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Operators with Wentzell boundary conditions and the Dirichlet‐to‐Neumann operator
Author(s) -
Binz Tim,
Engel KlausJochen
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800064
Subject(s) - mathematics , trace operator , semi elliptic operator , operator (biology) , dirichlet distribution , mathematical analysis , semigroup , boundary value problem , bounded function , pure mathematics , finite rank operator , shift operator , boundary (topology) , approximation property , poincaré–steklov operator , dirichlet boundary condition , differential operator , elliptic operator , banach space , compact operator , neumann boundary condition , mixed boundary condition , robin boundary condition , elliptic boundary value problem , programming language , repressor , computer science , transcription factor , biochemistry , chemistry , extension (predicate logic) , gene
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 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 ∂ X . Here we assume that the (abstract) “trace” operator L : X → ∂ Xis bounded that is typically satisfied if X is a space of continuous functions. Concrete applications are made to various second order differential operators.