### A characterization of sets of functions and distributions on ${\mathbb{R}}^{n}$ described by constraints on the gradient.

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Existence results for a class of one-dimensional abstract variational problems with volume constraints are established. The main assumptions on their energy are additivity, translation invariance and solvability of a transition problem. These general results yield existence results for nonconvex problems. A counterexample shows that a naive extension to higher dimensional situations in general fails.

We prove a result for the existence and uniqueness of the solution for a class of mildly nonlinear complementarity problem in a uniformly convex and strongly smooth Banach space equipped with a semi-inner product. We also get an extension of a nonlinear complementarity problem over an infinite dimensional space. Our last results deal with the existence of a solution of mildly nonlinear complementarity problem in a reflexive Banach space.

We prove the partial ${C}^{1,\beta}$-regolarity up to the free boundary of the $p$-harmonic maps which minimize the $p$-energy $\int {\left|Du\right|}^{p}dx$.

We study integrals of the form ${\int}_{\Omega}f\left(d\omega \right)$, where $1\le k\le n$, $f:{\Lambda}^{k}\to \mathbb{R}$ is continuous and $\omega $ is a $\left(k-1\right)$-form. We introduce the appropriate notions of convexity, namely ext. one convexity, ext. quasiconvexity and ext. polyconvexity. We study their relations, give several examples and counterexamples. We finally conclude with an application to a minimization problem.