Non-linear operators of divergence form on the Sierpinski gasket

Abstract

We consider non-linear operators L of divergence type on the Sierpinski gasket. We prove the Harnack inequality for non-negative L-harmonic functions on the Sierpinski gasket. The proof is based on Moser’s method, and we construct a new measure λ by combining a Hausdorff measure and an energy measure, which plays an essential role at key steps. The Harnack inequality is achieved by involving the Caccioppoli type inequality and the weighted Sobolev inequality which give the local boundedness and the Weak Harnack inequality of solutions with respect to the measure λ.