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Operators with Wentzell boundary conditions and the Dirichlet-to-Neumann operator. (English) Zbl 1422.35014

Summary: 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 \(\partial X\). Our approach is based on similarity transformations and perturbation arguments and allows to split \(A\) into an operator \(A_{00}\) with Dirichlet-type boundary conditions on a space \(X_0\) of states having “zero trace” and the operator \(N\). If \(A_{00}\) 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 \(\partial X\). Here we assume that the (abstract) “trace” operator \(L:X \to \partial X\) is bounded that is typically satisfied if \(X\) is a space of continuous functions. Concrete applications are made to various second order differential operators.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
47D06 One-parameter semigroups and linear evolution equations
47F05 General theory of partial differential operators

References:

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