Partial Regularity Results for Quasimonotone Elliptic Systems with General Growth

Abstract

We present a partial Hölder regularity result for the gradient of solutions to quasimonotone systems:

on bounded domains in the weak sense. Here certain continuity, uniformly strictly quasimonotonicity, growth conditions are imposed on the coefficients, including an asymptotic Uhlenbeck behaviour close to the origin, while the inhomogeneous term satisfies controllable growth conditions. The result is achieved along a two-scale regime: degenerate and non-degenerate. In particular, we will use approximation lemmas, Diening et al. [J. Diff. Equ. 253 (2012)(7), 1943–1958; SIAM J. Math. Anal. 44 (2012)(5), 3594–3616], that simplify and unify the proof in the power growth case and allow us to consider also the general growth case.