The Dirichlet-to-Neumann operator via hidden compactness

We show that to each symmetric elliptic operator of the form [ $A$ =

\sum \partial_k , a_{kl} , \partial_l + c ] on a bounded Lipschitz domain $Ω \subset R^{d}$ one can associate a self-adjoint Dirichlet-to-Neumann operator on $L_{2} (\partial Ω)$ , 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 their coefficients converge uniformly and the second-order limit operator in $L_{2} (Ω)$ has the unique continuation property. We also consider semigroup convergence.

We show that to each symmetric elliptic operator of the form [ $A$ =

\sum \partial_k , a_{kl} , \partial_l + c ] on a bounded Lipschitz domain $Ω \subset R^{d}$ one can associate a self-adjoint Dirichlet-to-Neumann operator on $L_{2} (\partial Ω)$ , 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 their coefficients converge uniformly and the second-order limit operator in $L_{2} (Ω)$ has the unique continuation property. We also consider semigroup convergence.

文献摘要