For essentially convex interaction kernels, a threshold condition on an auxiliary density guarantees existence, uniqueness, and compact support of minimizers in R^d and curved half-spaces; a counterexample shows the threshold fails for superharmonic kernels.
Riesz equilibrium on a ball in the external field of a point charge
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abstract
We investigate the Riesz energy minimization problem on a $d$-dimensional ball in the presence of an external field created by a point charge above the ball in $\R^{d+1}$, $d\geq1$. Both cases of an attractive charge and a repulsive charge are considered. The notion of a signed equilibrium measure is one of the main tools in the present study. For the case of a positive (repulsive) charge, the determination of the support of the equilibrium measure is a nontrivial question. We solve it in the one-dimensional case by making use of iterated balayage, a method already applied in logarithmic potential theory. Here we use a modified version of it, in order to handle the phenomenon of mass loss, characteristic of the Riesz balayage of positive measures. Moreover, we also consider minimization of Coulomb energy on the ball in dimension $d\geq2$, and of logarithmic energy on the segment in dimension 1. Different techniques are used for these two cases.
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Existence of minimizers for interaction energies with external potentials
For essentially convex interaction kernels, a threshold condition on an auxiliary density guarantees existence, uniqueness, and compact support of minimizers in R^d and curved half-spaces; a counterexample shows the threshold fails for superharmonic kernels.