Gelfand problem on a large spherical cap

TL;DR

This paper investigates the minimal solution to the Gelfand problem on a spherical cap, analyzing its asymptotic behavior as the cap enlarges to the entire sphere, using new eigenvalue estimates.

Contribution

It provides a sharp estimate of the torsion function on spherical caps and explores the solution's asymptotics as the cap approaches a full sphere.

Findings

Derived a sharp estimate of the torsion function in terms of the principal eigenvalue.

Analyzed the asymptotic behavior of solutions as the spherical cap expands.

Connected the solution behavior to geometric properties of the spherical cap.

Abstract

We study the behaviour of the minimal solution to the Gelfand problem on a spherical cap under the Dirichlet boundary conditions. The asymptotic behaviour of the solution is discussed as the cap approaches the whole sphere. The results are based on the sharp estimate of the torsion function of the spherical cap in terms of the principle eigenvalue which we derive in this work.