# Gauss variational problem

A variational problem, first studied by C.F. Gauss (1840) , 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), 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) 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 $K$.