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, p. 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
@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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 8},
     number = {3},
     year = {2009},
     pages = {469-507},
     zbl = {1197.49043},
     mrnumber = {2581424},
     language = {en},
     url = {http://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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