Continuity of solutions to a basic problem in the calculus of variations

Clarke, Francis

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 3, pp. 511-530.

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: the lower (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 data (instead of $C^2$). We prove in particular that the solution is locally Lipschitz in $\Omega$. In certain cases, as when $\Gamma$ is a polyhedron or else of class $C^{1,1}$, we obtain in addition a global Hölder condition on $\overline{\Omega }$.

Classification : 49J10, 35J20

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