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

  • Marco Biroli Politecnico di Milano
  • Paola Vernole University of Rome ”la Sapienza”
Keywords: Nonlinear potential theory, Harnack inequality, Nonlinear elliptic problems


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 >.

Author Biographies

Marco Biroli, Politecnico di Milano
Dipartimento di Matematica ”F. Brioschi”
Politecnico di Milano,
Piazza Leonardo da Vinci 32, 20133 Milano Italy
Paola Vernole, University of Rome ”la Sapienza”
Dipartimento di Matematica
University of Rome ”la Sapienza”,
Piazzale Aldo Moro 5, 00182 Roma Italy