Harnack inequality for harmonic functions relative to a nonlinear p-homogeneous Riemannian Dirichlet form

Abstract

We consider a measure valued map α(u) defined on D where D is a subspace of L^p(X,m) with X a locally compact Hausdorff topological space with a distance under which it is a space of homogeneous type. Under assumptions of convexity, Gateaux differentiability and other assumptions on α which generalize the properties of the energy measure of a Dirichlet form, we prove the Holder continuity of the local solution u of the problem  ∫Xµ(u,v)(dx) = 0  for each v belonging to a suitable space of test functions, where µ(u,v) =< α'(u),v >.

Keywords

• Nonlinear potential theory
• Harnack inequality
• Nonlinear elliptic problems