Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A theory of traces and the divergence theorem

Moritz Schönherr, Friedemann Schuricht

TL;DR

This work addresses the limitation of classical Gauss-Green formulas on irregular domains by developing a general trace theory in which traces are linear functionals on spaces like $\mathcal{L}^ fty(U)$ and $\mathcal{W}^{1,\infty}(U)$. The approach uses finitely additive boundary densities and density measures to localize traces near the boundary, enabling Gauss-Green formulas for open sets and sets of finite perimeter with vector fields whose distributional divergence is a Radon measure. Core results include representation theorems for traces with boundary measures $\lambda$ and $\mu$ in variants (G,L,C), and divergence theorems that involve boundary terms concentrated near $\partial\Omega$ through normal and density measures. The framework extends Sobolev/BV trace theory, handles cracks and shocks, and yields weak solutions to boundary-value problems on arbitrary domains, with applications in continuum mechanics and PDE theory.

Abstract

We introduce a general approach to traces that we consider as linear continuous functionals on some function space where we focus on some special choices for that space. This leads to an integral calculus for the computation of the precise representative of an integrable function and of the trace of a Sobolev or BV function. For integrable vector fields with distributional divergence being a measure, we also obtain Gauss-Green formulas on arbitrary Borel sets. It turns out that a second boundary integral is needed in general. The advantage of the integral calculus is that neither a normal field nor a trace function on the boundary is needed. The Gauss-Green formulas are also available for Sobolev and BV functions. Finally, for any open set the existence of a weak solution of a boundary value problem is shown as application of the trace theory.

A theory of traces and the divergence theorem

TL;DR

This work addresses the limitation of classical Gauss-Green formulas on irregular domains by developing a general trace theory in which traces are linear functionals on spaces like and . The approach uses finitely additive boundary densities and density measures to localize traces near the boundary, enabling Gauss-Green formulas for open sets and sets of finite perimeter with vector fields whose distributional divergence is a Radon measure. Core results include representation theorems for traces with boundary measures and in variants (G,L,C), and divergence theorems that involve boundary terms concentrated near through normal and density measures. The framework extends Sobolev/BV trace theory, handles cracks and shocks, and yields weak solutions to boundary-value problems on arbitrary domains, with applications in continuum mechanics and PDE theory.

Abstract

We introduce a general approach to traces that we consider as linear continuous functionals on some function space where we focus on some special choices for that space. This leads to an integral calculus for the computation of the precise representative of an integrable function and of the trace of a Sobolev or BV function. For integrable vector fields with distributional divergence being a measure, we also obtain Gauss-Green formulas on arbitrary Borel sets. It turns out that a second boundary integral is needed in general. The advantage of the integral calculus is that neither a normal field nor a trace function on the boundary is needed. The Gauss-Green formulas are also available for Sobolev and BV functions. Finally, for any open set the existence of a weak solution of a boundary value problem is shown as application of the trace theory.
Paper Structure (13 sections, 53 theorems, 837 equations, 5 figures)

This paper contains 13 sections, 53 theorems, 837 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries about measures
  3. Measures and integration
  4. Some weakly absolutely continuous measures
  5. Theory of traces
  6. General traces
  7. Traces of vector fields with divergence measure
  8. Representation of traces
  9. Proofs
  10. Divergence theorems
  11. Divergence measure fields
  12. Normal measures
  13. Sobolev functions and BV functions

Key Result

Proposition 2.1

Let $\mathcal{A}$ be an algebra on $\Omega$ and let $\mu \in \operatorname{ba} (\Omega,\mathcal{A})$. Then there is a unique decomposition where we call $\mu_c$$\sigma$-additive part and $\mu_p$pure part of $\mu$.

Figures (5)

  • Figure 1: Tent function $\chi^{{\it \Gamma}}_\delta$ of ${\it \Gamma}$ and $\delta$.
  • Figure 2: The open set $U$.
  • Figure 3: The figure shows the graph of $\varphi_+$ and $\varphi_-$ where the dashed graph is a translation of $\varphi_-$.
  • Figure 4: The figure shows the boundary $\partial{\Omega}$ (solid curve), its neighborhood $(\partial{\Omega})_\delta$ (dashed curves), the segment $[x,y]$ in $C_{j_x}$, and the polygonal curve $P$ in $\overline\Omega$.
  • Figure 5: Contributions to $\nu(B)$ for two versions of $B$

Theorems & Definitions (114)

  • Proposition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • proof
  • Example 2.8
  • Example 2.9
  • ...and 104 more