Fractional estimates for non-differentiable elliptic systems with general growth

Abstract

In this paper we study the regularity of weak solutions of the elliptic system -div(A(x,∇u)) = b(x,∇u) with non-standard ϕ-growth condition. Here ϕ is a given Orlicz function. We are interested in the case where A and b are not differentiable with respect to x but only Hölder continuous with exponent α. We show that the natural quantity V(∇u) is locally in the Nikolskiĭ space Nα, 2. From this it follows that the set of singularities of V(∇u) has Hausdorff dimension less or equal n – 2α, where n is the dimension of the domain Ω. One of the main features of our technique is that it handles the case of the p-Laplacian for 1 < p < ∞ in a unified way. There is no need to use different approaches for the cases p ≤ 2 and p ≥ 2.