Regularity of the Green Potential for the Laplacian with Robin Boundary Condition



Let Ω be a bounded Lipschitz domain in \( \mathbb{R}_n \) and let ν be the outward unit normal for Ω. For λ ∈ [0,∞], the Poisson problem for the Laplacian \( \Delta = \sum\limits_{i = 1}^n {\partial _i^2 } \) in Ω with homogeneous Robin boundary condition reads
$$ \left\{ {\begin{array}{*{20}c} {\Delta u = f\,{\rm in}\,\Omega } \hfill \\ {\partial _v u + \lambda \ {\rm Tr}\ u = 0\,{\rm on}\,\partial \Omega ,} \hfill \\ \end{array}} \right. $$
where \( \partial _v u \) denotes the normal derivative of u on ∂Ω and Tr stands for the boundary trace operator. In the case when λ = ∞, the boundary condition in (23.1) should be understood as Tr u = 0 on ∂Ω. The solution operator to (23.1) (i.e., the assignment \( f \mapsto u \)) is naturally expressed as
$$ G_\lambda f\left( x \right): = \begin{array}{*{20}c} {\int_\Omega {G_\lambda \left( {x,y} \right)f\left( x \right)dy,} } & {x \in \Omega } \\ \end{array}, $$
where \( G_\lambda \) is the Green function for the Robin Laplacian. That is, for each \( x \in \Omega ,\ G_\lambda \) satisfies
$$ \left\{ {\begin{array}{*{20}c} {\Delta _y G_\lambda (x,y) = \delta _x (y),\,y \in \Omega ,} \hfill \\ {\begin{array}{*{20}c} {\partial _{v(y)} G_\lambda \left( {x,y} \right) + \lambda G_\lambda \left( {x,y} \right) = 0,} \hfill & {y \in \partial \Omega ,} \hfill \\ \end{array}} \hfill \\ \end{array}} \right. $$
where \( \delta _x \) is the Dirac distribution with mass at x. The scope of this chapter is to investigate mapping properties of the operator \( \nabla \mathbb{G}_\lambda \) when acting on \( L_1 \left( \Omega \right) \) Lebesgue space of integrable functions in Ω. In this regard, weak-Lp spaces over Ω, which we denote by \( L^{p,\infty } (\Omega ) \) play an important role (for a precise definition see Section 23.2). The following theorem summarizes the regularity results for \( G_\lambda \) and \( \mathbb{G}_\lambda \) proved in this chapter.


Green Function Hardy Space Lebesgue Space Lipschitz Domain Lorentz Space 
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© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.University of MissouriColumbiaUSA
  2. 2.Worcester Polytechnic InstituteWorcesterUSA

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