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

Chapter

## Abstract

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

## Keywords

Green Function Hardy Space Lebesgue Space Lipschitz Domain Lorentz Space
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