The integrability of negative powers of the soution to the Saint Venant problem

Abstract

We initiate the study of the finiteness condition Z u(x) dx  C(, ) < +1where ✓ R n is an open set and u is the solution of the Saint Venant problem 1u = 1 in , u = 0 on @. The central issue which we address is that of determining the range of values of the parameter > 0 for which the aforementioned condition holds under various hypotheses on the smoothness of and demands on the nature of the constant C(, ). Classes of domains for which our analysis applies include bounded piecewise C 1 domains in R n , n 2, with conical singularities (in particular polygonal domains in the plane), polyhedra in R 3 , and bounded domains which are locally of class C 2 and which have (finitely many) outwardly pointing cusps. For example, we show that if u N is the solution of the Saint Venant problem in the regular polygon N with N sides circumscribed by the unit disc in the plane, then for each 2 (0, 1) the following asymptotic formula holds:One of the original motivations for addressing the aforementioned issues was the study of sublevel set estimates for functions v satisfying v(0) = 0, rv(0) = 0 and 1v c > 0.

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].…”

Variable step mollifiers and applications

“…and use this function θ instead of η in the construction of the operator T . We briefly discuss how such a function θ can indeed be constructed: In [CMMR14] it was shown that for a bounded, open set Ω, whose boundary ∂Ω has zero Lebesgue measure and upper Minkowski dimension strictly less then N − α, for some 0 < α < N , it holds that (5.5)…”

On Sublevel Set Estimates and the Laplacian

“…Revisiting Carbery's estimate. A natural question is whether it is possible to weaken the assumptions in Carbery's estimate (this was also discussed by Carbery-Maz'ya-Mitrea-Rule [7] by very different means). We note that, with the inequality of arithmetic and geometric mean, we obtain…”