Lower bounds on the norms of extension operators for Lipschitz domains

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.