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

Scheven, Christoph; Schmidt, Thomas

Scheven, Christoph ; Schmidt, Thomas

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 $u : ℝ^{n} \supset Ω \to ℝ^{N}$ . 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 $R (u) : = {x \in Ω : u is Lipschitz near x}$ we prove that $R (u)$ is dense in $Ω$ , but we demonstrate for $n \geq 3$ by an example that $Ω ∖ R (u)$ may have positive measure. In contrast, for $n = 2$ one has $R (u) = Ω$ .

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

MR 2581424 | Zbl 1197.49043

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 = {archive.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://archive.numdam.org/item/ASNSP_2009_5_8_3_469_0/

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