Showing papers in "Annales De L Institut Henri Poincare-analyse Non Lineaire in 1990"
TL;DR: In this article, the existence and regularity of diffeomorphisms in bounded open sets was studied. But the authors focused on the existence of diffEomorphisms and not on the regularity.
Abstract: Let Ω ⊂ ℝn a bounded open set and f > 0 in Ω ¯ satisfying ∫ Ω f ( x ) d x = meas Ω . We study existence and regularity of diffeomorphisms u : Ω ¯ → Ω ¯ such that
{ det ∇ u ( x ) = f ( x ) , x ∈ Ω u ( x ) = x , x ∈ ∂ Ω .
389 citations
TL;DR: In this paper, the authors extend the Van der Waals-Cahn-Hilliard theory of phase transitions to the case of a mixture of n non-interacting fluids, and study the asymptotic behavior as e → 0 + of minimizers of the energy functionals.
Abstract: In this paper we extend the Van der Waals-Cahn-Hilliard theory of phase transitions to the case of a mixture of n non-interacting fluids. By describing the state of the mixture as given by a vector density function u = (u1, …, un), the problem consists in studying the asymptotic behaviour as e → 0 + of minimizers of the energy functionals:
E e ( u ) = ∫ Ω | e 2 | D u | 2 + W ( u ) | dx under the volume constraint ∫ Ω u ( x ) d x = m , with m ∈ Rn fixed. The function W, which represents the Gibbs free energy, is non-negative and vanishes only in a finite number of points α1, …, αk ∈ Rn. The result is that the minimizers asymptotically approach a configuration which corresponds to a partition of the container Ω into k subsets whose boundaries satisfy a minimality condition.
258 citations
TL;DR: In this paper, various extensions to general linear or nonlinear, elliptic or parabolic operators of a celebrated result due to G. Talenti are studied, involving the solutions of conveniently symmetrized problems, using Schwarz spherical symmetrization.
Abstract: We study various extensions to general linear or nonlinear, elliptic or parabolic operators of a celebrated result due to G. Talenti. We give several comparison results for solutions of such problems involving the solutions of conveniently symmetrized problems, using Schwarz spherical symmetrization.
190 citations
TL;DR: In this paper, a class of matrices infinies (matrices whose coefficients sont a decroissance exponentielle ou polynomiale en seloignant de la diagonale) are defined.
Abstract: Resume Nous etudions dans cet article deux classes de matrices infinies (matrices dont les coefficients sont a decroissance exponentielle ou polynomiale en s’eloignant de la diagonale). Nous montrons qu’elles possedent des proprietes du type « calcul symbolique » : L’inversibilite sur l 2 implique que les coefficients de la matrice inverse ont la meme propriete de decroissance. De plus, l’inversion est une operation « locale » : Une perturbation des coefficients de A n’est pas ressentie sur son inverse loin de la zone de perturbation. Nous fournissons deux exemples d’applications concernant les proprietes des bases orthonormees d’ondelettes.
160 citations
TL;DR: In this article, a general notion of G-convergence for sequences of maximal monotone operators of the form h → + ∞, of the solutions u h to the equations and of their momenta a h (x, D u h ).
Abstract: A general notion of G-convergence for sequences of maximal monotone operators of the form is introduced in terms of the asymptotic behavior, as h → + ∞, of the solutions u h to the equations and of their momenta a h ( x , D u h ). The main results of the paper are the local character of the G-convergence and the G-compactness of some classes of nonlinear monotone operators.
127 citations
TL;DR: In this article, the authors propose a condition necessaire and suffisante portant sur cette fonction for qu’elle engendre effectivement une analyse multiresolution.
Abstract: Resume Suivant un raisonnement du a S. Mallat, on peut reconstituer une analyse multiresolution de L2(ℝn) par la donnee simple d’une fonction m0 possedant la propriete de « quadrature mirror filter ». Nous proposons ici une condition necessaire et suffisante portant sur cette fonction pour qu’elle engendre effectivement une analyse multiresolution. Des applications sont ensuite presentees pour illustrer le sens de ce critere. Ainsi, dans le cas unidimensionel, nous construisons une base orthonormee d’ondelettes de la classe de Schwartz dont les elements sont arbitrairement proches d’un signal progressif. Dans le cas multidimensionel, nous presentons une generalisation non triviale de cette construction.
109 citations
TL;DR: In this paper, the authors prove several first and high order inverse mapping theorems for set-valued maps from a complete metric space to a Banach space and study the stability of the open mapping principle.
Abstract: We prove several first and high order inverse mapping theorems for set-valued maps from a complete metric space to a Banach space and study the stability of the open mapping principle. The obtained results allow to investigate questions of controllability of finite and infinite dimensional control systems, necessary conditions for optimality, implicit function theorem, stability of constraints with respect to a parameter. Applications to problems of optimization, control theory and nonsmooth analysis are provided.
101 citations
TL;DR: In this article, the authors characterize the maps in H 1 (B 3, S 2 ) which can be approximated by smooth ones and show that these maps can be represented by smooth maps.
Abstract: We characterize the maps in H 1 (B 3 , S 2 ) which can be approximated by smooth ones.
94 citations
TL;DR: In this article, it was shown that the functional complexity of concave x = ∫ 0 T g ( t, x ( t ) ) d t + ∫ T h ( t, x ′ ( t )) d t attains a minimum under the condition that g be concave in x.
Abstract: We show that the functional
I ( x ) = ∫ 0 T g ( t , x ( t ) ) d t + ∫ 0 T h ( t , x ′ ( t ) ) d t attains a minimum under the condition that g be concave in x.
89 citations
TL;DR: In this paper, the authors construct maps u0 : B3 → S2 such that the Cauchy problem "find u : B 3 × [0, + ∞) → S 2 such that u(x, 0) = u0(x) in B 3, u t − Δ u = u | ∇ u | 2, u = 0 on ∂B3 × [ 0, + ǫ)
Abstract: We construct maps u0 : B3 → S2 such that the Cauchy problem « find u : B3 × [0, + ∞) → S2 such that u(x, 0) = u0(x) in B3, u t − Δ u = u | ∇ u | 2 , u = u0 on ∂B3 × [0, + ∞) » has infinitely many weak solutions.
72 citations
TL;DR: In this paper, the critical exponent for the k-th Hessian operator and the solvability of the associated Dirichlet problem with sub-critical nonhomogeneous term is discussed.
Abstract: The critical exponent for the k-th Hessian operator (k=2,…,n) is determined and the solvability of the associated Dirichlet problem with sub-critical nonhomogeneous term is discussed.
TL;DR: In this article, the existence of a T-periodic solution for a class of Hamiltonian systems which includes the N-body one was shown to be true for any given T. The authors also proved that the solution they find is not a simultaneous collision one.
Abstract: In this paper we prove the existence of a T-periodic solution (for any given T) for a class of Hamiltonian systems which includes the N-body one. We also prove that the solution we find is not a simultaneous collision one.
TL;DR: In this article, the existence of a homoclinic orbit of the second-order Hamiltonian system was proved under the strong force condition of Gordon and the uniqueness of a global maximum of V.
Abstract: We consider the second order Hamiltonian system: (HS) q .. + V ′ ( q ) = 0 where q = (q1, …, qN) ∈ RN(N ≧ 3) and V:RN\{e} → R(e ∈ RN) is a potential with a singularity, i.e., |V(q)| → ∞ as q →e. We prove the existence of a homoclinic orbit of (HS) under suitable assumptions. Our main assumptions are the strong force condition of Gordon [8] and the uniqueness of a global maximum of V.
TL;DR: Weak continuity for sequences of Jacobians of vector-valued functions in W 1,n(Ω; Rn) is established via a maximal function method developed by Acerbi and Fusco [1] as discussed by the authors.
Abstract: Weak continuity for sequences of Jacobians of vector-valued functions in W1,n(Ω; Rn) in the sense of Chacon’s Biting Lemma [13] is established via a maximal function method developed by Acerbi and Fusco [1]. A « div-curl » Lemma in the same spirit is also proved. The results are applied to the existence problem of nonlinear elastostatics and to a problem involving rank-one connections appearing in the theory of phase transitions.
TL;DR: In this article, it was shown that the point singularity at the origin of the unit ball B1 in ℝn and a Riemannian manifold M is removable provided the p-energy functional is sufficiently small.
Abstract: For the unit ball B1 in ℝn and a Riemannian manifold M we consider mappings u: B1 – {0} → M of class
C 1 ( B 1 − { 0 } , M ) ∩ H 1 , p ( B 1 , ℝ k ) which are stationary points of the p-energy functional
for some exponent p ≧ 2. We shall prove that the point singularity at the origin is removable provided the p-energy is sufficiently small. There are no a priori assumptions on the image of u in M.
TL;DR: For weak solutions of equations of the type of nonlinear filtration in RN × (0, T), 0 < T < ∞, this article proved precise sup-estimates and local and global Harnack type inequalities.
Abstract: For weak solutions of equations of the type of nonlinear filtration in RN × (0, T), 0 < T < ∞, we prove precise sup-estimates and local and global Harnack type inequalities. These estimations permit to identify the initial traces and describe the behavior of such solutions as |x| → ∞.
The main point is to introduce a new approach, free of the specific features of the porous medium equation such as homogeneity, scaling, quasi-convexity, etc. This approach on one hand allows generalizations to a large variety of equations and on other yields new results on gradient averages.
TL;DR: In this paper, it was shown that the value function of a deterministic unbounded control problem is a viscosity solution and the maximum viscoity subsolution of a family of Bellman Equations; in particular, the one given by the hamiltonian, generally discontinuous, associated formally to the problem by analogy with the bounded case.
Abstract: We prove that the value function of a deterministic unbounded control problem is a viscosity solution and the maximum viscosity subsolution of a family of Bellman Equations; in particular, the one given by the hamiltonian, generally discontinuous, associated formally to the problem by analogy with the bounded case. In some cases, we show that this equation is equivalent to a first-order Hamilton-Jacobi Equation with gradient constraints for which we give several existence and uniqueness results. Finally, we indicate other applications of these results to first-order H. J. Equations, to some cheap control problems and to uniqueness results in the nonconvex Calculus of Variations.
TL;DR: In this paper, it was shown that the initial value problem for a regularized one-dimensional scalar conservation law converges along rays to the solution of a certain Riemann problem for the hyperbolic conservation law, even when this conservation law is not genuinely nonlinear.
Abstract: It is shown that as time approaches infinity, the solution of the initial value problem for a regularized one-dimensional scalar conservation law converges along rays to the solution of a certain Riemann problem for the hyperbolic conservation law, even when this conservation law is not genuinely nonlinear.
TL;DR: In this paper, it was shown that for any two points a, b ∈ ℝ4 there exists a geodesic, with respect to g, joining a and b.
Abstract: Let g = g(z) (z = (z0, …, z3) ∈ ℝ4) be a Lorentz metric (with signature +, −, −, −) on the space-time manifold ℝ4. Suppose that g is stationary, i.e. g does not depend on z0. Then we prove, under some other mild assumptions on g, that for any two points a, b ∈ ℝ4 there exists a geodesic, with respect to g, joining a and b.
TL;DR: In this paper, it was shown that every stationary solution is necessarily stable in the presence of large external forces (see Section 2.1.1) in the Euler coordinates.
Abstract: In [1], § 5 we gave a necessary and sufficient condition for the existence of stationary solutions for the system (1.1) in the presence of arbitrarily large external forces f(x). Here we prove, see theorem 2.1 , that every stationary solution is necessarily stable. We use Euler coordinates, since a proof in Lagrange coordinates (see for instance [2] , [3] , [4] ) should present greater obstacles.
TL;DR: In this paper, it was shown that there is always a break of symmetry along the branch of radial solutions if the thickness of the radial solution is sufficiently small, where ρ is an annulus and ρ > 1.
Abstract: In this article we consider the problem
(Pλ) − Δ u = u p + λ u in Ω u = 0 on ∂ Ω u > 0 where Ω is an annulus and p > 1. We prove there is always a break of symmetry along the branch of radial solutions if the thickness of Ω is sufficiently small.
TL;DR: In this article, it was shown that isolated point singularities of the Yang-Mills-Higgs equations on a vector bundle over a 2-dimensional manifold are removable by a smooth gauge transformation.
Abstract: We prove that under a holonomy decay condition; with L1 growth of curvature and integral growth bounds on the Higgs field (depending on the sign of the coupling constant) that isolated point singularities of the Yang-Mills-Higgs equations on a vector bundle over a 2-dimensional manifold are removable by a smooth gauge transformation.
TL;DR: The trace de Krein this article generalises celles de Morse-Ekeland et de Conley et Zehnder, which generalise celles of Hilbert et Minkowski.
Abstract: Resume On presente quelques notions et resultats de la theorie des espaces de Krein, qui generalisent a la fois les espaces de Hilbert et de Minkowski. On introduit en particulier la « trace de Krein » de certains operateurs. Ce travail est ensuite applique a une definition de l’index d’un systeme hamiltonien lineaire periodique, dont on montre qu’elle generalise celles de Morse-Ekeland et de Conley et Zehnder, qui sont donc equivalentes.
TL;DR: In this article, the authors consider the Cauchy problem for general partial differential equations of first order and show that it admits locally a smooth solution for a solution of class C1.
Abstract: We consider the Cauchy problem for general partial differential equations of first order. It is well known that it admits locally a smooth solution. When we extend a solution of class C1, what kinds of phenomena may appear? The aim of this paper is to see what may happen in this extension. Our method depends principally on the analysis of characteristic curves.
TL;DR: Brezis and Browder as discussed by the authors proved the following Theorem, which extends previous results of H. Brezis et al. and F. E. Browder: if w ∈ W 0 m, p ( Ω ), w ≥ 0 a. in Ω and T ϵ W−m,p′, T = μ + h where μ is a positive Radon measure and h ∈ L loc 1 (Ω ) is such that hw ≥ − |Φ| a. e.
Abstract: The first Section of this paper is devoted to prove the following Theorem, which extends previous results of H. Brezis and F. E. Browder: Let w ∈ W 0 m , p ( Ω ) , w ≥ 0 a. e. in Ω and T ϵ W−m,p′, T = μ + h where μ is a positive Radon measure and h ∈ L loc 1 ( Ω ) is such that hw ≥ − |Φ| a. e. in Ω for some Φ ϵ L1(Ω); then w belongs to L1(Ω; dμ), hw belongs to L1(Ω) and 〈 T , w 〉 = ∫ Ω w d μ + ∫ Ω h w d x . The second and third Sections deal with applications of this Theorem to the study of two unilateral problems.
TL;DR: In this article, it was shown that there are no non-trivial (potential) energy stable minimal cones in ℝn × n × n+ with singularity at 0, if 2 ≦ n ≦ 5.
Abstract: We show that there are no non-trivial (potential) energy stable minimal cones in ℝn × ℝ+ with singularity at 0, if 2 ≦ n ≦ 5. The sharpness of this result is demonstrated by proving that a certain six dimensional cone in ℝ7 is stable. Moreover, we extend all results to the more general α-energy functional.
TL;DR: In this article, a condition Dirichlet n’a pas de solution positive quand λ est assez grand sous des conditions assemz generales sur f et g.
Abstract: Resume On montre que l’equation −Δu = f(u(x)) − λg(x) sur Ω avec la condition Dirichlet n’a pas de solution positive quand λ est assez grand sous des conditions assez generales sur f et g.
TL;DR: In this article, the existence and unicity of eigenvalues and corresponding eigenfunctions concerning the complex Monge-Ampere operator det ( ∂ 2 u / ∂ z j ∂ Z ¯ k ) and right-hand side of the forme F(z, u) is studied in a bounded strictly pseudoconvex domain of.
Abstract: A problem of existence and unicity of eigenvalues and corresponding eigenfunctions concerning the complex Monge-Ampere operator det ( ∂ 2 u / ∂ z j ∂ z ¯ k ) and right-hand side of the forme F(z, u) is studied in a bounded strictly pseudoconvex domain of .
TL;DR: In this paper, the authors present a generalisation of Arrow-Debreu's theory of nonconvexite ensembles of production to the theory of nonsmooth production.
Abstract: Resume Le but de cet article est de presenter la theorie mathematique de l’equilibre economique dans le cadre classique traite par K. Arrow et G. Debreu, ainsi qu’une de ses generalisations recentes qui prend en compte les « non-convexites » dans le secteur de la production. Cette generalisation est importante sur le plan economique car la convexite des ensembles de production, hypothese faite par Arrow-Debreu, semble etre l’exception plutot que la regle; en particulier, la presence de rendements croissants (une forme particuliere de non-convexite) dans des secteurs de production comme l’electricite ou les chemins de fer est reconnue depuis le siecle dernier. Sur le plan mathematique, la generalisation au cadre non convexe necessite l’utilisation de techniques plus sophistiquees que dans la theorie classique; d’une part, des outils d’« analyse non differentiable » developpes durant les dix dernieres annees et, d’autre part, des outils de topologie differentielle et de la theorie de Morse.
TL;DR: In this article, the existence of multiple critical points for functionals whose potential operators preserve an order structure was studied using Morse type arguments, and it was shown that the presence of local minima of a functional which are ordered in a special way forces the functional to have many additional critical points.
Abstract: We are studying the existence of multiple critical points for functionals whose potential operators preserve an order structure. By using Morse type arguments we prove that the existence of local minima of a functional Φ which are ordered in a special way « forces » Φ to have many additional critical points. We also show how these abstract results apply to a concrete situation.