Domination of nonlinear semigroups generated by regular, local Dirichlet forms

Abstract

In this article we study perturbations of local, nonlinear Dirichlet forms on arbitrary topological measure spaces. As a main result, we show that the semigroup generated by a local, regular, nonlinear Dirichlet form \({\mathcal {E}}\) dominates the semigroup generated by another local functional \({\mathcal {F}}\) if, and only if, \({\mathcal {F}}\) is a specific zero order perturbation of \({\mathcal {E}}\). On the way, we prove a nonlinear version of the Riesz–Markov representation theorem, we define an abstract boundary of a topological measure space, and apply the notion of nonlinear capacity. The main result helps to classify the perturbations that lie between Neumann and Dirichlet boundary conditions.