In this paper, we consider a class of infinite dimensional stochastic impulsive evolution inclusions. We prove existence of solutions and study properties of the solution set. It is also indicated how these results can be used in the study of control systems driven by vector measures.

Dans cet article nous proposons différents algorithmes pour résoudre une nouvelle classe de problèmes variationels non convexes. Cette classe généralise plusieurs types d’inégalités variationnelles (Cho et al. (2000), Noor (1992), Zeng (1998), Stampacchia (1964)) du cas convexe au cas non convexe. La sensibilité de cette classe de problèmes variationnels non convexes a été aussi étudiée.

Dans cet article nous proposons différents algorithmes pour résoudre une nouvelle classe de problèmes variationels non convexes. Cette classe généralise plusieurs types d'inégalités variationnelles (Cho et al. (2000), Noor (1992), Zeng (1998), Stampacchia (1964)) du cas convexe au cas non convexe. La sensibilité de cette classe de problèmes variationnels non convexes a été aussi étudiée.

Inequalities of Korn's type involve a positive constant, which depends on the domain, in general. A question arises, whether the constants possess a positive infimum, if a class of bounded two-dimensional domains with Lipschitz boundary is considered. The proof of a positive answer to this question is shown for several types of boundary conditions and for two classes of domains.

We show how an operation of inf-convolution can be used to approximate convex functions with C1 smooth convex functions on Riemannian manifolds with nonpositive curvature (in a manner that not only is explicit but also preserves some other properties of the original functions, such as ordering, symmetries, infima and sets of minimizers), and we give some applications.

We prove that the higher integrability of the data $f,f_0$ improves on the integrability of minimizers $u$ of functionals $ℱ$, whose model is $${\int }_{\Omega }\left[{\left|Du\right|}^p+{(det\left(Du\right))}^2-\langle f,Du\rangle +\langle f_0,u\rangle \right]dx,$$ where $u:\Omega \subset {ℝ}^n\to {ℝ}^n$ and $p\ge 2$.

Given a separable metric locally compact space $\mathrm{\Omega }$, and a positive finite non-atomic measure $\lambda$ on $\mathrm{\Omega }$, we study the integral representation on the space of measures with bounded variation $\mathrm{\Omega }$ of the lower semicontinuous envelope of the functional $$F(u)={\int }_{\mathrm{\Omega }}f(x,y)d\lambda \mathit{\hspace{1em}\hspace{1em}}u\in L^1(\mathrm{\Omega },\lambda ,{ℝ}^n)$$ with respect to the weak convergence of measures.

We study the integral representation properties of limits of sequences of integral functionals like $\int f(x,Du)\mathrm{d}x$ under nonstandard growth conditions of $(p,q)$-type: namely, we assume that$${\left|z\right|}^{p\left(x\right)}\le f(x,z)\le {L(1+|z|}^{p\left(x\right)}).$$Under weak assumptions on the continuous function $p\left(x\right)$, we prove $\Gamma$-convergence to integral functionals of the same type. We also analyse the case of integrands $f(x,u,Du)$ depending explicitly on $u$; finally we weaken the assumption allowing $p\left(x\right)$ to be discontinuous on nice sets.

We study the integral representation properties of limits of sequences of integral functionals like  $\int f(x,Du)\mathrm{d}x$  under nonstandard growth conditions of (p,q)-type: namely, we assume that $${\left|z\right|}^{p\left(x\right)}\le f(x,z)\le {L(1+|z|}^{p\left(x\right)}).$$ Under weak assumptions on the continuous function p(x), we prove Γ-convergence to integral functionals of the same type. We also analyse the case of integrands f(x,u,Du) depending explicitly on u; finally we weaken the assumption allowing p(x) to be discontinuous on nice sets.