The Dirichlet-to-Neumann operator via hidden compactness

We show that to each symmetric elliptic operator of the form $\mathcal{A} = - \sum \partial_k \, a_{kl} \, \partial_l + c$ on a bounded Lipschitz domain $Ω\subset \mathbb{R}^d$ 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 $\mathcal{A}$. 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.

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