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

We consider multidimensional variational integrals for vector-valued functions u : R(n) superset of Omega -> R(N). Assuming that the integrand satisfies the standard smoothness, convexity and growth assumptions only near infinity we investigate the partial regularity of minimizers (and generalized minimizers) u. Introducing the open set R(u) := {x is an element of Omega) : u is Lipschitz near x}, we prove that R(u) is dense in Omega, but we demonstrate for n >= 3 by an example that Omega \ R(u) may have positive measure. In contrast, for n = 2 one has R(u) = Omega. Additionally, we establish analogous results for weak solutions of quasilinear elliptic systems.