Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure
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 $R\left(u\right):=\left\{x\in \Omega \phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}u\phantom{\rule{4pt}{0ex}}\text{is}\phantom{\rule{4pt}{0ex}}\text{Lipschitz}\phantom{\rule{4pt}{0ex}}\text{near}\phantom{\rule{4pt}{0ex}}x\right\}$ 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.

Classification : 49N60,  35B65,  35H99
@article{ASNSP_2009_5_8_3_469_0,
author = {Scheven, Christoph and Schmidt, Thomas},
title = {Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {469--507},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 8},
number = {3},
year = {2009},
zbl = {1197.49043},
mrnumber = {2581424},
language = {en},
url = {www.numdam.org/item/ASNSP_2009_5_8_3_469_0/}
}
Scheven, Christoph; Schmidt, Thomas. Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 3, pp. 469-507. http://www.numdam.org/item/ASNSP_2009_5_8_3_469_0/

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