Math 2.0

Potential theory Potentials, energy and capacity Problem

Gauss variational problem

Builds on Compact setNewtonian potentialEnergy of a measureCapacity of a set

Problem

Let n≥3n \ge 3, let K⊆ℝnK \subseteq \R^n be a compact set of positive capacity, and let f ⁣:K→ℝf \colon K \to \R be a continuous function. For a positive Borel measure μ\mu on KK, let Uμ(x)=∫K∣x−y∣2−n dμ(y)U^\mu(x) = \int_K \abs{x - y}^{2-n} \, d\mu(y) be its Newtonian potential. The Gauss integral of μ\mu is Gf(μ)=∫K(Uμ−2f)dμ,G_f(\mu) = \int_K \left( U^\mu - 2 f \right) d\mu, that is, the energy of μ\mu minus twice the integral of ff. The problem asks for a positive Borel measure μ\mu on KK with μ(K)=1\mu(K) = 1 that makes Gf(μ)G_f(\mu) as small as possible.

Remarks

Under these hypotheses a minimizing measure exists and is unique. Existence follows from the weak-star compactness of the probability measures on KK and the lower semicontinuity of the energy. Uniqueness follows from the strict positive definiteness of the Newtonian energy on signed measures of finite energy. For f=0f = 0 the problem is to minimize the energy among unit measures on KK, and its solution is the equilibrium measure of KK. The minimizer μ\mu is characterized by two inequalities with one constant cc: Uμ−f≥cU^\mu - f \ge c on KK outside a set of capacity zero, and Uμ−f≤cU^\mu - f \le c on the support of μ\mu. Gauss treated the problem for forces that follow the inverse square law. Rigorous proofs of existence rely on the measure theory that was developed later.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Capacity of a set, concept, no evidence levelCapacity of a setconceptCompact set, concept, no evidence levelCompact setconceptEnergy of a measure, concept, no evidence levelEnergy of a measureconceptNewtonian potential, concept, no evidence levelNewtonian potentialconceptGauss variational problem, problem, no evidence levelGauss variational pro…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Capacity of a set concept
  2. Compact set concept
  3. Energy of a measure concept
  4. Newtonian potential concept
  5. Gauss variational problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Capacity_of_a_set
About this page
Tag94GSMEWPTN
URIm2://94GSMEWPTN
Kindproblem
WritingW0 stub
MSC31B15, 31A15
StatusThis copy of the site has no build data. Run m2 status 94GSMEWPTN.
AboutEquilibrium measure
UsesCompact set, Newtonian potential, Energy of a measure, Capacity of a set

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