The Dirichlet-to-Neumann operator via hidden compactness

By W. Arendt, James B. Kennedy, A. T. Elst, Manfred Sauter. 2013

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Abstract We show that to each symmetric elliptic operator of the form A = − ∑ ∂ k a k l ∂ l + c on a bounded Lipschitz domain Ω ⊂ R d one can associate a self-adjoint Dirichlet-to-Neumann operator on L 2 ( ∂ 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 L 2 ( Ω ) has the unique continuation property. We also consider semigroup convergence.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/db3fdbc021bfaf32b94e2a65edf587684de7fcf7

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Citations in the corpus

Boundary Value Problems, Weyl Functions, and Differential Operators (2020, background): “Furthermore, the recent contributions [35, 36, 37, 38, 39, 40] by Arendt, ter Elst, and coauthors form an interesting series of papers on elliptic differential operators and Dirichlet-to-Neumann operators based mainly on form methods.”
Eventually positive semigroups of linear operators (2015, background): “Neumann operator is dened by D ’:= @u @ ; where is the outer unit normal to @ . A proper construction of D as the generator of a strongly continuous analytic semigroup on L2(@ ) can be found in [4,7]. It is shown in [12] that the semigroup e tD is not positive, but only eventually positive for cer”