Calculus of Variations and Geometric Measure Theory
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J. Kristensen - G. Mingione

Boundary regularity of minima

created by mingione on 29 Sep 2012


Published Paper

Inserted: 29 sep 2012
Last Updated: 29 sep 2012

Journal: Rend. Lincei, Mat. Appl.
Volume: 19
Pages: 265-277
Year: 2008


Let $u\colon \Omega \to \mathcal R^N$ be any given solution to the Dirichlet variational problem \[ \min_{w} \int_{\Omega} F(x,w,Dw)\ dx\qquad w\equiv u_0 \ \ \mbox{on}\ \ \partial \Omega \,, \] where the integrand $F(x,w,Dw)$ is strongly convex in the gradient variable $Dw$, and suitably H\"older continuous with respect to $(x,u)$. We prove that almost every boundary point, in the sense of the usual surface measure of $\partial \Omega$, is a regular point for $u$. This means that $Du$ is H\"older continuous in a relative neighborhood of the point. The existence of even one of such regular boundary points was an open problem for the general functionals considered here, and known only under certain very special structure assumptions.

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