Existência de minimizantes para integrais n-dimensionais não-convexos

Primeiro demonstra-se a existência de minimizantes para o integral múltiplo ∫ Ω ℓ∗∗ ( u (x) , ρ1 (x, u(x))∇u (x) ) ρ2 (x, u(x)) d x on W 1;1 u@ (Ω) , onde Ω ⊂ Rd é aberto e limitado, u : Ω → R pertence ao espaço de Sobolev u@ (·) + W1;1 0 (Ω), u@ (·) ∈ W1;1 (Ω) ∩ C0 ( Ω ) ; ℓ : R×Rd → [0,∞] é superlinear L⊗B−mensurável, ρ1(·, ·), ρ2(·, ·) ∈ C0 (Ω×R) ambos > 0 e ℓ∗∗(·, ·) é apenas sci em (·, 0). Também se considera o caso ∫ Ω L∗∗ (x, u(x),∇u(x) ), embora com hipóteses mais complexas, mas é igualmente possível ter L(x, ·, ξ) não-sci para ξ ̸= 0; Por último demonstra-se a existência de minimizantes radialmente simétricos, i.e. uA(x) = UA ( |x| ), uniformemente contínous para o integral múltiplo ∫ BR L∗∗ ( u(x), |Du(x) | ) d x na bola BR ⊂ Rd, u : Ω → Rm pertence ao espaço de Sobolev A + W1;1 0 (BR, Rm ), L∗∗ : Rm×R → [0,∞] é convexa, sci e superlinear, L∗∗ ( S, · ) é par; note-se também que enquanto no caso escalar, m = 1, apenas necessitamos de mais uma hipótese : ∃ min L∗∗ (R, 0 ), no caso vectorial, m > 1, L∗∗ (·, ·) também tem de satisfazer uma restrição geométrica, a qual chamamos quasi − escalar; sendo o exemplo mais simples de uma função quasi − escalar o caso biradial L∗∗ ( | u(x) | , |Du(x) | ); ABSTRACT: First it is proved the existence of minimizers for the multiple integral ∫ Ω ℓ∗∗ ( u (x) , ρ1 (x, u(x))∇u (x) ) ρ2 (x, u(x)) d x on W 1;1 u@ (Ω) , where Ω ⊂ Rd is open bounded, u : Ω → R is in the Sobolev space u@ (·) + W1;1 0 (Ω), with boundary data u@ (·) ∈ W1;1 (Ω) ∩ C0 ( Ω ) ; and ℓ : R×Rd → [0,∞] is superlinear L⊗B − measurable with ρ1(·, ·), ρ2(·, ·) ∈ C0 (Ω×R) both > 0 and ℓ∗∗(∫ ·, ·) only has to be lsc at (·, 0). The case Ω L∗∗ (x, u(x),∇u(x) ) is also treated, though with less natural hypotheses, but still allowing L(x, ·, ξ) non − lsc for ξ ̸= 0; Lastly it is proved the existence of uniformly continuous radially symmetric minimizers uA(x) = UA ( |x| ) for the multiple integral ∫ BR L∗∗ ( u(x), |Du(x) | ) d x on a ball BR ⊂ Rd, among the vector Sobolev functions u(·) in A + W1;1 0 (BR, Rm ), using a convex lsc L∗∗ : Rm×R → [0,∞] with L∗∗ ( S, · ) even and superlinear; but while in the scalar m = 1 case we only need one more hypothesis : ∃ min L∗∗ (R, 0 ), in the vectorial m > 1 case L∗∗ (·, ·) also has to satisfy a geometric constraint, which we call quasi − scalar; the simplest example being the biradial case L∗∗ ( | u(x) | , |Du(x) | )

Constrained Nonsmooth Problems of the Calculus of Variations

The paper is devoted to an analysis of optimality conditions for nonsmooth multidimensional problems of the calculus of variations with various types of constraints, such as additional constraints at the boundary and isoperimetric constraints. To derive optimality conditions, we study generalised concepts of differentiability of nonsmooth functions called codifferentiability and quasidifferentiability. Under some natural and easily verifiable assumptions we prove that a nonsmooth integral functional defined on the Sobolev space is continuously codifferentiable and compute its codifferential and quasidifferential. Then we apply general optimality conditions for nonsmooth optimisation problems in Banach spaces to obtain optimality conditions for nonsmooth problems of the calculus of variations. Through a series of simple examples we demonstrate that our optimality conditions are sometimes better than existing ones in terms of various subdifferentials, in the sense that our optimality conditions can detect the non-optimality of a given point, when subdifferential-based optimality conditions fail to disqualify this point as non-optimal.Comment: A number of small mistakes and typos was corrected in the second version of the paper. Moreover, the paper was significantly shortened. Extended and improved versions of the deleted sections on nonsmooth Noether equations and nonsmooth variational problems with nonholonomic constraints will be published in separate submission

Multiscale homogenization of convex functionals with discontinuous integrand

This article is devoted to obtain the $\Gamma$-limit, as $\epsilon$ tends to zero, of the family of functionals $$F_{\epsilon}(u)=\int_{\Omega}f\Bigl(x,\frac{x}{\epsilon},..., \frac{x}{\epsilon^n},\nabla u(x)\Bigr)dx$$, where $f=f(x,y^1,...,y^n,z)$ is periodic in $y^1,...,y^n$, convex in $z$ and satisfies a very weak regularity assumption with respect to $x,y^1,...,y^n$. We approach the problem using the multiscale Young measures.Comment: 18 pages; a slight change in the title; to be published in J. Convex Anal. 14 (2007), No.

Quelques contributions au calcul des variations et aux équations elliptiques

Les travaux de recherche qui sont présentés font partie de trois thématiques différentes. Le premier sujet concerne les systèmes d’une classe d'équations aux dérivées partielles, dites «implicites» dans la littérature. Ces problèmes sont complètement non linéaires. Les équations scalaires, où l'inconnue est une fonction, admettent en général, une infinité de solutions. On développe des méthodes variationnelles pour sélectionner des solutions avec des critères de régularités. On traite aussi les cadres vectoriels, où l'inconnue est une application, en présentant des théorèmes d’existence et quelques applications.Les problèmes isopérimétriques font partie de la deuxième thématique de recherche. On traite des inégalités isopérimétriques pour des problèmes aux valeurs propres non linéaires, ainsi que la version quantitative de l'inégalité isopérimétrique classique. On étudie aussi les propriétés de symétrie desminimiseurs d’un problème variationnel non coercitif sur une boule, en montrant une rupture de symétrie, en fonction de l’un des paramètres qui définissent le problème.La mauvaise coercitivité est aussi liée au troisième axe de recherche présenté. On analyse des résultats d’existence et régularité de solutions de certains problèmes elliptiques, définis à travers un opérateur elliptique à coercitivité dégénérée. On montre en particulier les effets régularisants dequelques termes d'ordre inférieur pour les problèmes de Dirichlet correspondants, en fonction de la régularité de la source

Well-Behavior, Well-Posedness and Nonsmooth Analysis

AMS subject classification: 90C30, 90C33.We survey the relationships between well-posedness and well-behavior. The latter notion means that any critical sequence (xn) of a lower semicontinuous function f on a Banach space is minimizing. Here “critical” means that the remoteness of the subdifferential ∂f(xn) of f at xn (i.e. the distance of 0 to ∂f(xn)) converges to 0. The objective function f is not supposed to be convex or smooth and the subdifferential ∂ is not necessarily the usual Fenchel subdifferential. We are thus led to deal with conditions ensuring that a growth property of the subdifferential (or the derivative) of a function implies a growth property of the function itself. Both qualitative questions and quantitative results are considered