Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sobolev inequalities for nonlinear Dirichlet forms

Ralph Chill, Burkhard Claus

TL;DR

The paper addresses the problem of linking Sobolev-type inequalities with isocapacitary inequalities for a broad class of nonlinear Dirichlet forms. It develops a nonlinear Dirichlet form framework with Dirichlet spaces and capacities Cap_{𝔇}, proving that Sobolev embeddings into spaces such as $L^∞$, $L^{p,w}$, and $L^p$ are equivalent to corresponding isocapacitary inequalities. Through elliptic regularity arguments (including Stampacchia-type methods) and growth-type conditions, it derives boundedness of solutions to nonlinear elliptic problems and establishes $L^2$–$L^p$ smoothing for the associated semigroups. These results generalize classical connections between capacity and Sobolev embeddings to nonlinear and nonlocal operators, with implications for regularity and semigroup theory in nonlinear settings.

Abstract

In this short note we show an equivalence between Sobolev type inequalities and so called isocapacitary inequalities in the context of a large class of nonlinear Dirichlet forms, their associated Dirichlet spaces and their associated capacities.

Sobolev inequalities for nonlinear Dirichlet forms

TL;DR

The paper addresses the problem of linking Sobolev-type inequalities with isocapacitary inequalities for a broad class of nonlinear Dirichlet forms. It develops a nonlinear Dirichlet form framework with Dirichlet spaces and capacities Cap_{𝔇}, proving that Sobolev embeddings into spaces such as , , and are equivalent to corresponding isocapacitary inequalities. Through elliptic regularity arguments (including Stampacchia-type methods) and growth-type conditions, it derives boundedness of solutions to nonlinear elliptic problems and establishes – smoothing for the associated semigroups. These results generalize classical connections between capacity and Sobolev embeddings to nonlinear and nonlocal operators, with implications for regularity and semigroup theory in nonlinear settings.

Abstract

In this short note we show an equivalence between Sobolev type inequalities and so called isocapacitary inequalities in the context of a large class of nonlinear Dirichlet forms, their associated Dirichlet spaces and their associated capacities.
Paper Structure (5 sections, 14 theorems, 64 equations)

This paper contains 5 sections, 14 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on nonlinear Dirichlet forms
  3. Isocapacitary inequalities
  4. Elliptic Regularity
  5. $L^2$-$L^p$-regularization of semigroups

Key Result

Theorem 2.2

Let $\mathcal{E}: {L^2(X,m)} \rightarrow {\mathbb R} \cup \{ \infty\}$ be a densely defined, convex, lower semicontinuous functional. Then $- \partial \mathcal{E}$ generates a strongly continuous semigroup $(T(t))_{t\geq 0}$ of contractions on ${L^2(X,m)}$ in the sense that for all $u_0 \in {L^2(X,m and in fact a strong solution on $(0,\infty )$ in the sense that $u\in H^1_{loc} ((0,\infty ); {L^2

Theorems & Definitions (28)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4: Cl23
  • Theorem 2.5
  • Definition 2.6
  • Theorem 2.7: Cl23
  • Theorem 2.8: Cl23
  • Theorem 2.9: Cl23
  • Lemma 3.1
  • ...and 18 more