Harnack inequality for harmonic functions relative to a nonlinear image-homogeneous Riemannian Dirichlet form Original Research Article

We consider a Riemannian (p-homogeneous) Dirichlet functional Φ(u)=∫Xμ(u)(dx)Φ(u)=∫Xμ(u)(dx) Turn MathJax on (p>1p>1) defined on D, where D is a dense subspace of Lp(X,m)Lp(X,m) and X is a locally compact Hausdorff topological space endowed with the distance d connected with Φ(u)Φ(u) (see Section 2 for the definitions). We denote by View the MathML sourcea(u,v)=∫Xμ˜(u,v)(dx) the Dirichlet form related to Φ(u)Φ(u). We prove a Harnack type inequality for positive harmonic function relative to the form a(u,v)a(u,v); as a consequence we obtain also the Hölder continuity of harmonic function relative to the form a(u,v)a(u,v).