The Dirichlet-to-Neumann operator via hidden compactness

We show that to each symmetric elliptic operator of the formA=−∑∂kakl∂l+c on a bounded Lipschitz domain Ω⊂Rd one can associate a self-adjoint Dirichlet-to-Neumann operator on L2(∂Ω), which may be multi-valued if 0 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 L2(Ω) has the unique continuation property. We also consider semigroup convergence.