Lower bounds on the norms of extension operators for Lipschitz domains

Abstract

Let Ω⊂ Rd be a bounded or an unbounded Lipschitz domain. In this note we address the problem of continuation of functions from the Sobolev space H1(Ω) up to functions in the Sobolev space H1(Rd) 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 Ω and Rd \Ω . Another estimate of this norm is also given, where spectral properties of Schrodinger operators with the δ -interaction supported on the hypersurface ∂Ω are involved. General results are illustrated with examples. Mathematics subject classification (2010): 47A30, 46E35, 35P15, 47B38.