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b j (x)u x j (x) + c(x)u(x) − f(x) = 0 (0.5)
is a special case **of** (0.1). **Of** course, the Dirichlet problem for (0.5) has a unique strong (here meaning W loc 2,n pointwise a.e.) solution, which by [4], Theorem 2.10 and Proposition 2.9, is a unique viscosity solution. However, it is an interesting artifact **of** the history **of** the subject that there seems to be no quotable direct statement **of** the existence **of** viscosity solutions **of** the Dirichlet problem in the literature covering this case. However, there are “good solutions” - see e.g. Cerutti, Fabes and Manselli [6] - and “good solutions” are standard viscosity solutions in the continuous coefficient linear case. The current note handles the general Isaacs’ **equations** in a similar manner, and puts the matter in some perspective. Moreover, we treat the parabolic analogue as well. Finally, existence is proved for the measurable coefficient case in both the **elliptic** and parabolic settings.

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order to avoid any further technicalities, and in order to ease the reading **of** our paper, we chose to focus on the case **of** a system **of** 2 × 2 linear hyperbolic **equations**. This paper is organized as follows. Section 2 presents the system **of** linear hyperbolic **equations**, the sliding mode method and introduces the notion **of** solutions that will be used all along the paper. Section 3 gathers the main results **of** the paper, namely an existence theorem and a global asymptotic result. Section 4 is devoted to the proof **of** the main theorems. Section 5 illustrates via nu- merical simulations the efficiency **of** our sliding mode **control**. Finally, Section 6 collects some remarks and in- troduces some future research lines to be followed. Notation. The set **of** non-negative real numbers is denoted in this paper by R+. When a function f only depends on the time variable t (resp. on the space variable x), its derivative is denoted by ˙ f (resp. f 0 ). Given any subset **of** R denoted by Ω (R+ or an interval, for instance), L p (Ω; R n ) denotes the set **of** (Lebesgue) measurable functions f1, . . ., fn such that, for i = {1, . . . , n}, R

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where λ > 0 is a positive scalar and f ∈ L 2 (Γ) is complex valued on the **boundary** Γ = ∂Ω. The
resonance occurs at the transition between the propagative and non propagative regions **of** the domain.
It is characterized by a coefficient α ∈ C 2 (Ω) that changes sign inside the domain Ω, typically over a
closed curve denoted as Σ. In this study, Σ 6= ∅ is totally enclosed in the domain Ω, and α behaves as a signed distance to Σ = {α = 0}. The equation (1.1) can be seen as two separate degenerate ellipitic **equations**, one based in {α < 0}, the other in {α > 0} with remaining compact terms, both coupled at Σ.

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A j = df j , (2.68)
where the f j : R N → R N were fluxes in a system **of** conservation laws, by substituting the approximate solution ansatz into (1.3). In that case for each j the N matrix components **of** ∂ u A j are given by the Hessian matrices **of** the N components **of** f j . The symmetry **of** those matrix components implied that the analogue **of** our equation (2.63) did not involve any **of** the σ m,k . Thus, the **boundary** problem satisfied by v decoupled from the other **equations** and v could be determined first, independently **of** the σ m,k . In this paper we do not assume that the A j satisfy (2.68), so we do not obtain the same decoupling. In the strictly hyperbolic case the equation (2.63) shows that decoupling occurs even without the condition (2.68), since k = k 0 = 1 then and the integrals in (2.63) are zero.

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Let us also comment on the corresponding degenerate inhomoge- neous problem divA∇u = f, with divA∇u denoting the left hand side in ( 1 ). A study **of** such **equations** would be **of** interest in its own right, but could also prove useful in the study **of** **boundary** value problems **of** the type we study here. Such applications were implemented in [ 17 , 16 ] in the non-degenerate setting where the coeﬃcients **of** the op- erator were assumed to satisfy a Carleson measure condition in place **of** t-independence. In [ 17 ] a duality argument reduced the desired es- timate for solutions to the homogeneous equation to an estimate on a solution to an inhomogeneous equation, which could subsequently be proved. While such investigations in the degenerate setting would certainly make the theory more complete, studying the inhomogeneous equation is not our goal here.

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Introduction
Hyperbolic systems have been studied for several centuries, as their importance in representing physical phenomena is undeniable. From gaz dynamics to population evolution through wave **equations** and fluid dynamics they are found in many areas. As they represent the propagation phenomena **of** numerous physical or industrial systems [1, 14, 19], the issue **of** their controllability and stability is a major concern, with both theoretical and practical interest. If the question **of** controllability has been well-studied [20], the problem **of** stabilization under **boundary** **control**, however, is only well known in the particular case **of** an absence **of** source term. However, in many case neglecting the source term is a crude approximation and reduces greatly the analysis, in particular because it implies that the system can be reduced to decoupled **equations** or slightly coupled **equations** (see [11] for instance). For most physical **equations** the source term cannot therefore be neglected and the steady-states we aim at stabilizing can be non-uniform with potentially large variations **of** amplitude (e.g. Saint-Venant **equations**, see [5] Chapter 5 or [17], Euler **equations**, see [12] or [15], Telegrapher **equations**, etc.). Taking into account these nonuniform steady-states and stabilizing them is impossible when not taking the source term into account, although it is an important issue in many applications. In presence **of** a source term some results exist for the H 2 norm (and actually H p , p ≥ 2),

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− 1
2 ∇(a(x)∇u(x)) + λ(x)u(x) = f(x), x ∈ D
α(x)u(x) + β(x) ∂u(x) ∂n = g(x), x ∈ ∂D (2.1) in a domain D divided in subdomains in which both the diffusion coefficient a(x) > 0 and the damping coefficient λ(x) > 0 are constant. The positive coefficients α and β (which cannot vanish simultaneously) may depend on x which is often the case in real applications. For example in electrical impedance tomography applied to breast cancer, the tumors are modelized by Dirichlet conditions, the electrodes by Robin conditions and the rest **of** the skin by Neumann conditions. We assume that this equation has a unique solution which essentially means that we are not in the pure Neumann case that is α = 0 and λ = 0 everywhere. Our Monte Carlo method is based on the evolution **of** a particle and **of** its score along a path that goes from one subdomain to another until it is killed due to the **boundary** conditions or to the damping term. It is constituted **of** two main steps: a walk inside each subdomain with Dirichlet **boundary** conditions and a replacement when hitting an interface between subdomains or the **boundary** **of** D. The validity **of** the algorithm is obtain thanks to the double randomization principal.

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This almost-periodic functional framework has been previously used to construct approximate solutions to semilinear systems in the context **of** Wiener algebras by [JMR94] for the Cauchy problem and [Wil96] for the **boundary** value problem, as well as for **quasilinear** systems in the context **of** Bohr-Besicovich spaces, notably by [JMR95] for the Cauchy problem. In this work we attempt to achieve the next step, namely to obtain a similar result as the one **of** [JMR95], for **quasilinear** **boundary** value problems. We adapt the functional framework **of** [JMR95] to the context **of** **boundary** value problems, by considering functions that are quasi-periodic with respect to the tangential fast variables and almost-periodic with respect to the normal fast variable. Concerning the regularity, we choose a Sobolev **control** for the (slow and fast) tangential variables, and a uniform **control** for the normal variables. The leading profile **of** the WKB expansion is then obtained as the solution **of** a **quasilinear** problem which takes into account the potentially infinite number **of** resonances between the phases. We solve this **quasilinear** problem in a classical way by proving estimates without loss **of** regularity. The example **of** gas dynamics is used all along the paper to illustrate the general assumptions that will be made during the analysis. The main difference between this paper and [JMR95] is the absence **of** symmetry in the problem. Indeed, starting with an evolution problem in time, we modify it to obtain a propagation problem in the normal variable x d , with respect to which the system is not hyperbolic. In [JMR95], these symmetries are used for the a priori estimates to handle the resonance terms that appear in the **equations**. Even though it is relatively easy in our problem to create symmetries for the self-interaction terms, it is more delicate for the resonance terms, which, unlike the case **of** [CGW11], are in infinite number. The last assumption **of** the paper is made to deal with this issue.

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−1,p for p > 2.
As for the Dirichlet problem, we can basically treat it with the duality principle that Regularity solvability with L p data is equivalent to Dirichlet solvability for L p ′ data **of** the dual system. While the Regularity to Dirichlet direction has been known since [ KP ] for real symmetric **equations**, the converse is fairly recent for general systems (some partial results for real symmetric **equations** in Lipschitz domains are in [ S ]) and requires to incorporate square functions in the formulation **of** the Dirichlet problem. This was proved in full generality in [ AR ] for p = 2 and then in [ HKMP2 ] for **equations** and 1 + n ≥ 3 and p 6= 2 (Both articles allow some tdependence as well). We reprove and strengthen it even with the hypotheses there and also extend it to H 1 for Regularity vs BMO (or VMO) for Dirichlet. The Dirichlet problem is stated only with a square function estimate and no non-tangential maximal **control** which in fact comes as a priori information. We shall use in this part a recent result obtained by one **of** us together with S. Stahlhut [ AS ].

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− ∂u
∂t + ∆ − → p u = f (x, t, u, ∇u) in Ω × (0, T ) , (1.3) where Ω is a domain in R n , T is a positive real number, f is a continuous function, and ∆ − → p
is as in (1.1). Anisotropic **equations** like (1.2) and (1.3) have strong physical background. They emerge, for instance, from the mathematical description **of** the dynamics **of** fluids with different conductivities in different directions. We refer to the extensive books by Antontsev– D´ıaz–Shmarev [3] and Bear [9] for discussions in this direction. They also appear in biology, see Bendahmane–Karlsen [10] and Bendahmane–Langlais–Saad [12], as a model describing the spread **of** an epidemic disease in heterogeneous environments.

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Remark 1.2. By the Ekeland variational principle [5] we can prove that for λ ∈ (0, Λ 0 ) there exists a ground state solution to (1.1) which will be denoted by w 0 . The proof is similar to that in [8].
Remark 1.3. Noting that if u is a solution **of** the problem (1.1), then −u is also a solution **of** the problem (1.1) with −λ instead **of** λ. Without loss **of** generality, we restrict our study to the case λ ≥ 0.

Unité de recherche INRIA Lorraine, Technopôle de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LÈS NANCY Unité de recherche INRIA Rennes, Irisa, [r]

order have been recently proved to be a valuable tool in the modeling **of** many phenomena in various fields **of** science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, **control**, porous me- dia, electromagnetics, etc. (see [5, 11, 12, 14, 21, 22, 26]). There has been a significant progress in the investigation **of** fractional differential and partial differential **equations** in recent years; see the monographs **of** Kilbas et al [17], Miller and Ross [23], Samko et al [30] and the papers **of** Delbosco and Rodino [4], Diethelm et al [5, 6, 7], El-Sayed [8, 9, 10], Kaufmann and Mboumi [15], Kilbas and Marzan [16], Mainardi [21], Momani and Hadid [24], Momani et al [25], Podlubny et al [29], Yu and Gao [31] and Zhang [32] and the references therein. Very recently some basic theory for the initial **boundary** value problems **of** fractional differential **equations** involving a Riemann–Liouville differential op- erator **of** order 0 < α ≤ 1 has been discussed by Lakshmikantham and Vatsala [18, 19, 20]. In a series **of** papers (see [1, 2, 3]) the authors considered some classes **of** initial value problems for functional differential **equations** involving Riemann–Liouville and Caputo fractional derivatives **of** order 0 < α ≤ 1.

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(x ∈ R n ). (3.5)
Oﬀ-diagonal estimates are used to show that all H s,p ψ,L (quasi-)norms are equivalent in a certain range **of** parameters σ, τ that depends on s, p and dimension. This provides us with a space that does not depend **of** the speciﬁc choice **of** such ψ. Hence, we drop ψ in the notation. When s and p vary, they form complex interpolation scales. Moreover, holomorphic bounded functions **of** L act continuously on such spaces. We use the suﬃx pre to emphasize that they might be non complete spaces and some care is to be taken with respect to that.

It seems that the **control** **of** Cauchy system for **elliptic** operators is globally an open problem. Lions in [14] proposed a method **of** approximation by penalization and obtained a singular optimality system, under a supplementary hypothe- sis **of** Slater type. In [15], Sougalo and Nakoulima analyzed the Cauchy problem using a regularization method, consisting in viewing a singular problem as a limit **of** a family **of** well- posed problems. They have obtained a singular optimality system for the considered **control** problem, also assuming the Slater condition. Unfortunately, the recent paper by Massengo Mophou and Nakoulima [16] is the same as the one by Sougalo and Nakoulima (1998) using the same old references, and nothing new is brought.

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− r N −1 u 0 (r) 0
≥ 2 λ r N −1
in any interval (r 0 , r 1 ) where u(r) > 0. It readily follows that there must exist a first r 1 > r 0 with u(r 1 ) = 0. The conditions u = 0 in terms **of** System (17) is nothing but the surface {z = x p−1 }. A consequence is that W u (O) ∩ {z = x p−1 } is a curve in a neighborhood **of** O, which locally splits W u (O) into two components. Trajectories which lie below this surface are eventually forced to cross it. This **of** course also holds for a trajectory representing the unstable manifold **of** P . The validity **of** (iii) thus follows. Let us conclude the proof **of** Theorem 2. The value z = λ ∗ represents exactly the value for which the unique trajectory corresponding to the sin- gular solution crosses the surface defined by the zero **boundary** value for the solution **of** (13). The rest **of** the theorem follows in exactly the same way as

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[ 8 ] G . Tarantello ; On nonhomogeneous el liptic **equations** involving critical Sobolev exponent , Ann . Inst . H . Poincar ´ e Anal . Non . Lin e ´ aire , 9 ( 1 992 ) 28 1 - 304 .
[ 9 ] S . Terracini ; On positive entire solutions to a class **of** **equations** with singular coefficient and critical exponent , Adv . Differential **Equations** , 1 ( 1 996 ) 241 - 264 . [ 10 ] Z . Wang , H . Zhou ; Solutions for a nonhomogeneous el liptic problem involving critical So bol ev -

When dealing with reentry **boundary** conditions (including periodic **boundary** conditions, spec- ular reflections, diffuse reflections, generalized or mixed type **boundary** conditions), many progress have been made in the recent years in the understanding **of** the spectral features **of** one-dimensional models [14, 15, 16, 17, 18, 19, 20]. However, to our knowledge, for higher dimensions, only few partial results are available in the literature [21, 22, 23, 24], dealing in particular with very pecu- liar shapes **of** the spatial domain. Our paper deals with the following two problems concerning multidimensional transport **equations** with bounce-back (reverse) **boundary** conditions in convex bounded domains:

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