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A look into some of the fine properties of functions with bounded $\mathscr A$-variation

Adolfo Arroyo-Rabasa, Anna Skorobogatova

TL;DR

This work extends the classical fine properties of $BV$-functions to the broader setting of $BV^{\mathcal{A}}$, where $\mathcal{A}$ is a $k$th-order constant-coefficient operator with finite-dimensional kernel (complex-elliptic). By representing $\mathcal{A}u$ as a finite Radon measure, the authors establish one-sided $L^{\frac{n}{n-1}}$ approximate limits on Lipschitz hypersurfaces and countably rectifiable sets, along with sharp trace characterizations (exterior and interior) and a scale-dependent continuity framework. They prove that the Lebesgue discontinuity set has zero $(n-1)$-dimensional Riesz capacity and develop an order-reduction decomposition that reduces higher-order problems to first-order methods, enabling $k$th-order $L^p$-differentiability of $BV^{\mathcal{A}}$ maps. The paper also analyzes the jump parts for complex-elliptic operators, provides a complete description of higher-order jumps, and offers removable-singularity results and sharp lower bounds on the number of equations required for complex-elliptic operators, thereby broadening the BV theory to a wide class of PDE-constrained variational problems with potential applications in regularity analysis and PDE-constrained optimization.

Abstract

We establish certain fine properties for functions of bounded $\mathscr A$-variation known in the classical $BV$ setting. Here, $\mathscr A$ is a $k$th order constant-coefficient homogeneous linear differential operator with a finite-dimensional kernel (also known as a complex-elliptic operator). We prove that if $\mathscr Au$ can be represented by a finite Radon measure, then the potential $u$ has one-sided $L^p$-approximate limits on Lipschitz hypersurfaces, and, more generally, on countably rectifiable sets of codimension one. We use this to give pointwise characterizations of the (functional) interior and exterior traces. We also establish a quantitative scale-dependent continuity result, which allows us to prove that the Lebesgue discontinuity set has zero $(n-1)$-dimensional Riesz capacity. Lastly, we introduce a decomposition that reduces the complexity of analyzing $k$th-order operators to that of first-order methods and allows us to establish the $k$th order $L^p$-differentiability of $BV^{\mathscr A}$ maps.

A look into some of the fine properties of functions with bounded $\mathscr A$-variation

TL;DR

This work extends the classical fine properties of -functions to the broader setting of , where is a th-order constant-coefficient operator with finite-dimensional kernel (complex-elliptic). By representing as a finite Radon measure, the authors establish one-sided approximate limits on Lipschitz hypersurfaces and countably rectifiable sets, along with sharp trace characterizations (exterior and interior) and a scale-dependent continuity framework. They prove that the Lebesgue discontinuity set has zero -dimensional Riesz capacity and develop an order-reduction decomposition that reduces higher-order problems to first-order methods, enabling th-order -differentiability of maps. The paper also analyzes the jump parts for complex-elliptic operators, provides a complete description of higher-order jumps, and offers removable-singularity results and sharp lower bounds on the number of equations required for complex-elliptic operators, thereby broadening the BV theory to a wide class of PDE-constrained variational problems with potential applications in regularity analysis and PDE-constrained optimization.

Abstract

We establish certain fine properties for functions of bounded -variation known in the classical setting. Here, is a th order constant-coefficient homogeneous linear differential operator with a finite-dimensional kernel (also known as a complex-elliptic operator). We prove that if can be represented by a finite Radon measure, then the potential has one-sided -approximate limits on Lipschitz hypersurfaces, and, more generally, on countably rectifiable sets of codimension one. We use this to give pointwise characterizations of the (functional) interior and exterior traces. We also establish a quantitative scale-dependent continuity result, which allows us to prove that the Lebesgue discontinuity set has zero -dimensional Riesz capacity. Lastly, we introduce a decomposition that reduces the complexity of analyzing th-order operators to that of first-order methods and allows us to establish the th order -differentiability of maps.

Paper Structure

This paper contains 35 sections, 18 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. The jump part for elliptic operators
  4. Characterization of traces for complex elliptic operators
  5. Characterization of the jump part for complex-elliptic operators
  6. Approximate continuity properties
  7. Statements for higher-order operators
  8. $L^p$-differentiability
  9. Jumps of higher-order gradients
  10. Related statements
  11. Other properties of complex-elliptic operators
  12. Removable singularities
  13. Minimal number of equations for a complex-elliptic operator
  14. Structure of paper
  15. Preliminaries
  16. ...and 20 more sections

Key Result

Theorem 1

Let $\mathcal{A}$ be a first-order elliptic operator and let $u\in BV^\mathcal{A}(\Omega)$. Then, $J_u \subset \Theta_u$ and there exists a countably $\mathcal{H}^{n-1}$-rectifiable set $G_u \subset J_u$ (with orientation $\nu_u$) satisfying and such that

Theorems & Definitions (43)

  • Definition 1: Approximate continuity
  • Definition 2: $\mathbb{R}$-elliptic and $\mathbb{C}$-elliptic
  • Definition 3
  • Definition 4: Approximate jump
  • Theorem 1: Jump part characterization I
  • Remark 1.1
  • Theorem 2: One-sided limits on interior rectifiable sets
  • Corollary 2.1: Characterization of the exterior trace
  • Corollary 2.2: Interior trace
  • Remark 2.1
  • ...and 33 more