Comparison methods for semilinear elliptic problems on Riemannian manifolds with a Ricci lower bound
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910085367726080 |
|---|---|
| author | Espinar, José M. González-Ibáñez, Fernán Marín, Diego A. |
| author_facet | Espinar, José M. González-Ibáñez, Fernán Marín, Diego A. |
| contents | In the first part of the article we develop a comparison method for positive solutions of the semilinear Dirichlet problem $Δu+f(u)=0$ on domains $Ω\subset \mathcal M^n$ of a Riemannian manifold $(\mathcal{M}^n,g)$ with a Ricci lower bound $\operatorname{Ric}_g\ge (n-1)k\,g$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28479 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Comparison methods for semilinear elliptic problems on Riemannian manifolds with a Ricci lower bound Espinar, José M. González-Ibáñez, Fernán Marín, Diego A. Differential Geometry Analysis of PDEs 35N25, 35J61, 53C21, 58J05 In the first part of the article we develop a comparison method for positive solutions of the semilinear Dirichlet problem $Δu+f(u)=0$ on domains $Ω\subset \mathcal M^n$ of a Riemannian manifold $(\mathcal{M}^n,g)$ with a Ricci lower bound $\operatorname{Ric}_g\ge (n-1)k\,g$. 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. |
| title | Comparison methods for semilinear elliptic problems on Riemannian manifolds with a Ricci lower bound |
| topic | Differential Geometry Analysis of PDEs 35N25, 35J61, 53C21, 58J05 |
| url | https://arxiv.org/abs/2603.28479 |