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arXiv · 2510.15098

The role of the curvature of a surface in the shape of the solutions to elliptic equations

Francesca Gladiali·Massimo Grossi·Luigi Provenzano
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Abstract

We prove uniqueness and non-degeneracy of the critical point of positive, semi-stable solutions of $-\Delta u=f(u)$ with Dirichlet boundary conditions for a class of star-shaped domains of the sphere and of the hyperbolic plane satisfying a geometric condition. In the spherical case, this condition is weaker than convexity, while in the hyperbolic case it is weaker than horoconvexity. Finally, we construct examples showing that this geometric condition is indeed optimal.

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Francesca Gladiali, Massimo Grossi, Luigi Provenzano. 2025-10-16. The role of the curvature of a surface in the shape of the solutions to elliptic equations. https://arxiv.org/abs/2510.15098

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