Haut PDF Boundary control of quasilinear elliptic equations

Boundary control of quasilinear elliptic equations

Boundary control of quasilinear elliptic equations

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Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations

Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations

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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Convexity estimates for nonlinear elliptic equations and application to free boundary problems

Convexity estimates for nonlinear elliptic equations and application to free boundary problems

of ∂ n u = 0 by L.A. Caffarelli and J. Spruck [7]. The exterior problem in Ω = IR 2 \O with O convex and a ≡ 1 has been studied for λ > 0 by B. Kawohl [22] and R.S. Hamilton [14], and for λ = 0 by B. Kawohl [21]. Let us also mention two related results on convex rings [7, 8] and, for general questions on the convexity of the level sets, [20]. We can also quote a recent paper by L.A. Caffarelli and J. Salazar [6] for the equation ∆u + cu = 0 and results by A. Henrot and H. Shahgholian [15, 16, 17] (which rely on a lower bound on the gradient), but for which the extension to general quasilinear operators has not yet be done. Concerning estimates on the curvature and the use of the Fr´echet formula, one may refer to [25] (in the case of the Laplace operator). The results of this article were announced in [10].
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Boundary sliding mode control of a system of linear hyperbolic equations: a Lyapunov approach

Boundary sliding mode control of a system of linear hyperbolic equations: a Lyapunov approach

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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Degenerate elliptic equations for resonant wave problems

Degenerate elliptic equations for resonant wave problems

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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Resonant leading order geometric optics expansions for quasilinear hyperbolic fixed and free boundary problems

Resonant leading order geometric optics expansions for quasilinear hyperbolic fixed and free boundary problems

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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Boundary value problems for degenerate elliptic equations and systems

Boundary value problems for degenerate elliptic equations and systems

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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Exponential stability of general 1-D quasilinear systems with source terms for the C 1 norm under boundary conditions

Exponential stability of general 1-D quasilinear systems with source terms for the C 1 norm under boundary conditions

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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Stochastic finite differences for elliptic diffusion equations in stratified domains

Stochastic finite differences for elliptic diffusion equations in stratified domains

− 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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WEAKLY NONLINEAR MULTIPHASE GEOMETRIC OPTICS FOR HYPERBOLIC QUASILINEAR BOUNDARY VALUE PROBLEMS: CONSTRUCTION OF A LEADING PROFILE

WEAKLY NONLINEAR MULTIPHASE GEOMETRIC OPTICS FOR HYPERBOLIC QUASILINEAR BOUNDARY VALUE PROBLEMS: CONSTRUCTION OF A LEADING PROFILE

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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Boundary layers, Rellich estimates and extrapolation of solvability for elliptic systems

Boundary layers, Rellich estimates and extrapolation of solvability for elliptic systems

−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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Strong maximum principles for anisotropic elliptic and parabolic equations

Strong maximum principles for anisotropic elliptic and parabolic equations

− ∂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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Nonhomogeneous elliptic equations involving critical Sobolev exponent and weight

Nonhomogeneous elliptic equations involving critical Sobolev exponent and weight

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.

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Second Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems

Second Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems

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]

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Boundary value problems for fractional differential equations

Boundary value problems for fractional differential equations

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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Hardy spaces for boundary value problems of elliptic systems with block structure

Hardy spaces for boundary value problems of elliptic systems with block structure

(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.

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Optimal Control of the Ill-Posed Cauchy Elliptic Problem

Optimal Control of the Ill-Posed Cauchy Elliptic Problem

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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Geometry of phase space and solutions of semilinear elliptic equations in a ball

Geometry of phase space and solutions of semilinear elliptic equations in a ball

− 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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Nonhomogeneous elliptic equations with decaying cylindrical potential and critical exponent.

Nonhomogeneous elliptic equations with decaying cylindrical potential and critical exponent.

[ 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 -

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Spectral analysis of transport equations with bounce-back boundary conditions

Spectral analysis of transport equations with bounce-back boundary conditions

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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