View source for Gauss variational problem

A variational problem, first studied by C.F. Gauss (1840) [[#References|[1]]], which may be formulated in modern terms as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g0435701.png" /> be a positive measure in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g0435702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g0435703.png" />, of finite energy (cf. [[Energy of measures|Energy of measures]]), and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g0435704.png" /></td> </tr></table>

define the [[Newton potential|Newton potential]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g0435705.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g0435706.png" />. Out of all measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g0435707.png" /> with compact support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g0435708.png" /> it is required to find a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g0435709.png" /> giving the minimum of the integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g04357010.png" /></td> </tr></table>

which is the scalar product (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g04357011.png" />) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g04357012.png" />; for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g04357013.png" /> may be taken to be a homogeneous mass distribution over a sphere with centre in the coordinate origin that includes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g04357014.png" />.

====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g04357015.png" /> is the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g04357016.png" /> on the convex cone of all positive measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g04357017.png" />, of finite energy, and with support contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043570/g04357018.png" />. 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>