Abstract

We initiate the study of the finiteness condition ∫Ωu-Bdx≤C 0 for which the aforementioned condition holds under various hypotheses on the smoothness of Ω and demands on the nature of the constant C. Classes of domains for which our analysis applies include bounded piecewise C1 domains in Rn, n ≥ 2, with conical singularities, polyhedra in R3, and bounded domains which are locally of class C2 and which have 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 B ϵ the following asymptomatic formula holds: ∫ΩNuN-Bdx=4Bπ/1/B+O as N→∞. One of the original motivations for addressing the aforementioned issues was the study of sublevel set estimates for functions v satisfying v = 0, Vv and ∆v≥C>0