The torsion function of convex domains of high eccentricity

TL;DR

This paper investigates the torsion function in convex planar domains with high eccentricity, providing approximations and bounds for its shape and comparing it to the first eigenfunction of the Laplacian.

Contribution

It introduces an approximation method for the torsion function using perturbation of rectangles and establishes sharp bounds on its Hessian and level set shape.

Findings

Derived an approximation for the torsion function in elongated convex domains.

Established sharp bounds on the Hessian and shape of level sets near the maximum.

Compared the behavior of the torsion function with the first Laplacian eigenfunction.

Abstract

The torsion function of a convex planar domain has convex level sets, but explicit formulae are known only for rectangles and ellipses. Here we study the torsion function on convex planar domains of high eccentricity. We obtain an approximation for the torsion function by viewing the domain as a perturbation of a rectangle in order to define an approximate Green's function for the Laplacian. For a class of convex domains we use this approximation to establish sharp bounds on the Hessian and the infinitesimal shape of the level sets around its maximum. We also use these results to compare the behaviour of the torsion function and the first eigenfunction of the Dirichlet Laplacian around their respective maxima.