On Holder continuity of solutions for a class of nonlinear elliptic systems with p-growth via weighted integral techniques
Abstract
We consider weak solutions of nonlinear elliptic systems in a W-1,W- p-setting which arise as Euler-Lagrange equations of certain variational integrals with pollution term, and we also consider minimizers of a variational problem. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the independent and the dependent variables.
We impose new structural conditions on the nonlinearities which yield C-alpha-regularity and C-alpha-estimates for the solutions. These structure conditions cover variational integrals like integral F(del u) dx with potential F(del u) := F (Q(1)(del u),..., Q(N) (del u)) and positive definite quadratic forms Q(i) in del u defined as Q(i) (del u) = Sigma(alpha beta) a(i)(alpha beta) del u(alpha). del u(beta).
A simple example consists in F (xi(1), xi(2)) := vertical bar xi(1)vertical bar(p/2) + vertical bar xi(2)vertical bar(p/2) or F(xi(1), xi(2)) := vertical bar xi(1)vertical bar(p/4) + vertical bar xi(2)vertical bar(p/4). Since the quadratic forms Q(i) need not to be linearly dependent, our result covers a class of nondiagonal, possibly nonmonotone elliptic systems.
As a by-product, we also prove a kind of Liouville theorem. As a new analytical tool, we use a weighted integral technique with singular weights in an L-p-setting for the proof and establish a weighted hole-filling inequality in a setting where Green-function techniques are not available.