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

Abstract

We initiate the study of the finiteness condition Ω u(x) -β dx ≤ C(Ω, β) < +∞ where Ω ⊆ R n is an open set and u is the solution of the Saint Venant problem ∆u = -1 in Ω, u = 0 on ∂Ω.The central issue which we address is that of determining the range of values of the parameter β > 0 for which the aforementioned condition holds under various hypotheses on the smoothness of Ω and demands on the nature of the constant C(Ω, β).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 uN is the solution of the Saint Venant problem in the regular polygon ΩN with N sides circumscribed by the unit disc in the plane, then for each β ∈ (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, ∇v(0) = 0 and ∆v ≥ c > 0.

Locations

• ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE - View - PDF
• Edinburgh Research Explorer (University of Edinburgh) - View - PDF