Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure - Scheven, Christoph; Schmidt, Thomas (Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 2009)
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.
UnivIS-Import:2015-03-09:Pub.2009.nat.dma.lma2.asympt