The omnidirectional trace in H 1 ($\Omega$)

TL;DR

This paper introduces a new concept of omnidirectional trace for functions in Sobolev spaces, enabling boundary value analysis without regularity assumptions and facilitating variational problem solutions.

Contribution

It defines the omnidirectional trace for H 1 functions, proves its properties, and demonstrates its usefulness in boundary value problems without boundary regularity constraints.

Findings

Omnidirectional trace exists for a broad class of Sobolev functions.

The set of functions with omnidirectional trace is closed in H 1.

This trace satisfies an integration-by-parts formula involving boundary points.

Abstract

We first prove that all the functions in L 2 whose directional derivative is in L 2 have a directional trace on the boundary of any open bounded domain, without assumptions on its regularity. This enables us to define the omnidirectional trace of the elements of the Sobolev space H 1 ($\Omega$) for which there exists a function on the boundary that is almost everywhere equal, with respect to the directional measure, to the directional trace, regardless of the direction. The set of all these elements of H 1 ($\Omega$), denoted by H 1 tr ($\Omega$), is shown to be closed, and to always contain the closure in H 1 ($\Omega$) of the set C 0 ($\Omega$)$\cap$ H 1 ($\Omega$) (it is always equal to this set in the 1D case, and can be strictly greater in higher dimensions). The omnidirectional trace always satisfies an integration-by-parts formula, which combines the values of the trace on opposite points of the boundary. Examples show that this notion enables the resolution of variational problems involving the values at the boundary of the domain.