The integrability of negative powers of the solution of the Saint Venant problem

• Anthony Carbery ^[1] ; Vladimir Maz'ya ^[2] ; Marius Mitrea ^[3] ; David Rule ^[2]
  1. [1] University of Edinburgh

     University of Edinburgh

     Reino Unido

     
  2. [2] Linköping University

     Linköping University

     Linköpings S:t Lars, Suecia

     
  3. [3] University of Missouri at Columbia, USA

  Mostrar afiliaciones +
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 13, Nº 2, 2014, págs. 465-531
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We initiate the study of the finiteness condition Z  u(x)−" dx  C(, ") < +1 where  ✓ Rn 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 C1 domains in Rn, n % 2, with conical singularities (in particular polygonal domains in the plane), polyhedra in R3, and bounded domains which are locally of class C2 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 N with N sides circumscribed by the unit disc in the plane, then for each " 2 (0, 1) the following asymptotic formula holds:

    Z N uN (x)−" dx = 4"⇡ 1 − " + O(N"−1) as N !1.

    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.