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Dirichlet and Neumann boundary conditions: What is in between?

In: Nonlinear Evolution Equations and Related Topics

Author

Listed:
  • Wolfgang Arendt

    (Universität Ulm, Abteilung Angewandte Analysis)

  • Mahamadi Warma

    (Universität Ulm, Abteilung Angewandte Analysis)

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.

Suggested Citation

  • Wolfgang Arendt & Mahamadi Warma, 2003. "Dirichlet and Neumann boundary conditions: What is in between?," Springer Books, in: Wolfgang Arendt & Haïm Brézis & Michel Pierre (ed.), Nonlinear Evolution Equations and Related Topics, pages 119-135, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-7924-8_6
    DOI: 10.1007/978-3-0348-7924-8_6
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