A regularity theorem for stationary measures
Analysis of PDEs
2026-06-26 v1
Abstract
We investigate a variational problem for eigenvalues of the Laplace-Beltrami operator on smooth manifolds with respect to Radon measures belonging to a suitable class; we are motivated by conformal eigenvalues in dimension two. Our main result is a regularity result for stationary measures with respect to outer variations. More precisely, we prove that any sufficiently regular stationary measure is absolutely continuous with respect to the classical volume measure and that its density is induced by an harmonic map. Our result has some interesting applications to Steklov eigenvalues on subdomains.
Cite
@article{arxiv.2606.28211,
title = {A regularity theorem for stationary measures},
author = {Luigi Ambrosio and Giovanni Siclari},
journal= {arXiv preprint arXiv:2606.28211},
year = {2026}
}