Lower bounds on the norms of extension operators for Lipschitz domains

TL;DR

This paper investigates the lower bounds of extension operator norms for Lipschitz domains, linking them to spectral properties of Robin Laplacians and Schrödinger operators with delta interactions, providing theoretical estimates and examples.

Contribution

It introduces new lower bound estimates for extension operator norms based on spectral analysis of Robin Laplacians and Schrödinger operators with delta interactions.

Findings

Lower bounds are expressed via spectral properties of Robin Laplacians.

Additional bounds involve Schrödinger operators with delta interactions.

Results are illustrated with specific examples.

Abstract

Let $\Omega\subset\dR^d$ be a bounded or an unbounded Lipschitz domain. In this note we address the problem of continuation of functions from the Sobolev space $H^1(\Omega)$ up to functions in the Sobolev space $H^1(\dR^d)$ via a linear operator. The minimal possible norm of such an operator is estimated from below in terms of spectral properties of self-adjoint Robin Laplacians on domains $\Omega$ and $\dR^d\setminus\ov\Omega$. Another estimate of this norm is also given, where spectral properties of Schr\"odinger operators with the $\delta$-interaction supported on the hypersurface $\partial\Omega$ are involved. General results are illustrated with examples.