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The integrability of negative powers of the solution of the Saint Venant problem
Edinburgh University, UK.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
University of Missouri, USA.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-8976-8299
2014 (English)In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. XIII, no 2, 465-531 p.Article in journal (Refereed) Published
Abstract [en]

We initiate the study of the finiteness condition∫ Ω u(x) −β dx≤C(Ω,β)<+∞ whereΩ⊆R n is an open set and u is the solution of the Saint Venant problem Δu=−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 classC 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 Ω N with N sides circumscribed by the unit disc in the plane, then for each β∈(0,1) the following asymptotic formula holds: % {eqnarray*} \int_{\Omega_N}u_N(x)^{-\beta}\,dx=\frac{4^\beta\pi}{1-\beta} +{\mathcal{O}}(N^{\beta-1})\quad{as}\,\,N\to\infty. {eqnarray*} % 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 Δv≥c>0 .

Place, publisher, year, edition, pages
Scuola Normale Superiore , 2014. Vol. XIII, no 2, 465-531 p.
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Natural Sciences
Identifiers
URN: urn:nbn:se:liu:diva-108526ISI: 000339985500008Scopus ID: 2-s2.0-84908458320OAI: oai:DiVA.org:liu-108526DiVA: diva2:730709
Available from: 2014-06-29 Created: 2014-06-29 Last updated: 2017-12-05Bibliographically approved

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Maz'ya, VladimirRule, David

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