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Journal of Convex Analysis 22 (2015), No. 1, 219--246
Copyright Heldermann Verlag 2015

Partial Hölder Continuity of Minimizers of Functionals Satisfying a General Asymptotic Relatedness Condition

Mikil Foss
Dept. of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.

Christopher S. Goodrich
Dept. of Mathematics, Creighton Preparatory School, Omaha, NE 68114, U.S.A.
and: Dept. of Mathematics, University of Rhode Island, Kingston, RI 02881, U.S.A.


We consider the partial H\"{o}lder continuity of minimizers of functionals of the form $$v\mapsto\int_{\Omega}f(x,v,Dv)\ dx,$$ where $\Omega\subseteq\mathbb{R}^n$ is open and bounded. In our setting the integrand $f\ : \ \Omega\times\mathbb{R}^N\times \mathbb{R}^{N\times n}\rightarrow\mathbb{R}$ is not necessarily continuous in any of its three arguments. In particular, due to the use of a suitable asymptotic relatedness condition, $f$ possesses continuity and convexity only as the norm of its third argument tends to infinity. Since, in particular, $v$ is possibly vector-valued, this provides a generalization of certain existing regularity results in the literature and helps to further build a low-order regularity theory.

Keywords: Partial regularity, Morrey regularity, Hoelder continuity.

MSC: 49N60; 46E35

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