Dirichlet and Neumann boundary conditions: What is in between?
Wolfgang Arendt, Mahamadi Warma · Birkhäuser Basel eBooks · 2003
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.