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 $1\le k\le n$, $f:{\Lambda }^k\to ℝ$ 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.

In this article, we present a detailed study of the complex calculus of variations introduced in [M. Gondran: Calcul des variations complexe et solutions explicites d’équations d’Hamilton–Jacobi complexes. C.R. Acad. Sci., Paris 2001, t. 332, série I]. This calculus is analogous to the conventional calculus of variations, but is applied here to ${𝐂}^n$ functions in $𝐂$. It is based on new concepts involving the minimum and convexity of a complex function. Such an approach allows us to propose explicit solutions...

We study the problem of minimizing ${\int }_{\Omega }F\left(Du\left(x\right)\right)dx$ over the functions $u\in W^{1,1}\left(\Omega \right)$ that assume given boundary values $\phi$ on $\Gamma :=\partial \Omega$. The lagrangian $F$ and the domain $\Omega$ are assumed convex. A new type of hypothesis on the boundary function $\phi$ is introduced: thelower (or upper) bounded slope condition. This condition, which is less restrictive than the familiar bounded slope condition of Hartman, Nirenberg and Stampacchia, allows us to extend the classical Hilbert-Haar regularity theory to the case of semiconvex (or semiconcave) boundary...