Continuity of solutions of a problem in the calculus of variations |
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Authors: | Pierre Bousquet |
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Institution: | 1. LATP UMR6632 Universit?? Aix-Marseille 1, Marseille Cedex, France
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Abstract: | We study the problem of minimizing ${\int_{\Omega} L(x,u(x),Du(x))\,{\rm d}x}$ over the functions ${u\in W^{1,p}(\Omega)}$ that assume given boundary values ${\phi}$ on ???. We assume that L(x, u, Du)?=?F(Du)?+?G(x, u) and that F is convex. We prove that if ${\phi}$ is continuous and ?? is convex, then any minimum u is continuous on the closure of ??. When ?? is not convex, the result holds true if F(Du)?=?f(|Du|). Moreover, if ${\phi}$ is Lipschitz continuous, then u is H?lder continuous. |
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