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

Scheven, Christoph; Schmidt, Thomas

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

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.

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