Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The extended Dirichlet space and criticality theory for nonlinear Dirichlet forms

Marcel Schmidt, Ian Zimmermann

Abstract

In this paper we establish the existence of the extended Dirichlet space for nonlinear Dirichlet forms under mild conditions. We employ it to introduce and characterize criticality (recurrence) and subcriticality (transience) and establish basics of a potential theory.

The extended Dirichlet space and criticality theory for nonlinear Dirichlet forms

Abstract

In this paper we establish the existence of the extended Dirichlet space for nonlinear Dirichlet forms under mild conditions. We employ it to introduce and characterize criticality (recurrence) and subcriticality (transience) and establish basics of a potential theory.

Paper Structure

This paper contains 20 sections, 51 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Nonlinear Dirichlet forms, resolvents and the Green operator
  3. The modular space and the extended Dirichlet space
  4. The modular space
  5. The extended Dirichlet space
  6. Excessive functions
  7. Invariant sets and irreducibility
  8. Criticality theory
  9. The Green operator and Hardy inequalities
  10. Weak Hardy and Poincaré inequalities
  11. Criticality theory
  12. Equilibrium potentials and a glimpse at potential theory
  13. Examples
  14. Variable exponent Dirichlet forms on Riemannian manifolds
  15. $p$-energies induced by strongly local regular Dirichlet forms
  16. ...and 5 more sections

Key Result

Proposition 2.3

Let ${\mathcal{E}} \colon L^2(\mu) \to (-\infty,\infty]$ be convex, lower semicontinuous and proper.

Theorems & Definitions (144)

  • Definition 2.1: The Beurling-Deny criteria and nonlinear Dirichlet forms
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 3.1
  • Lemma 3.2: Luxemburg seminorms are compatible with contractions
  • proof
  • Lemma 3.3: Continuity of certain contractions
  • proof
  • Remark 3.4
  • ...and 134 more