Nonlinear Mean-Value Formulas on Fractal Sets

Abstract

In this paper we study the solutions to nonlinear mean-value formulas on fractal sets. We focus on the mean-value problem 12maxq∈Vm,p{f(q)}+12minq∈Vm,p{f(q)}−f(p)=0 in the Sierpiński gasket with prescribed values f(p1), f(p2) and f(p3) at the three vertices of the first triangle. For this problem we show existence and uniqueness of a continuous solution and analyze some properties like the validity of a comparison principle, Lipschitz continuity of solutions (regularity) and continuous dependence of the solution with respect to the prescribed values at the three vertices of the first triangle.