*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

**Links:**
Link to the published version

**Abstract:**

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.

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