On pointmass Riesz external fields on the real axis
Abstract
The purpose of this work is twofold. First, we aim to extend for $0<s<1$ the results of one of the authors about equilibrium measures in the real axis in external fields created by pointmass charges for the case of logarithmic potentials ($s=0$). Our second motivation comes from the work of the other two authors on Riesz $s$equilibrium problems on finitely many intervals on the real line in the presence of external fields. They have shown that when the signed equilibrium measure has concave positive part on every interval, then the $s$equilibrium support is also a union of finitely many intervals, with one of them at most included in each of the initial intervals. As the positive part of the signed equilibrium for pointmass external fields is not necessarily concave, the investigation of the corresponding $s$equilibrium support is of comparative interest. Moreover, we provide simple examples of compactly supported equilibrium measures in external fields $Q$ not satisfying the usual requirements about the growth at infinity, that is, $\displaystyle \lim_{x\rightarrow \infty}\,Q(x) = \infty\,.$ Our main tools are signed equilibrium measures and iterated balayage algorithm in the context of Riesz $s$equilibrium problems on the real line. As these techniques are not mainstream work in the field and can be applied in other contexts we highlight their use here.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.03618
 Bibcode:
 2019arXiv190503618B
 Keywords:

 Mathematics  Classical Analysis and ODEs