Operators with Wentzell boundary conditions and the Dirichlet‐to‐Neumann operator

Tim Binz and Klaus‐Jochen Engel

Mathematische Nachrichten, 2019, vol. 292, issue 4, 733-746

Abstract: In this paper we relate the generator property of an operator A with (abstract) generalized Wentzell boundary conditions on a Banach space X and its associated (abstract) Dirichlet‐to‐Neumann operator N acting on a “boundary” space ∂X. Our approach is based on similarity transformations and perturbation arguments and allows to split A into an operator A00 with Dirichlet‐type boundary conditions on a space X0 of states having “zero trace” and the operator N. If A00 generates an analytic semigroup, we obtain under a weak Hille–Yosida type condition that A generates an analytic semigroup on X if and only if N does so on ∂X. Here we assume that the (abstract) “trace” operator L:X→∂X is bounded that is typically satisfied if X is a space of continuous functions. Concrete applications are made to various second order differential operators.

Date: 2019
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https://doi.org/10.1002/mana.201800064

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