A density result for homogeneous Sobolev spaces on planar domains

TL;DR

This paper proves that smooth Sobolev functions are dense in homogeneous Sobolev spaces on bounded simply connected planar domains, enhancing understanding of function approximation in these spaces.

Contribution

It establishes the density of smooth Sobolev functions in homogeneous Sobolev spaces for planar domains, a result previously unconfirmed in this setting.

Findings

Smooth Sobolev functions are dense in homogeneous Sobolev spaces on planar domains.

The result applies to bounded simply connected planar domains.

This advances the theory of function approximation in Sobolev spaces.

Abstract

We show that in a bounded simply connected planar domain $\Omega$ the smooth Sobolev functions $W^{k,\infty}(\Omega)\cap C^\infty(\Omega)$ are dense in the homogeneous Sobolev spaces $L^{k,p}(\Omega)$.