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Báo cáo hóa học: " Singular integrals of the compositions of LaplaceBeltrami and Green’s operators"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Singular integrals of the compositions of LaplaceBeltrami and Green’s operators | Fang and Ding Journal of Inequalities and Applications 2011 2011 74 http www.journalofinequalitiesandapplications.eom content 2011 1 74 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Singular integrals of the compositions of LaplaceBeltrami and Green s operators Ru Fang 1 and Shusen Ding2 Correspondence fangr@hit.edu. cn department of Mathematics Harbin Institute of Technology Harbin 150001 P.R.China Full list of author information is available at the end of the article Springer Abstract We establish the Poincaré-type inequalities for the composition of the LaplaceBeltrami operator and the Green s operator applied to the solutions of the non-homogeneous A-harmonic equation in the John domain. We also obtain some estimates for the integrals of the composite operator with a singular density. Keywords Poincaré-type inequalities differential forms A-harmonic equations the Laplace-Beltrami operator Green s operator 1 Introduction The purpose of the article is to develop the Poincaré-type inequalities for the composition of the Laplace-Beltrami operator A dd d d and Green s operator G over the d-John domain. Both operators play an important role in many fields including partial differential equations harmonic analysis quasiconformal mappings and physics 1-6 . We first give a general estimate of the composite operator A o G. Then we consider the composite operator with a singular factor. The consideration was motivated from physics. For instance when calculating an electric field we will deal with the integral p Í 1_ f Í1Í vt_ _4__fl A A crln o vo A Í í c A 11 A r.TO on ci A A 1 A í c rh o i nroncol AÍ ori . r 4ns0 Jd p x r-x 3 x where r x is a charge density and x is the integral variable. It is singular if r e D. Obviously the singular integrals are more interesting to us because of their wide applications in different fields of mathematics and physics. In this article we assume that M is a bounded convex domain and B is a ball .