Comparison methods for semilinear elliptic problems on Riemannian manifolds with a Ricci lower bound

Abstract

In the first part of the article we develop a comparison method for positive solutions of the semilinear Dirichlet problem $\hat{I}”u+f(u)=0$ on domains $\hat{I}\text{c◯}\subset {\mathcal{M}}^n$ of a Riemannian manifold $({\mathcal{M}}^n,g)$ with a Ricci lower bound ${\mathrm{Ric}}_g\ge (n-1)kg$. Assuming admissibility and structural conditions on $f$, we prove a sharp pointwise gradient comparison, with a rigid characterization of the equality case. As applications, we derive an explicit isoperimetric-type inequality and a quantitative hot-spot localization estimate under natural convexity assumptions. In the second part, on ${\mathbb{S}}^n$ we show that isoparametric foliations produce non-rotational $f$-extremal domains, and that these examples descend to smooth quotients under free isometric actions preserving the foliation.

42 pages, 5 figures