The Torsion Function of Convex Domains of High Eccentricity

Abstract

The torsion function of a convex planar domain Ω 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.

A dimension-free Hermite–Hadamard inequality via gradient estimates for the torsion function

“…This equation has several different names: the torsion function [2] or the Saint Venant problem [8] in the theory of elasticity, the landscape function [10] in the theory of localization of eigenfunctions of elliptic operators and half of the expected lifetime of Brownian motion before exiting the domain Ω in probability theory. It is also one of the elliptic 'benchmark' PDEs which are usually the first testcase for new results [3,16,23]. We establish a new result for this equation; it is an interesting question to which extent it can be generalized to other equations.…”

“…There might be more than one point on the boundary that is of minimal distance and in each of these points there might be more than one supporting hyperplane -in case of ambiguity, any choice is admissible. We select a time T > 0 and study the (1) expected lifetime of particles colliding with the plane within T units of time, (2) the likelihood of a particle colliding with the plane within T units of time, (3) and what happens to the surviving articles. The first step is easy because of domain monotonicity: the likelihood of a Brownian motion impacting the boundary is made smaller if we enlarge the domain: in the first step we will thus not use any information about the convex domain except for the fact that Ω is in the upper half space defined by the supporting hyperplane.…”

Optimal Trapping for Brownian Motion: a Nonlinear Analogue of the Torsion Function

“…In particular, it is not very difficult to show that for b → 0 it converges to the torsion function (and thus has convex level sets on convex domains); conversely, for b → ∞, the interpretation as a drift-diffusion suggests that the solution should be of the form [14]) and should also have convex level sets on convex domains; one could wonder whether this is then also true in the intermediate regime b = 1. There are several other results about level sets [7,24,30,32] that may be interpreted as a stepping stones to a more complete theory of level sets of elliptic PDEs, we believe that −∆u − b · |∇u| = 1 might be another natural test case.…”

On Efficiency and Localisation for the Torsion Function

“…. Further examples demonstrating different behaviour of the torsion function, and the first eigenfunction of the Dirichlet Laplacian around their respective maxima for elongated convex planar domains have been constructed in [3].…”

“…Letting concludes the proof. To prove (108) we use [15, Theorem 1.1 (i)] for 2 to see that 2 3 . We conclude that the supremum in Eq.…”