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Published in:
2019 | OriginalPaper | Chapter
3. The Theory of Potential
Authors : Marin Marin, Andreas Öchsner
Published in: Essentials of Partial Differential Equations
Publisher: Springer International Publishing
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Abstract
The Newtonian potential, or the potential of volume, associated to the Laplace equation \(\Delta u=0\) is, by definition, the following improper integral: where \(r=r_{\xi x}=\left| \overline{\xi x}\right| =\sqrt{ \sum \limits _{i=1}^n\left( x_i-\xi _i\right) ^2}\), and \(\omega _n\) is the area of the unit sphere from the n-dimensional space \(\mathrm{I}\!\mathrm{R}^n\).
$$\begin{aligned} U(\xi )=-\frac{1}{(n-2)\omega _n}\int _{\Omega }\frac{f(x)}{r^{n-2}}\mathrm {d}x, \end{aligned}$$