# A simple partial regularity proof for minimizers of variational integrals

created by schmidt on 22 Feb 2012
modified on 05 Mar 2012

[BibTeX]

Published Paper

Inserted: 22 feb 2012
Last Updated: 5 mar 2012

Journal: NoDEA, Nonlinear Differ. Equ. Appl.
Volume: 16
Number: 1
Pages: 109-129
Year: 2009
We consider multi-dimensional variational integrals $F[u]:=\int_\Omega f(\cdot,u,Du)\,{\rm d}x \qquad\text{for }u\colon\mathbb{R}^n\supset\Omega\to\mathbb{R}^N\,,$ where the integrand $f$ is a strictly convex function of its last argument. We give an elementary proof for the partial $C^{1,\alpha}$-regularity of minimizers of $F$. Our approach is based on the method of $A$-harmonic approximation, avoids the use of Gehring’s lemma, and establishes partial regularity with the optimal Hölder exponent $\alpha$ in a single step.