View source for Gauss variational problem

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A variational problem, first studied by [[Gauss, Carl Friedrich|C.F. Gauss]] (1840) [[#References|[1]]], which may be formulated in modern terms as follows. Let $\mu$ be a positive measure in a Euclidean space $\mathbf R^n$, $n\geq3$, of finite energy (cf. [[Energy of measures|Energy of measures]]), and let

$$U^\mu(x)=\int\frac{1}{|x-y|^{n-2}}d\mu(y)$$

define the [[Newton potential]] $U^\mu$ of $\mu$. Out of all measures $\lambda$ with compact support $K\subset\mathbf R^n$ it is required to find a measure $\mu_0$ giving the minimum of the integral

$$\int(U^\lambda-2U^\mu)d\lambda,$$

which is the scalar product ($\lambda-2\mu,\lambda$) in the pre-Hilbert space of measures of finite energy.

The importance of the Gauss variational problem consists in the fact that the equilibrium measure (cf. [[Robin problem|Robin problem]]) may be obtained as a solution of the Gauss variational problem for a certain choice of $\mu$; for example, $\mu$ may be taken to be a homogeneous mass distribution over a sphere with centre in the coordinate origin that includes $K$.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  C.F. Gauss,   "Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehung- und Abstössungs-Kräfte" , ''Werke'' , '''5''' , K. Gesellschaft Wissenschaft. Göttingen  (1877)  pp. 195–242</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.S. [N.S. Landkov] Landkof,   "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Brélot,   "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR>
</table>



====Comments====
The measure $\mu_0$ is the projection of $\mu$ on the convex cone of all positive measures $\lambda$, of finite energy, and with support contained in $K$. See also [[#References|[a1]]], Chapt. I.XIII for a treatment of this subject.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Doob,   "Classical potential theory and its probabilistic counterpart" , Springer  (1984)</TD></TR></table>