The support of the equilibrium measure for a class of external fields on a finite interval.
The determination of the support of the equilibrium in the external field is a key step in the solution of many extremal problems in analysis. Unfortunately, it is also an essentially non-trivial one. There are not so many general results describing the properties of the support in terms of the equilibrium compact or the external field. Some of them are well known (for instance, when a convex and smooth external field is acting on the real line). One of the main results of this paper is a rather general sufficient condition on the external field on an interval ensuring that the support of the corresponding equilibrium measure is the union of at most two intervals. It is applied further to a non-smooth analogue of a monomial field of the form \(Q(x)=-c \, \text{sign}(x) | x| ^\alpha \), with \(c>0\) and \(\alpha \geq 1\), satisfying the conditions of the main theorem.
scientific article; zbMATH DE number 1867907