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On the singular set of $\operatorname{BV}$ minimizers for non-autonomous functionals

Lukas Fußangel, Buddhika Priyasad, Paul Stephan

TL;DR

This work analyzes the regularity of generalized BV minimizers for non-autonomous convex variational integrals with linear growth on bounded Lipschitz domains, establishing higher gradient integrability and singular-set dimension bounds. The authors develop a vanishing-viscosity scheme via Ekeland’s principle to construct an approximating sequence $u_k$ and obtain fractional Besov–Nikolskii estimates for $Du_k$ through second-order finite differences, leveraging Hölder continuity in the first variable and $\mu$-ellipticity. The main results show that, for $1<\mu<1+\frac{\alpha}{n}$, generalized minimizers belong to $W^{1,p}_{\mathrm{loc}}(\Omega)$ for all $1\le p<\frac{(3-\mu)n}{2n-\alpha}$, and that if $1<\mu<\frac{3n}{3n-\alpha}$ then the singular set has Hausdorff dimension at most $n-\frac{\alpha}{2}$. The analysis extends partial regularity theory to non-autonomous linear-growth functionals, providing sharp one-dimensional sharpness results and linking fractional differentiability to geometric measure bounds via the measure-density lemma.

Abstract

We investigate regularity properties of minimizers for non-autonomous convex variational integrands $F(x, \mathrm{D} u)$ with linear growth, defined on bounded Lipschitz domains $Ω\subset \mathbb{R}^n$. Assuming appropriate ellipticity conditions and Hölder continuity of $\mathrm{D}_zF(x,z)$ with respect to the first variable, we establish higher integrability of the gradient of minimizers and provide bounds on the Hausdorff dimension of the singular set of minimizers.

On the singular set of $\operatorname{BV}$ minimizers for non-autonomous functionals

TL;DR

This work analyzes the regularity of generalized BV minimizers for non-autonomous convex variational integrals with linear growth on bounded Lipschitz domains, establishing higher gradient integrability and singular-set dimension bounds. The authors develop a vanishing-viscosity scheme via Ekeland’s principle to construct an approximating sequence and obtain fractional Besov–Nikolskii estimates for through second-order finite differences, leveraging Hölder continuity in the first variable and -ellipticity. The main results show that, for , generalized minimizers belong to for all , and that if then the singular set has Hausdorff dimension at most . The analysis extends partial regularity theory to non-autonomous linear-growth functionals, providing sharp one-dimensional sharpness results and linking fractional differentiability to geometric measure bounds via the measure-density lemma.

Abstract

We investigate regularity properties of minimizers for non-autonomous convex variational integrands with linear growth, defined on bounded Lipschitz domains . Assuming appropriate ellipticity conditions and Hölder continuity of with respect to the first variable, we establish higher integrability of the gradient of minimizers and provide bounds on the Hausdorff dimension of the singular set of minimizers.

Paper Structure

This paper contains 18 sections, 13 theorems, 131 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Convex variational integrals on $\operatorname{BV}$.
  3. Context and main results
  4. Main ideas of the proofs
  5. Organization of the paper
  6. Preliminaries
  7. General Notation
  8. Functions of bounded variation
  9. Application of Reshetnyak's Theorem
  10. Besov spaces and embeddings
  11. The space $\operatorname{W}^{-1,1}(\Omega;\mathbb{R}^{N})$
  12. The measure density lemma
  13. Improved Sobolev regularity - Proof of Theorem \ref{['maintheorem']}
  14. Implementation of the Ekeland approximation
  15. Fractional estimates for the approximating sequence
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $F: \Omega \times \mathbb{R}^{N\times n} \to \mathbb{R}$ with $F \in \operatorname{C}(\bar{\Omega}\times\mathbb{R}^{N \times n})$ and $F(x,\cdot) \in \operatorname{C}^2(\mathbb{R}^{N \times n})$ for every $x \in \Omega$. Assume that $F$ satisfies the following conditions: Then, for $1 < \mu < 1+ \frac{\alpha}{n}$, any generalized minimizer $u \in \mathrm{GM}(\mathscr F; u_0)$ of variationalpr

Figures (1)

  • Figure 1: Numerical simulation of Example \ref{['counterexample']} with parameters $\mu = 1.4$, $\alpha = 0.25$ and $M=20$.

Theorems & Definitions (23)

  • Theorem 1.1: Sobolev regularity
  • Proposition 1.2
  • Theorem 1.3: Dimension bound
  • Theorem 2.1: Reshetnyak
  • Lemma 2.2: Bildhauer03book
  • Definition 2.3
  • Lemma 2.4
  • Proposition 2.5: Mingione03, Beck
  • Lemma 3.1
  • Lemma 3.2: GmeinederKristensen19BD, BeckSchmidt13
  • ...and 13 more