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Non-uniformly elliptic variational problems on BV

Lisa Beck, Franz Gmeineder, Mathias Schäffner

TL;DR

This work addresses regularity for relaxed minimizers of convex functionals with linear growth on BV, a regime that sits at the endpoint between standard superlinear $(p,q)$-growth and pure linear growth. By employing Lebesgue–Serrin–Marcellini relaxation and a carefully designed Ekeland-vanishing viscosity scheme, the authors derive universal $ ext{W}^{1,1}$-regularity and higher gradient integrability under a non-uniform ellipticity condition parameterized by $(oldsymbol{ extmu},q)$, with $q+oldsymbol{ extmu}<2+ rac{2}{n-1}$ and dimension-dependent lower bounds on $oldsymbol{ extmu}$. In two dimensions, a stronger set of assumptions yields local $C^{1,oldsymbol{eta}}$-regularity for relaxed minimizers. The results bridge the gap between the linear-growth endpoint and the superlinear regime, provide a sharp gradient-integrability framework, and include a dimension reduction for the singular set, all while avoiding integral representations of the relaxed functional and relying on BV-based analysis and measure-theoretic techniques. The methods have potential implications for endpoint growth problems such as $L ext{log}L$-type growth and other non-uniformly elliptic BV-variational problems.

Abstract

We establish $\mathrm{W}^{1,1}$-regularity and higher gradient integrability for relaxed minimizers of convex integral functionals on $\mathrm{BV}$. Unlike classical examples such as the minimal surface integrand, we only require linear growth from below but not necessarily from above. This typically comes with a non-uniformly degenerate elliptic behaviour, for which our results extend the presently available bounds from the superlinear growth case in a sharp way.

Non-uniformly elliptic variational problems on BV

TL;DR

This work addresses regularity for relaxed minimizers of convex functionals with linear growth on BV, a regime that sits at the endpoint between standard superlinear -growth and pure linear growth. By employing Lebesgue–Serrin–Marcellini relaxation and a carefully designed Ekeland-vanishing viscosity scheme, the authors derive universal -regularity and higher gradient integrability under a non-uniform ellipticity condition parameterized by , with and dimension-dependent lower bounds on . In two dimensions, a stronger set of assumptions yields local -regularity for relaxed minimizers. The results bridge the gap between the linear-growth endpoint and the superlinear regime, provide a sharp gradient-integrability framework, and include a dimension reduction for the singular set, all while avoiding integral representations of the relaxed functional and relying on BV-based analysis and measure-theoretic techniques. The methods have potential implications for endpoint growth problems such as -type growth and other non-uniformly elliptic BV-variational problems.

Abstract

We establish -regularity and higher gradient integrability for relaxed minimizers of convex integral functionals on . Unlike classical examples such as the minimal surface integrand, we only require linear growth from below but not necessarily from above. This typically comes with a non-uniformly degenerate elliptic behaviour, for which our results extend the presently available bounds from the superlinear growth case in a sharp way.
Paper Structure (28 sections, 31 theorems, 319 equations)

This paper contains 28 sections, 31 theorems, 319 equations.

Table of Contents

  1. Introduction
  2. Setting and LSM-extensions
  3. Main results, context and strategy of proof
  4. Structure of the paper
  5. Preliminaries
  6. General notation
  7. Function spaces
  8. Functions of bounded variation.
  9. Negative Sobolev spaces
  10. A Campanato-type lemma
  11. Weighted Lebesgue spaces
  12. Exponential Orlicz classes
  13. Functionals of measures
  14. Ekeland's variational principle
  15. Miscellaneous estimates
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb R^{n}$ be open and bounded with Lipschitz boundary. Moreover, let $F\in\operatorname{C}^{2}(\mathbb R^{N\times n})$ be a variational integrand with eq:1qgrowth and eq:muell1q, where $1\leq\mu,q<\infty$ satisfy Then, for any $u_{0}\in\operatorname{W}^{1,q}(\Omega;\mathbb R^{N})$, every relaxed minimizer $u\in\operatorname{BV}(\Omega;\mathbb R^{N})$ of $\mathscr{F}$ subjec

Theorems & Definitions (70)

  • Theorem 1.1: Universal higher gradient integrability
  • Theorem 1.2: Universal $\operatorname{C}^{1,\alpha}$-regularity in $n=2$ dimensions
  • Remark 1.3: Comparison with $(p,q)$- and $(\mu,s,q)$-growth conditions
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5: Recession function
  • Proposition 2.6: Reshetnyak's lower semicontinuity theorem
  • ...and 60 more