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

2018/01/16 by Binz, Tim, Engel, Klaus-Jochen
#34G10 #47D06 #47E05 #47F05 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1801.05261

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 what is typically satisfied if X is a space of continuous functions. Concrete applications are made to various second order differential operators.

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