Energy Spaces, Dirichlet Forms and Capacities in a Nonlinear Setting

TL;DR

This paper extends the theory of Dirichlet forms and capacities to a nonlinear setting, constructing energy spaces and establishing properties like lattice structure and quasicontinuity for these forms.

Contribution

It introduces a framework for nonlinear Dirichlet forms, constructs associated energy spaces, and generalizes key properties known from the bilinear case.

Findings

Energy space for nonlinear Dirichlet forms is a lattice.

Defined capacity and quasicontinuity in the nonlinear setting.

Extended classical results to nonlinear Dirichlet forms.

Abstract

In this article we study lower semicontinuous, convex functionals on real Hilbert spaces. In the first part of the article we construct a Banach space that serves as the energy space for such functionals. In the second part we study nonlinear Dirichlet forms, as defined by Cipriani and Grillo, and show, as it is well known in the bilinear case, that the energy space of such forms is a lattice. We define a capacity and introduce the notion quasicontinuity associated with these forms and prove several results, which are well known in the bilinear case.