The Laplace Equation as the Prototype of an Elliptic Partial Differential Equation of Second Order

Abstract

In this section Ω is a bounded domain in ${\mathbb{R}}^{d}$ for which the divergence theorem holds; this means that for any vector field V of class ${C}^{1}(\Omega ) \cap {C}^{0}(\bar{\Omega }),$

$$\int\limits_{\Omega }\mathrm{div}V (x)\mathrm{d}x =\int\limits_{\partial \Omega }V (z) \cdot \nu (z)do(z),$$

(2.1.1)

where the dot $\cdot$ denotes the Euclidean product of vectors in ${\mathbb{R}}^{d}$, ν is the exterior normal of ∂Ω, and do(z) is the volume element of ∂Ω. Let us recall the definition of the divergence of a vector field $V = ({V }^{1},\ldots,{V }^{d}) : \Omega \rightarrow {\mathbb{R}}^{d}$:

$$\mathrm{div}V (x) :=\sum\limits_{i=1}^{d}\frac{\partial {V }^{i}} {\partial {x}^{i}} (x).$$

In order that (2.1.1) holds, it is, for example, sufficient that ∂Ω be of class C ¹.