Torsion and ground state maxima: close but not the same

TL;DR

This paper investigates whether the maximum points of solutions to a semilinear Poisson equation are independent of the nonlinearity, providing counterexamples and exploring the relationship between torsion and eigenfunction maxima.

Contribution

It constructs counterexamples on specific domains showing the conjecture fails and highlights the unexpected proximity of maxima between torsion functions and eigenfunctions.

Findings

Counterexamples on half-disk and right triangle domains.

Maxima of torsion function and eigenfunction are closely located.

Open problem: quantify the closeness in terms of domain and nonlinearity.

Abstract

Could the location of the maximum point for a positive solution of a semilinear Poisson equation on a convex domain be independent of the form of the nonlinearity? Cima and Derrick found certain evidence for this surprising conjecture. We construct counterexamples on the half-disk, by working with the torsion function and first Dirichlet eigenfunction. On an isosceles right triangle the conjecture fails again. Yet the conjecture has merit, since the maxima of the torsion function and eigenfunction are unexpectedly close together. It is an open problem to quantify this closeness in terms of the domain and the nonlinearity.