Partial expansion of a Lipschitz domain and some applications

TL;DR

This paper demonstrates how to locally expand a Lipschitz domain near a boundary segment and applies this to vector Sobolev space decomposition and finite element projectors with partial boundary conditions.

Contribution

It introduces a method for partial domain expansion of Lipschitz domains and applies it to Sobolev space decomposition and finite element space construction.

Findings

Lipschitz domain can be expanded near a boundary part enclosed by a C1 curve.

Regular decomposition of vector Sobolev spaces with partial boundary conditions.

Construction of low-regularity projectors into finite element spaces with partial boundary conditions.

Abstract

We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C1 curve. The expanded domain as well as the extended part are both Lipschitz. We apply this result to prove a regular decomposition of standard vector Sobolev spaces with vanishing traces only on part of the boundary. Another application in the construction of low-regularity projectors into finite element spaces with partial boundary conditions is also indicated.