LOWER BOUNDS ON THE NORMS OF EXTENSION OPERATORS FOR LIPSCHITZ DOMAINS

Let Omega subset of R-d. Rd 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(R-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 R-d \ (Omega) over bar. Another estimate of this norm is also given, where spectral properties of Schrodinger operators with the delta-interaction supported on the hypersurface. partial derivative Omega are involved. General results are illustrated with examples.