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 Nikolskiȋ space N α,2 . From this it follows that the set of singularities of V(∇u) has Hausdorff dimension less or equal n − 2 α, 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

Fractional estimates for non-differentiable elliptic systems with general growth

Diening L, Ettwein F. Fractional estimates for non-differentiable elliptic systems with general growth. Forum Mathematicum. 2008;20(3):523-556

C1,αC^{1,\alpha}-regularity for electrorheological fluids in two dimensions

Diening L, Ettwein F, Růžička M. $C^{1,\alpha}$-regularity for electrorheological fluids in two dimensions. NoDEA. Nonlinear Differential Equations and Applications. 2007;14(1-2):207-217