Inclusions différentielles et problèmes variationnels

In this thesis we deal with three different but connected questions. Firstly (cf. Chapter 2) we make a systematic study of the generalized notions of convexity for sets. We study the notions of polyconvex, quasiconvex and rank one convex set. We remark that these notions are nowadays well known in the context of functions, but not in the context of sets. Following the classical approach, we give precise definitions of generalized convex sets and we study several issues, in this generalized sense, as the concept of convex hull, Carathéodory and separation theorems and the notion of extremal point. Secondly we have studied some differential inclusions of the form The method we have used to solve this kind of problems is the Baire categories method developed by Dacorogna-Marcellini [14]. Known sufficient conditions for this problem are connected to the generalized convex hull of the set E. In Chapter 3, we compute the rank one convex hull of some matrix sets to obtain, in Chapter 4, existence results. Namely, we have considered the problem of finding u : Ω ⊂ Rn → RN with Dirichlet boundary condition such that Φ (Du(x)) ∈ {α, β}, a.e.  x ∈ Ω, Φ being an arbitrary quasi-affine function. We have also considered the problem of finding u : Ω ⊂ Rn → Rn such that where λ1(Du) ≤...≤ λn(Du) are the singular values of Du ∈ Rn×n. Finally, in Chapter 5, we deal with several minimizing problems of the form Denoting by Qf  the quasiconvex envelope of f, we verify that solving the equation Qf(Du(x)) = f(Du(x)), a.e.  x ∈ Ω is, under some conditions, sufficient to ensure the existence of solution of (P). The differential inclusions that we consider in Chapter 4 are helpful to solve some equations of the form (2) and thus, it allows us to solve problems of type (P). In particular, we have considered the problem (P) with f(ξ) = g(Φ(ξ)), ∀ ξ ∈ RN×n Φ being an arbitrary quasi-affine function.