On the Trace Operator for Functions of Bounded $\mathbb{A}$-Variation

TL;DR

This paper characterizes when functions of bounded $ ext{A}$-variation have well-defined boundary traces, linking this property to the $ ext{C}$-ellipticity of the differential operator, and applies it to boundary value problems.

Contribution

It establishes the equivalence between the existence of boundary traces and $ ext{C}$-ellipticity of the operator $ ext{A}$ for functions of bounded $ ext{A}$-variation, extending previous results.

Findings

Trace existence iff $ ext{A}$ is $ ext{C}$-elliptic

Trace construction avoids fundamental theorem of calculus

Application to Dirichlet problems for variational functionals

Abstract

In this paper, we consider the space $\mathrm{BV}^{\mathbb A}(\Omega)$ of functions of bounded $\mathbb A$-variation. For a given first order linear homogeneous differential operator with constant coefficients $\mathbb A$, this is the space of $L^1$--functions $u:\Omega\rightarrow\mathbb R^N$ such that the distributional differential expression $\mathbb A u$ is a finite (vectorial) Radon measure. We show that for Lipschitz domains $\Omega\subset\mathbb R^{n}$, $\mathrm{BV}^{\mathbb A}(\Omega)$-functions have an $L^1(\partial\Omega)$-trace if and only if $\mathbb A$ is $\mathbb C$-elliptic (or, equivalently, if the kernel of $\mathbb A$ is finite dimensional). The existence of an $L^1(\partial\Omega)$-trace was previously only known for the special cases that $\mathbb A u$ coincides either with the full or the symmetric gradient of the function $u$ (and hence covered the special cases $\mathrm{BV}$ or $\mathrm{BD}$). As a main novelty, we do not use the fundamental theorem of calculus to construct the trace operator (an approach which is only available in the $\mathrm{BV}$- and $\mathrm{BD}$-setting) but rather compare projections onto the nullspace as we approach the boundary. As a sample application, we study the Dirichlet problem for quasiconvex variational functionals with linear growth depending on $\mathbb A u$.