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

Scheven C, Schmidt T (2009)

Publication Language: English

Publication Type: Journal article, Original article

Publication year: 2009

Journal

Annali della Scuola Normale Superiore di Pisa-Classe di Scienze

Book Volume: VIII

Pages Range: 469-507

Journal Issue: 3

URI: http://cvgmt.sns.it/paper/1766/

DOI: 10.2422/2036-2145.2009.3.04

Abstract

We consider multidimensional variational integrals for vector-valued functions u : ℝn ⊃ Ω → ℝN. Assuming that the integrand satisfies the standard smoothness, convexity and growth assumptions only near ∞ we investigate the partial regularity of minimizers (and generalized minimizers) u. Introducing the open set R(u) : = {x ∈ Ω : u is Lipschitz near x], we prove that R(u) is dense in Ω, but we demonstrate for n ≥ 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.

Authors with CRIS profile

Thomas Schmidt Department Mathematik

Involved external institutions

Universität Duisburg-Essen (UDE) DE Germany (DE)

How to cite

APA:

Scheven, C., & Schmidt, T. (2009). Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, VIII(3), 469-507. https://dx.doi.org/10.2422/2036-2145.2009.3.04

MLA:

Scheven, Christoph, and Thomas Schmidt. "Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure." Annali della Scuola Normale Superiore di Pisa-Classe di Scienze VIII.3 (2009): 469-507.

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