A theory of traces and the divergence theorem

TL;DR

This paper develops a unified integral calculus framework for traces and divergence theorems applicable to Sobolev and BV functions, enabling Gauss-Green formulas on arbitrary sets without requiring boundary normal fields.

Contribution

It introduces a novel approach to traces as linear functionals, leading to boundary integral formulas that do not depend on normal fields, extending divergence theorems to more general contexts.

Findings

Gauss-Green formulas on arbitrary Borel sets.

Trace theory applicable to Sobolev and BV functions.

Existence of weak solutions for boundary value problems.

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.