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 \({C}^{1}(\Omega ) \cap {C}^{0}(\bar{\Omega }),\)
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 \(V = ({V }^{1},\ldots,{V }^{d}) : \Omega \rightarrow {\mathbb{R}}^{d}\):
In order that (2.1.1) holds, it is, for example, sufficient that ∂Ω be of class C 1.
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Notes
- 1.
\({C}_{0}^{\infty }(\Omega ) :=\{ f \in {C}^{\infty }(\Omega )\), \(\mathrm{supp}(f) := \overline{\{x : f(x)\neq 0\}}\) is a compact subset of Ω}.
- 2.
Here, \(\|\partial \Omega \|\) denotes the measure of the boundary ∂Ω of Ω; it is given as \(\int\limits_{\partial \Omega }do(x)\).
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Jost, J. (2013). The Laplace Equation as the Prototype of an Elliptic Partial Differential Equation of Second Order. In: Partial Differential Equations. Graduate Texts in Mathematics, vol 214. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4809-9_2
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