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 (23.1) $$ \left\{ {\begin{array}{*{20}c} {\Delta u = f\,{\rm in}\,\Omega } \hfill \\ {\partial _v u + \lambda \ {\rm Tr}\ u = 0\,{\rm on}\,\partial \Omega ,} \hfill \\ \end{array}} \right. $$ 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 (23.2) $$ G_\lambda f\left( x \right): = \begin{array}{*{20}c} {\int_\Omega {G_\lambda \left( {x,y} \right)f\left( x \right)dy,} } & {x \in \Omega } \\ \end{array}, $$ where $$ G_\lambda $$ is the Green function for the Robin Laplacian. That is, for each $$ x \in \Omega ,\ G_\lambda $$ satisfies (23.3) $$ \left\{ {\begin{array}{*{20}c} {\Delta _y G_\lambda (x,y) = \delta _x (y),\,y \in \Omega ,} \hfill \\ {\begin{array}{*{20}c} {\partial _{v(y)} G_\lambda \left( {x,y} \right) + \lambda G_\lambda \left( {x,y} \right) = 0,} \hfill & {y \in \partial \Omega ,} \hfill \\ \end{array}} \hfill \\ \end{array}} \right. $$ 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.