Calculus of Variations and Geometric Measure Theory
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C. Scheven - T. Schmidt

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
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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