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

Scheven, Christoph¹ ; Schmidt, Thomas²

¹ Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany
² Department Mathematik, Friedrich-Alexander-Universität, Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054 Erlangen, Germany

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 3, pp. 469-507

Résumé

We consider multidimensional variational integrals for vector-valued functions $u:{ℝ}^n\supset \Omega \to {ℝ}^N$. 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 $R\left(u\right):=\{x\in \Omega :u\text{is}\text{Lipschitz}\text{near}x\}$ we prove that $R\left(u\right)$ is dense in $\Omega$, but we demonstrate for $n\ge 3$ by an example that $\Omega \setminus R\left(u\right)$ may have positive measure. In contrast, for $n=2$ one has $R\left(u\right)=\Omega$.

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

Détail

Classification : 49N60, 35B65, 35H99