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Rigidity for the hyperbolic Monge-Ampère equation

Chun-Chi Lin (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Some properties of nonlinear partial differential equations are naturally associated with the geometry of sets in the space of matrices. In this paper we consider the model case when the compact set  K is contained in the hyperboloid ℋ - 1 , where ℋ - 1 ⊂ 𝕄 sym 2 × 2 , the set of symmetric 2 × 2 matrices. The hyperboloid ℋ - 1 is generated by two families of rank-one lines and related to the hyperbolic Monge-Ampère equation det ∇ 2 u = - 1 . For some compact subsets K ⊂ ℋ - 1 containing a rank-one connection, we show the rigidity property of K by imposing...

Scaling laws for non-euclidean plates and the W 2 , 2 isometric immersions of riemannian metrics

Marta Lewicka, Mohammad Reza Pakzad (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ-convergence under the proper...

Scaling laws for non-Euclidean plates and the W2,2 isometric immersions of Riemannian metrics

Marta Lewicka, Mohammad Reza Pakzad (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ-convergence under the proper scaling....

Some Liouville theorems for PDE problems in periodic media

Luis Caffarelli (2003)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Liouville problems in periodic media (i.e. the study of properties of global solutions to PDE) arise both in homogenization and dynamical systems. We discuss some recent results for minimal surfaces and free boundaries.

Symmetry of minimizers with a level surface parallel to the boundary

Giulio Ciraolo, Rolando Magnanini, Shigeru Sakaguchi (2015)

Journal of the European Mathematical Society

We consider the functional ℐ Ω ( v ) = ∫ Ω [ f ( | D v | ) - v ] d x , where Ω is a bounded domain and f is a convex function. Under general assumptions on f , Crasta [Cr1] has shown that if ℐ Ω admits a minimizer in W 0 1 , 1 ( Ω ) depending only on the distance from the boundary of Ω , then Ω must be a ball. With some restrictions on f , we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these...

The concentration-compactness principle in the calculus of variations. The limit case, Part II.

Pierre-Louis Lions (1985)

Revista Matemática Iberoamericana

This paper is the second part of a work devoted to the study of variational problems (with constraints) in functional spaces defined on domains presenting some (local) form of invariance by a non-compact group of transformations like the dilations in RN. This contains for example the class of problems associated with the determination of extremal functions in inequalities like Sobolev inequalities, convolution or trace inequalities... We show how the concentration-compactness principle and method...

The concentration-compactness principle in the calculus of variations. The limit case, Part I.

Pierre-Louis Lions (1985)

Revista Matemática Iberoamericana

After the study made in the locally compact case for variational problems with some translation invariance, we investigate here the variational problems (with constraints) for example in RN where the invariance of RN by the group of dilatations creates some possible loss of compactness. This is for example the case for all the problems associated with the determination of extremal functions in functional inequalities (like for example the Sobolev inequalities). We show here how the concentration-compactness...

Un nuovo tipo di funzionale del calcolo delle variazioni

Ennio De Giorgi, Luigi Ambrosio (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In questo lavoro si studia una classe di funzionali che intervengono in molti problemi di Fisica Matematica e, in particolare, nel problema di trovare le configurazioni di equilibrio di una miscela di liquidi isotropi e cristalli liquidi.

Un théorème d'existence en théorie non linéaire des coques minces

Philippe G. Ciarlet, Daniel Coutand (1999)

Journées équations aux dérivées partielles

Les équations bidimensionnelles d'une coque non linéairement élastique «en flexion» ont été récemment justifiées par V. Lods et B. Miara par la méthode des développements asymptotiques formels appliquée aux équations de l'élasticité non linéaire tridimensionnelle. Ces équations se mettent sous la forme d'un problème de point critique d'une fonctionnelle dont l'intégrande est une expression quadratique en termes de la différence exacte entre les tenseurs de courbure des surfaces déformée et non déformée,...

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