# Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure

created by schmidt on 22 Feb 2012
modified on 07 Mar 2012

[BibTeX]

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
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 $R(u)=\Omega$.