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Boundary regularity of minima. (English) Zbl 1194.49048

Summary: Let u: \(\Omega \rightarrow \mathbb R^N\) be any given solution to the Dirichlet variational problem minw \(\int \Omega F(x,w,Dw)\) \(dx, w\equiv u_{0}\) on \(\partial \Omega \), where the integrand \(F(x,w,Dw)\) is strongly convex in the gradient variable \(Dw\), and suitably Hölder 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 Hölder 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.

MSC:

49N60 Regularity of solutions in optimal control
35J60 Nonlinear elliptic equations
49J10 Existence theories for free problems in two or more independent variables
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