Fractional estimates for non-differentiable elliptic systems with general growth
Lars Diening, F. Ettwein · Forum Mathematicum · 2008
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 N α, 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. 2000 Mathematics Subject Classification: 35J60; 35D10.