Dirichlet and Neumann boundary conditions: What is in between?

Wolfgang Arendt () and Mahamadi Warma ()
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Wolfgang Arendt: Universität Ulm, Abteilung Angewandte Analysis
Mahamadi Warma: Universität Ulm, Abteilung Angewandte Analysis

A chapter in Nonlinear Evolution Equations and Related Topics, 2003, pp 119-135 from Springer

Abstract: Abstract Given an admissible measure µon óΩ where Ω ⊂ ℝ n is an open set, we define a realizationA µ of the Laplacian in L 2 (12) with general Robin boundary conditions and we show that Aµ generates a holomorphic C 0-semigroup on L2(Ω) which is sandwiched by the Dirichlet Laplacian and the Neumann Laplacian semigroups. Moreover, under a locality and a regularity assumption, the generator of each sandwiched semigroup is of the form Δµ. We also show that if D(Δµ) contains smooth functions, then µ is of the form dµ=βbσ(where σ is the (n — 1)-dimensional Hausdorff measure and β a positive measurable bounded function on ∂Ω); i.e. we have the classical Robin boundary conditions.

Keywords: 31C15; 31C25; 34D05; 35A15; 35J10; 47D07; Dirichlet forms; Dirichlet; Neumann and Robin boundary conditions (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-7924-8_6

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DOI: 10.1007/978-3-0348-7924-8_6

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