The Dirichlet-to-Neumann operator via hidden compactness

摘要

We show that to each symmetric elliptic operator of the form on a bounded Lipschitz domain ? C R~a one can associate a self-adjoint Dirichlet-to -Neumann operator on L_2(??),which may be multi-valued if o is in the Dirichlet spectrum of A. To overcome the lack of coerciveness in this case, we employ a new version of the Lax-Milgram lemma based on an indirect ellipticity property that we call hidden compactness. We then establish uniform resolvent convergence of a sequence of Dirichlet-to-Neumann operators whenever the underlying coefficients converge uniformly and the second-order limit operator in L_2 (?)has the unique continuation property. We also consider semigroup convergence.