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Local Lipschitz continuity for energy integrals with fast growth and lower order terms

Andrea Torricelli

TL;DR

This work establishes local Lipschitz regularity for local minimizers of variational functionals of the form $\mathcal{F}(u)=\int_{\Omega} f(Du)+g(x,u)\,dx$ in the fast-growth regime with $u$-dependence. The authors extend techniques from prior slow-growth results by leveraging a priori gradient estimates and extensions of the Bounded Slope Condition (BSC) via dual structures related to the Fenchel transform $f^*$. They construct smooth, uniformly convex approximants $f_k$ and obtain Lipschitz bounds for the corresponding minimizers on small balls using barrier methods; passing to the limit yields a local minimizer of the original problem that is locally Lipschitz. The results rely on a detailed sequence of estimates, Euler–Lagrange and second variation formulas, and a Sobolev-type iterative scheme, with constants depending on the data and growth exponents. Overall, the paper advances regularity theory for nonstandard growth variational problems with $u$-dependent Lagrangians and has potential applications in elasticity and image processing.

Abstract

We consider integral functionals with fast growth and the lagrangian explicitly depending on $u$. We prove that the local minimizers are locally Lipschitz continuous.

Local Lipschitz continuity for energy integrals with fast growth and lower order terms

TL;DR

This work establishes local Lipschitz regularity for local minimizers of variational functionals of the form in the fast-growth regime with -dependence. The authors extend techniques from prior slow-growth results by leveraging a priori gradient estimates and extensions of the Bounded Slope Condition (BSC) via dual structures related to the Fenchel transform . They construct smooth, uniformly convex approximants and obtain Lipschitz bounds for the corresponding minimizers on small balls using barrier methods; passing to the limit yields a local minimizer of the original problem that is locally Lipschitz. The results rely on a detailed sequence of estimates, Euler–Lagrange and second variation formulas, and a Sobolev-type iterative scheme, with constants depending on the data and growth exponents. Overall, the paper advances regularity theory for nonstandard growth variational problems with -dependent Lagrangians and has potential applications in elasticity and image processing.

Abstract

We consider integral functionals with fast growth and the lagrangian explicitly depending on . We prove that the local minimizers are locally Lipschitz continuous.

Paper Structure

This paper contains 5 sections, 4 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Proof of Lemma \ref{['apriori_estimate']}
  4. Proof of Theorem \ref{['Lipschitz']}
  5. Acknowledgments

Key Result

Theorem 1.1

Given $u\in W_{loc}^{1,1}(\Omega)\cap L_{loc}^\infty(\Omega)$ local minimizer (see Definition local_min) of the functional functional. Assume that $f$ satisfies the growth assumptions H1-H3, and that $g$ satisfies hypothesis (G1)--(G4). Then $u$ is locally Lipschitz continuous in $\Omega$ and there with $\theta$ depending on $\alpha,\beta,$ and $n$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Lemma 1.2
  • Definition 2.1: Local minimizer
  • Definition 2.2: BSC
  • Proposition 2.3
  • proof
  • Lemma 2.4: EPT, Lemma 2.4