Generalisation of Hadamard’s inequality for convex functions to higher dimensions, and an application to the elastic torsion problem

The (generalised) torsion function u of a domain Ω ⊂ IR is a function which is zero on the boundary of the domain and whose Laplacian is minus one at every point in the interior of the domain. Denote by |Ω| the measure of Ω, xc its centroid. We establish, for convex Ω, 3 2(n+ 1)2 ≤ 1 (n + 1)2 |Ω| ∫ Ω u max Ω u ≤ |Ω|u(xc) ∫ Ω u ≤ |Ω| ∫ Ω u max Ω u ≤ 1 2 (n+ 1)(n+ 2). We generalise this to positive solutions u of the semilinear problem −∆u = uγ, 0 ≤ γ < 1, satisfying homogeneous Dirichlet boundary conditions. 1 Foundational materials 1.