 and WAC∪σ classes respectively, and defining f := T u ∈ ∞ ∞ C0 (Ω) for u ∈ C0 (Ω) gives [[u]] as the unique strong solution to TAC u = f for the class [f ]. −1 Now, given f ∈ L2 (Ω), one defines the image of the left inverse TAC (f ) in 1 the obvious way as the unique u ∈ WAC∪σ such that (2.10) holds. One can show that 1 TAC : (W, k · kWAC∪σ ) → (L2 (Ω), k · kL2 )

−1 is closed, and hence TAC is continuous by the closed graph theorem. Finally, there is 1 1 a continuous injection WAC∪σ ,→ L2 (Ω), that WAC∪σ inherits as a closed subspace 1,2 of W (Ω), which is compact by the Rellich theorem, which completes the proof of the following proposition.

Proposition 2.4. Let Ω be an admissible Tricomi domain. Then the operator 1 TAC |W : W ⊂ WAC∪σ → L2 (Ω),

where W is the subspace defined by (2.9), satisfies the following properties. −1 1 (a) TAC |W admits a continuous left inverse TAC : L2 (Ω) → W ⊂ WAC∪σ . −1 (b) TAC : L2 (Ω) → L2 (Ω) is a compact operator.

We have abused notation slightly in part (b) of the proposition by not indicating the compact injection mentioned above, but this will create no difficulties in what follows. It is this compactness property on L2 (Ω) which suggests the use of a dual variational method for the nonlinear problem. However, the identity (2.3) points to a fundamental asymmetry that results from the imposition of the boundary conditions on only a portion of the boundary. The closest statements to TAC being symmetric that one can make here are

8

1 1 u ∈ WAC∪σ , v ∈ WBC∪σ ,

hTAC u, viBC = hu, TBC viAC ,

(2.11)

or (TAC u, v)L2 = (u, TBC v)L2 ,

2 2 u ∈ WAC∪σ , v ∈ WBC∪σ ,

(2.12)

2 2 ∞ ∞ where WAC∪σ and WBC∪σ are W 2,2 (Ω) norm closures of CAC∪σ (Ω) and CBC∪σ (Ω) respectively (cf. [8] for (2.11) and [5] for (2.12)). In order to circumnavigate this asymmetry, we will consider from now on symmetric admissible Tricomi domains; that is, admissible domains in the sense of Definition 2.1 that are symmetric with respect to the y-axis. The statement of Proposition 2.2 clearly allows for such domains. On such symmetric domains, we will introduce a reflection operator in the obvious way which effectively symmetrizes the Tricomi operator TAC for use in the variational method. We consider the linear map

Φ : R2 → R2 with Φ(x, y) = (−x, y),

(2.13)

and the induced operator R : L2 (Ω) → L2 (Ω) with Ru = u ◦ Φ.

(2.14)

Clearly, R is a norm preserving, self-adjoint automorphism on L2 (Ω) such that 1 R2 = Id. Moreover, by restriction, R establishes an isomorphism between WAC∪σ 1 1 1 and WBC∪σ and for u ∈ WAC∪σ and v ∈ WBC∪σ one has 1 1 1 1 kRukWBC∪σ = kukWAC∪σ and kRvkWAC∪σ = kvkWBC∪σ .

(2.15)

One can easily verify the following proposition. Proposition 2.5. Let Ω be a symmetric admissible Tricomi domain and let R be the reflection operator defined by (2.13) and (2.14). Then the operator gotten by composing TAC with R 1 RTAC : W ⊂ WAC∪σ → L2 (Ω),

(2.16)

1 D(TAC ) = W = {w ∈ WAC∪σ : TAC w ∈ L2 (Ω)}.

(2.17)

where satisfies the following properties. (a) RTAC is a closed, densely defined operator which admits a continuous left inverse 1 (RTAC )−1 : L2 (Ω) → W ⊂ WAC∪σ . (b) (RTAC )−1 : L2 (Ω) → L2 (Ω) is a compact operator

9

(c) RTAC is symmetric in the sense that (RTAC u, v)L2 = (u, RTAC v)L2 ,

u, v ∈ W = D(RTAC ).

(2.18)

Proof: Parts (a) and (b) are immediate consequences of Proposition 2.4 and the continuity of R. For part (c), we begin by noting that if u, v ∈ W , then by the characterization (2.10) there exist functions f, g ∈ L2 (Ω) and sequences {uj }, {vj } ∞ in CAC∪σ (Ω) such that 1 1 lim kuj − ukWAC∪σ = lim kvj − vkWAC∪σ = 0

j→∞

j→∞

(2.19) lim kT uj − f kW −1

j→∞

BC∪σ

= lim kT vj − gkW −1 j→∞

= 0.

BC∪σ

It is clear that j ∈ N,

(RT uj , vj )L2 = (uj , RT vj )L2 ,

(2.20)

∞ since R is self-adjoint, Rvj ∈ CBC∪σ (Ω) allows one to apply (2.3), and R commutes ∞ with T on C (Ω). We claim that both sides of formula (2.20) define Cauchy sequences in R, and hence converge. Indeed, for the left hand member, the GCS inequality yields

|(RT uj , vj )L2 − (RT uk , vk )L2 | ≤ kRT (uj − uk )kW −1

AC∪σ

+

kRT uk kW −1

AC∪σ

1 kvj kWAC∪σ

1 kvj − vk kWAC∪σ .

(2.21)

−1 −1 Using the definitions of the norms on WAC∪σ and WBC∪σ , one can easily show that

kRwkW −1

AC∪σ

= kwkW −1

BC∪σ

and kRwkW −1

BC∪σ

= kwkW −1

,

AC∪σ

w ∈ L2 (Ω), (2.22)

and hence that the right hand side of (2.21) is equal to kT (uj − uk )kW −1

BC∪σ

1 kvj kWAC∪σ + kT (uk )kW −1

BC∪σ

1 kvj − vk kWAC∪σ .

By (2.19), one knows that {T uj } and {vj } are Cauchy sequences and bounded with respect to the needed norms, and hence the claim. The same argument works for the right hand member of (2.20). Finally, a similar argument using (2.19), (2.22), and the GCS inequality allows one to show that lim (RT uj , vj )L2 = (f, v)L2 = (RTAC u, v)L2

j→∞

10

and lim (uj , RT vj )L2 = (u, g)L2 = (u, RTAC v)L2

j→∞

and part (c) of the proposition follows. −1 As a final note, we remark that (RTAC )−1 = TAC ◦ R and that u = (RTAC )−1 v 2 ∞ for v ∈ L (Ω) means that there exists a sequence {uj } ⊂ CAC∪σ (Ω) such that 1 lim kuj − ukWAC∪σ = 0 and

j→∞

3

lim kT uj − RvkW −1

j→∞

= 0.

BC∪σ

The nonlinear results

In all that follows, Ω will be a symmetric admissible Tricomi domain so that all of the results of section 2 will apply. Our first task is to transport the L2 -based operator theory for (RTAC )−1 into Lp -based statements, which are well suited for the sublinear nonlinearity f (u) = sign(u)|u|p , with 0 < p < 1. More precisely, we will exploit mapping properties between E = L(p+1)/p (Ω) and E 0 = Lp+1 (Ω), where (p + 1)/p and p + 1 are conjugate exponents. Since p + 1 < 2 < (p + 1)/p and Ω is bounded, one has continuous inclusion maps i and j such that j

i

L(p+1)/p (Ω) ,→ L2 (Ω) ,→ Lp+1 (Ω).

(3.1)

We will denote by k·kq the norm on Lq (Ω) and we record the following proposition. Proposition 3.1. Let Ω be a symmetric admissible Tricomi domain and p ∈ (0, 1). Consider the operator K defined by K = j ◦ (RTAC )−1 ◦ i : L(p+1)/p (Ω) → Lp+1 (Ω),

(3.2)

−1

where (RTAC ) is the operator of Proposition 2.5 and i and j are the inclusion maps of (3.1). Then K satisfies the following properties. (a) K is a linear continuous operator; that is, there exists a constant C1 > 0 such that kKvkp+1 ≤ C1 kvk(p+1)/p ,

v ∈ L(p+1)/p (Ω).

(3.3)

(b) K is a compact operator. (c) K is a symmetric operator in the sense that Z Z vKw dxdy = wKv dxdy v, w ∈ L(p+1)/p (Ω). Ω

Ω

11

(3.4)

Proof. Parts (a) and (b) are immediate since K is the composition of (RTAC )−1 , which is compact as an operator on L2 (Ω), with the continuous inclusions i and j. For part (c), one notices that Z

vKw dxdy

Ω


v j ◦ (RTAC )−1 ◦ i (w) dxdy
R = Ω i(v) (RTAC )−1 ◦ i (w) dxdy,

=

R

Ω

(3.5)

since the inclusions i and j are just transposes of one another. Now, since i(v), i(w) ∈ L2 (Ω), Proposition 2.5(a) shows that there exist unique elements ϕ, ψ ∈ W = D(RTAC ) so that RTAC ϕ = i(v) and RTAC ψ = i(w), and hence (3.5) becomes Z Z Z vKw dxdy = ψRTAC ϕ dxdy = ϕRTAC ψ dxdy, Ω

Ω

(3.6)

Ω

where the last equality is theR symmetry formula (2.18). The right most member of (3.6) is nothing other than Ω wKv dxdy by the same reasoning, which completes the proof of the proposition. To utilize the dual variational method (cf. [1], [7]) we must define a functional whose critical points are tied to solutions of our problem (N ST ). To this end, we first note that f (s) = sign(s)|s|p ∈ C 0 (R) is an increasing function such that lims→±∞ f (s) = ±∞ and hence there exists an inverse function g ∈ C 0 (R), which is in fact g(t) = sign(t)|t|1/p . This function being continuous and obeying the bound |g(t)| ≤ |t|1/p gives rise to a continuous Nemitski operator g : L(p+1)/p (Ω) → Lp+1 (Ω) given by v 7→ g(v). Moreover, if we define the primitive G of g by Z t p G(t) = g(τ ) dτ = |t|(p+1)/p , p + 1 0 R the functional Ω G(v) dxdy has well understood properties on L(p+1)/p (Ω) (cf. Section 1.2 of [2]). These facts, together with Proposition 3.1, lead to the well defined functional J : L(p+1)/p (Ω) → R given by Z p 1 (p+1)/p J(v) = kvk(p+1)/p − vKv dxdy. (3.7) p+1 2 Ω Proposition 3.2. Let Ω be a symmetric admissible Tricomi domain and p ∈ (0, 1). Then the functional J defined by (3.7) satisfies J ∈ C 1 (L(p+1)/p (Ω), R) and 12

J 0 (v)[w] =

Z

sign(v)|v|1/p w dxdy −

Ω

Z

v, w ∈ L(p+1)/p (Ω), (3.8)

wKv dxdy,

Ω

0

where J denotes the Fr´echet derivative of J. Proof. We split the functional into two parts J(v) = J1 (v) − J2 (v) where Z p 1 (p+1)/p J1 (v) = kvk(p+1)/p and J2 (v) = vKv dxdy. p+1 2 Ω

(3.9)

That J1 ∈ C 1 (L(p+1)/p (Ω), R) and J10 (v)[w] gives the first term of (3.8) follows from standard nonlinear analysis (cf. Theorem 2.6 of [2]). For the piece J2 in (3.9), we note that for all v, w ∈ L(p+1)/p (Ω) one has Z Z J2 (v + w) − J2 (v) − wKv dxdy ≤ |wKw| dxdy, Ω

Ω

where one exploits the symmetry property (3.4). The Cauchy-Schwarz inequality and the continuity estimate (3.3) yields Z J2 (v + w) − J2 (v) − ≤ C1 kwk2 wKv dxdy (p+1)/p

v, w ∈ L(p+1)/p (Ω),

Ω

which shows that Ω wKv dxdy gives the Fr´echet derivative J20 (v)[w]. The continuity of J20 follows from the Cauchy-Schwarz inequality and (3.3), which completes the proof. R

From this proposition, it is clear that critical points of J correspond to weak solutions of the natural problem dual to (N ST ), a principle which we record in the following definition. Definition 3.3. Let Ω be a symmetric admissible Tricomi domain and p ∈ (0, 1). A function v ∈ L(p+1)/p (Ω) is a weak solution of Kv = sign(v)|v|1/p = g(v) if and only if v is a critical point of the functional J defined by (3.7). In fact, a slightly stronger statement is true since equation (3.8) shows that if J 0 (v)[w] = 0 for all w ∈ L(p+1)/p (Ω) one has hKv, wi = hsign(v)|v|1/p , wi,

w ∈ L(p+1)/p (Ω),

(3.10)

where h·, ·i is the natural pairing of Lp+1 (Ω) and L(p+1)/p (Ω), so there is equality of Kv and g(v) as elements in (L(p+1)/p (Ω))0 , but Kv, g(v) ∈ Lp+1 (Ω) and hence

13

by the injectivity of the natural map of Lp+1 (Ω) into (L(p+1)/p (Ω))0 , the equality (3.10) gives Kv = g(v) in Lp+1 (Ω).

(3.11)

Moreover, such critical points, if they exist, give rise to solutions of the original problem (N ST ) in the following way. Proposition 3.4. Let Ω be a symmetric admissible Tricomi domain and p ∈ (0, 1). Let v0 ∈ L(p+1)/p (Ω) be any critical point of the functional J defined by (3.7). Then u0 ∈ L2 (Ω) defined by u0 = (RTAC )−1 (i(v0 )),

(3.12)

or equivalently, j(u0 ) = Kv0 , gives a generalized solution to (N ST ) in the sense of Definition 1.1. Proof. We have from formula (3.11) the equality g(v0 ) = Kv0 = j(RTAC )−1 (i(v0 )) in Lp+1 (Ω).

(3.13)

Now, since f (s) = sign(s)|s|p gives rise to a continuous Nemitski operator f : Lp+1 (Ω) → L(p+1)/p (Ω) given by u 7→ f (u), and since f −1 (t) = g(t), one has from (3.13) the identity v0 = f (j(RTAC )−1 (i(v0 )) in L(p+1)/p (Ω), or i(v0 ) = if (j(RTAC )−1 (i(v0 )) in L2 (Ω),

(3.14)

using the inclusion i of (3.1). Then, setting u0 as defined by (3.12) into (3.14) gives RTAC u0 = (i ◦ f ◦ j)(u0 ) = f (u0 ) in L2 (Ω), where we note that u0 is the unique strong solution in W to RTAC u = i(v0 ) ∈ L2 (Ω) and that i ◦ f ◦ j = f on L2 (Ω) since i and j are just inclusion maps. Finally, an application of the reflection R to both sides gives (1.1), while (1.2) follows from the remark after Proposition 2.5, which completes the proof. It remains now to establish the existence of a non-trivial critical point v0 ∈ L(p+1)/p (Ω) to the functional J of (3.7). In fact, it turns out that one can establish the existence of a minimizer of J using well known techniques. Proposition 3.5. Let Ω be a symmetric admissible Tricomi domain and p ∈ (0, 1). Then there exists a minimizer v0 ∈ L(p+1)/p (Ω) of the functional J defined by (3.7); that is,

14

J(v0 ) =

min

J(v).

v∈L(p+1)/p

Proof. We will show that J is coercive and weakly lower semicontinuous on L(p+1)/p (Ω). Then, by a classical result (cf. [21], Theorem 1.2), J is bounded from below and the infimum of J must be achieved in L(p+1)/p (Ω). To show that J is coercive on L(p+1)/p (Ω), we note that by the estimate (3.3) one has a constant C1 > 0 such that J(v) ≥

p 1 (p+1)/p kvk(p+1)/p − C1 kvk2(p+1)/p . p+1 2

(3.15)

Now, if kvk(p+1)/p → +∞, then (3.15) shows that J(v) → +∞ since (p + 1)/p > 2, and hence the coercivity. Now, usingRthe compactness of K : L(p+1)/p (Ω) → Lp+1 (Ω), it is easy to show that J2 (v) := 12 Ω vKv dxdy is weakly lower semicontinuous. Indeed, let {vn } be any sequence in L(p+1)/p (Ω) such that vn converges weakly to v ∗ ∈ L(p+1)/p (Ω). By the compactness of K, one has {Kvn } converging to Kv ∗ strongly in the Lp+1 norm, and hence Z Z J2 (v ∗ ) = v ∗ Kv ∗ dxdy ≤ lim inf vn Kvn dxdy, n→+∞

Ω

Ω

which is the weak lower semicontinuity of J2 . Finally, it is well known that the Lq norm gives a weakly lower semicontinuous functional on Lq (Ω), and hence J will be weakly lower semicontinuous being the difference of two such functionals. This completes the proof. Having established the existence of a critical point v0 to the dual functional J, we would like to be certain that v0 is nontrivial. The following proposition was obtained jointly with Anna Maria Micheletti. Proposition 3.6. Let Ω be a symmetric admissible Tricomi domain and p ∈ (0, 1). Then there exists a vˆ ∈ L(p+1)/p (Ω) such that J(ˆ v ) < 0, and hence the minimizer v0 of J satisfies v0 6= 0 in L(p+1)/p (Ω). Proof. The result will follow the claim that: there exists v˜ ∈ L(p+1)/p (Ω) and a constant C2 > 0 such that Z v˜K v˜ dxdy = C2 > 0. (3.16) Ω

Indeed, given the claim, one defines the function h : [0, ∞) → R by the formula 1 h(t) = J(t˜ v ) = C3 t(p+1)/p − C2 t2 , 2

15

(p+1)/p

p where C3 = p+1 k˜ v k(p+1)/p is a positive constant. Since (p + 1)/p > 2, it is clear that there exists a t0 ∈ R such that h(t0 ) < 0, and hence defining vˆ = t0 v˜ will give the result. To prove the claim (3.16), we will construct a representative v˜ ∈ C0∞ (Ω) ⊂ (p+1)/p L (Ω) with the desired property. To begin with, one can select a ϕ˜ ∈ C0∞ (Ω) that satisfies the following conditions: (i) ϕ˜ 6≡ 0, (ii) the support of ϕ˜ is contained in Ω+ = Ω∩{y > 0}, and (iii) ϕ(−x, ˜ y) = ϕ(x, ˜ y). Then, since C0∞ (Ω) ⊂ W , as was observed in the discussion proceeding proposition 2.4, by defining v˜ = RT ϕ˜ one 1 has ϕ˜ as a smooth representative of the unique WAC∪σ class solving RTAC ϕ˜ = v˜ ∞ 2 for this v˜ ∈ C0 (Ω) ⊂ L (Ω). One then has, using the standard manipulations of the inclusions i and j and the ability to calculate the L2 class of RTAC (C0∞ (Ω)) by using RT , the identity

Z

v˜K v˜ dxdy =

Ω

Z

(RT ϕ) ˜ ϕ˜ dxdy =

Z

Ω

(T ϕ) ˜ Rϕ˜ dxdy =

Ω

Z

(T ϕ) ˜ ϕ˜ dxdy,

Ω

where we have also used the self-adjointness of R on L2 (Ω) and the property (iii) ∞ ∞ above. But now, ϕ˜ ∈ CAC∪σ (Ω) ∩ CBC∪σ (Ω) and so by the fundamental identity (2.3) gives Z Z Z
v˜K v˜ dxdy = ϕT ˜ ϕ˜ dxdy = y ϕ˜2x + ϕ˜2y dxdy, Ω

Ω

Ω

which must be strictly positive by properties (i) and (ii) above. This completes the proposition. We conclude with the outline of the proof of the main theorem whose details have been established above. Proof of Theorem 1.2 By Proposition 3.5, one has the existence of a critical point v0 ∈ L(p+1)/p (Ω), which minimizes J. Critical points v0 of J give rise to generalized solutions u0 = (RTAC )−1 (i(v0 )) ∈ L2 (Ω) of (N ST ) by Proposition 3.4. The critical point v0 cannot be trivial by Proposition 3.6, and hence u0 is nontrivial by the invertibility of the operator RTAC as given in Proposition 2.5(a).

Appendix In this appendix, we sketch the proof of the a priori estimates of Proposition 2.2. The argument follows that of Didenko [8] with some additional clarifications and transcription into our setting, where there are different sign conventions. For the estimate (2.6), one sets up the energy integral argument with the choice of coefficients a = −(1 + x), b = −h(1 + x), and c = 0, 16

(A.1)

where h is chosen to obey the bound in hypothesis (ii) of the proposition 0 < h < (3x0 /2)−1/3 ,

(A.2)

0 <  < 1/x0 ,

(A.3)

and  will obey the bounds

which ensures that a and b never vanish on Ω. The elliptic arc σ, written as as a C 2 graph y = g(x) for −x0 ≤ x ≤ x0 is assumed to obey the bound of hypothesis (ii) of the proposition −h < g 0 (x) < h for − x0 < x < x0 ,

(A.4)

which places a restriction on σ via the hypothesis (A.2). The idea is to estimate from above and below the quantity (Iu, T u)L2 where I is an integral operator giving the flow along the vector field D = a∂x + b∂y ,

(A.5)

with initial data imposed in such a way as to flow the vanishing of u on AC ∪ σ to the vanishing of v = Iu on BC ∪ σ. This is the key technical point of Didenko’s idea. One introduces the linear coordinate change (ξ, η) = Φ(x, y) = (y + h(x + x0 ), −y + h(x + x0 )),

(A.6)

which transforms (A.5) into D = 2ah∂ξ , and shows that Dη ≡ 0, Dξ 6= 0 on Ω. The hypotheses made on the boundary including (A.2) and the upper bound in (A.4) ensure that the boundary piece BC ∪ σ has the property that the coordinate lines {η = constant} intersect BC ∪ σ at at most one point, and hence BC ∪ σ is expressible as a graph ξ = ψ(η) for η ∈ I = [0, (3x0 /2)2/3 + hx0 ],

(A.7)

with ψ ∈ C 0 (I)∩C 1 (I\{η0 })∩C 2 (I\{η0 }) and η = η0 = 2hx0 at the point B(x0 , 0) 2 where there must be a corner in the boundary. Consequently, given u ∈ CAC∪σ (Ω) 0 1 2 there exists a unique v ∈ C (Ω) ∩ C (Ω\Γ) ∩ C (Ω\Γ) which solves  Dv = avx + bvy = u in Ω (A.8) v=0 on BC ∪ σ , where Γ = {(x, y) ∈ Ω : y − h(x + x0 ) = 2hx0 } is the flow into Ω along D of the point B(x0 , 0). Indeed, by defining v˜ by v(x, y) = v˜(Φ−1 (ξ, η)), one sees that v˜(ξ, η) = (2h)−1

Z

ξ

(a−1 u)(Φ−1 (t, η)) dt

ψ(η)

gives the required solution to (A.8) with the advertised regularity. 17

(A.9)

2 Now, choosing an arbitrary u ∈ CAC∪σ (Ω) with its associated v = Iu solving (A.8) one performs a standard energy integral analysis on Z ˜ I(Ω) = (Iu, T u)L2 = (v, T Dv)L2 = vT u dxdy Ω

which becomes, after applying the divergence theorem, the expression Z Z
1 1 ˜ I(Ω) = αvx2 + 2βvx vy + γvy2 dxdy + (v 2 + yvx2 )(a, b) ·~nAC ds, (A.10) 2 Ω 2 AC y where α = yax − (yb)y = −y + h(1 + x)

(A.11)

β = ay + ybx = −hy

(A.12)

γ = by − ax = 

(A.13)

and ~nAC = (1 − y)−1/2 (−1, −(−y)1/2 ) is the outward unit normal vector field on the characteristic AC. The boundary conditions v = 0 on BC ∪ σ and u = Dv = 0 on AC ∪ σ kill off most of the boundary integrals, where the novelty is to exploit the pair of linear conditions on v to annihilate the σ integral. The singularity along Γ creates no difficulty as one can consider (A.10) as a limit as δ → 0+ of integration over the subdomain Ωδ gotten by removing a strip of width 2δ about Γ from Ω. The hypotheses (A.2) and (A.3) together with the bound (i) |x| ≤ x0 on Ω ensure that the quantities (A.11) − (A.13) obey α > 0 and αγ − β 2 > 0 on Ω

(A.14)

and that (vy2 + yvx2 )(a, b) · ~nAC =

1 + x (1 + h2 y)(1 + h(−y)1/2 )vy2 ≥ 0 on AC. (A.15) (1 − y)1/2

In fact, to show that αγ − β 2 > 0, there is one additional condition on h which depends on , and hence one must first choose  suitably, which can be done in a way that preserves (A.3) and adds no restriction to h. From (A.14) and (A.15) one has the lower bound

(Iu, T u)L2 = (v, T Dv)L2 ≥ δ

Z

(vx2 + vy2 ) dxdy,

2 u ∈ CAC∪σ (Ω),

(A.16)

Ω

for some constant δ > 0. 1 1 For generic Tricomi domains, functions belonging to WAC∪σ (WBC∪σ ) vanish strongly enough on enough of the boundary to support Poincar´e inequalities: there exists C = C(Ω) > 0 such that

18

1 1 u ∈ WAC∪σ (WBC∪σ ).

kukL2 ≤ C(Ω)k∇ukL2 ,

(A.17)

These inequalities follow from integration along segments {x = constant} in Ω+ = Ω ∩ {y ≥ 0} and along segments {y = constant} in Ω− = Ω ∩ {y < 0} (cf. Lemma 1 IV.3.1 of [5]). Applying the inequality (A.17) to (A.16), where v ∈ WBC∪σ , gives (Iu, T u)L2 = (v, T Dv)L2 ≥ C1 kvk2W 1

2 u ∈ CAC∪σ (Ω).

,

BC∪σ

(A.18)

−1 2 1 1 Now, since u ∈ CAC∪σ (Ω) ⊂ WAC∪σ , T u ∈ WBC∪σ and Iu = v ∈ WBC∪σ , applying the GCS inequality to the left hand side of (A.18) gives 1 kIukWBC∪σ kT ukW −1

BC∪σ

≥ C1 kvk2W 1

,

BC∪σ

2 u ∈ CAC∪σ (Ω),

and, since v = Iu, the estimate kT ukW −1

BC∪σ

1 ≥ C1 kIukWBC∪σ ,

2 u ∈ CAC∪σ (Ω).

(A.19)

However, since D is a first order differential operator with smooth coefficients on Ω bounded, one knows that there exists C2 > 0 such that 1 1 kIukWBC∪σ = kvkWBC∪σ ≥ C2 kDvkL2 = C2 kukL2 ,

2 u ∈ CAC∪σ (Ω).

(A.20)

Therefore, one has from (A.19) − (A.20) the inequality kukL2 ≤ C3 kT ukW −1

BC∪σ

,

2 u ∈ CAC∪σ (Ω),

(A.21)

2 for some constant C3 > 0, which gives the desired estimate (2.6) since CAC∪σ (Ω) 1 is dense in WAC∪σ . The proof of the estimate (2.7) is exactly the same where one starts from

a = 1 − x, b = −h(1 − x), and c = 0,

(A.22)

where one makes use of the lower bound −h < g 0 (x) in the hypothesis (A.4) in this case. This completes the proof of the proposition.

Acknowledgments The authors wish to thank Professors M. Badiale and A. M. Micheletti for stimulating and useful discussions.

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References [1] A. AMBROSETTI and M. BADIALE, The dual variational principle and elliptic problems with discontinuous nonlinearities, J. Math. Anal. Appl. 140 (1989), 363-373. [2] A. AMBROSETTI and G. PRODI, “A Primer of Nonlinear Analysis”, Cambridge University Press, Cambridge, 1993. [3] S. AGMON, L. NIRENBERG, and M. H. PROTTER, A maximum principle for a class of hyperbolic equations and applications to equations of mixed elliptic-hyperbolic type, Comm. Pure Appl. Math. 6 (1953), 455-470. [4] A. K. AZIZ and M. SCHNEIDER, The existence of generalized solutions for a class of quasi-linear equations of mixed type, Jour. Math. Anal. Appl. 107 (1985), 425-445. [5] Y. M. BEREZANSKII, “Expansions in Eigenfunctions of Selfadjoint Operators”, Trans. Math. Monographs, Vol. 17, Amer. Math. Soc., Providence, R.I., 1968. [6] H. BERGER, G. WARNECKE, and W. WENDLAND, Finite elements for transonic potential flows, Numerical Methods for Partial Differential Equations 6 (1990), 17-42. [7] F. H. CLARKE, Periodic solutions of Hamilton’s equations and local minima of the dual action, Trans. Amer. Math. Soc. 287 (1985), 239-251. [8] V. P. DIDENKO, On the generalized solvability of the Tricomi problem, Ukrain. Math. J. 25 (1973), 10-18. [9] V. P. DIDENKO, A variational problem for equations of mixed type, Differential Equations 13 (1977), 29-32. [10] V. M. FILIPPOV, “Variational Principles for Nonpotential Operators”, Trans. Math. Monographs, Vol. 77, Amer. Math. Soc., Providence, R.I., 1989. [11] F. I. FRANKL’, On the problems of Chaplygin for mixed sub- and supersonic flows, Isv. Akad. Nauk. USSR Ser. Mat. 9 (1945), 121-143. [12] K. O. FRIEDRICHS, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 338-418. [13] N. N. GAIDAI, Existence of a spectrum for Tricomi’s operator, Differential Equations 17 (1981), 20-25.

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[14] P. LAX, On Cauchy’s problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure Appl. Math. 8 (1955), 615-633. [15] D. LUPO, A. M. MICHELETTI and K. R. PAYNE, in preparation. [16] C. S. MORAWETZ, Non-existence of transonic flow past a profile I, II, Comm. Pure Appl. Math. 9 (1956), 45-68, Comm. Pure Appl. Math. 10 (1957), 107-131. [17] C. S. MORAWETZ, The Dirichlet problem for the Tricomi equation, Comm. Pure Appl. Math. 23 (1970), 587-601. [18] K. R. PAYNE, Interior regularity for the Dirichlet problem for the Tricomi equation, J. Math. Anal. Appl. 199 (1996), 271-292. [19] A. G. POGDAEV, On the solvability of the generalized Tricomi problem for a nonlinear equation, Soviet Math. Dokl. 18 (1977), 1359-1363. [20] J. M. RASSIAS, On three new uniqueness theorems of the Tricomi problem for nonlinear mixed type equations, in “Mixed Type Equations”, TeubnerTexte Math., Vol. 90, Leipzig, 1986, 269-279. [21] M. STRUWE, “Variational Methods” , Springer Verlag, Berlin, 1990. [22] F. G. TRICOMI, Sulle equazioni lineari alle derivate parziali di secondo ordine, di tipo misto, Atti Acad. Naz. Lincei Mem. Cl. Fis. Mat. Nat. (5) 14 (1923), 134-247.

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