Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 8 (2009) no. 3, pp. 469-507.

We consider multidimensional variational integrals for vector-valued functions u : n Ω N . 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.

Classification: 49N60,  35B65,  35H99
Scheven, Christoph 1; Schmidt, Thomas 2

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
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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, Serie 5, Volume 8 (2009) no. 3, pp. 469-507. http://www.numdam.org/item/ASNSP_2009_5_8_3_469_0/

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