On Max–Min Mean Value Formulas on the Sierpinski Gasket

Abstract

In this paper, we study solutions to the max–min mean value problem ½ max q∈Vm,p {f(q)} + ½ min q∈Vm,p {f(q)} = f(p) in the Sierpinski Gasket with a prescribed Dirichlet datum at the three vertices of the first triangle. In the previous mean value, formula p is a vertex of one triangle at one stage in the construction of the Sierpinski Gasket and Vm,p is the set of vertices that are adjacent to p at that stage. For this problem, it was known that there are existence and uniqueness of a continuous solution, a comparison principle holds, and, moreover, solutions are Lipschitz continuous. Here we continue the analysis of this problem and prove that the solution is piecewise linear on the segments of the Sierpinski Gasket. Moreover, we also show for which values at the three vertices of the first triangle solutions to this mean value formula coincide with infinity harmonic functions.