*Published Paper*

**Inserted:** 22 feb 2012

**Last Updated:** 7 mar 2012

**Journal:** Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5)

**Volume:** 8

**Number:** 3

**Pages:** 469-507

**Year:** 2009

**Links:**
Link to the published version

**Abstract:**

We consider multidimensional variational integrals for vector-valued functions $u\colon\mathbb{R}^n\supset\Omega\to\mathbb{R}^N$. Assuming that the integrand satisfies the standard smoothness, convexity and growth assumptions only near $\infty$ we investigate the partial regularity of minimizers (and generalized minimizers) $u$. Introducing the open set \[R(u):=\{x\in\Omega\,:\,u\text{ is Lipschitz near }x\}\,,\] we prove that $R(u)$ is dense in $\Omega$, but we demonstrate for $n\ge3$ by an example that $\Omega\setminus R(u)$ may have positive measure. In contrast, for $n=2$ one has $R(u)=\Omega$.

Additionally, we establish analogous results for weak solutions of quasilinear elliptic systems.

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