Some Reaserches On Hardy Type Inequalities In Nonconvex Domain

As is known to all,Hardy inequality is one of the most important inequalities in the analysis,in mathematics and physics,it have important applications in many fields.In recent years,due to the needs of the development of mathematical equations,the study of various Hardy type inequalities caused the wide attention of both at home and abroad,this article use the techniques and methods of analysis,focus on a type of Hardy inequality on anisotropic Heisenberg group in nonconvex domain and Hardy type inequality corresponding to Greiner vector field.Overall goal of this paper is based on the latest information and international trend of possession,adopted new methods and techniques,systematically study Hardy inequality in the nonconvex domain.The main researches of this paper are as follows:(1)Hardy type inequalities on anisotropic Heisenberg group in nonconvex domain.There are plenty of literatures about Hardy type inequality of distance functions,so far,most of researches are on theconvex area or average convex area,Research on Hardy type inequality in nonconvex domain has just started,and results are rare.In this paper,by constructing the special vector function,obtained a type of Hardy inequality on the anisotropic Heisenberg group in nonconvex domain,thus promoted the previous related conclusions.(2)Hardy type inequality corresponding to Greiner vector field.Greiner vector field is promotion of standard left unchanged vector field on the Heisenberg group,in this paper,by constructing a new auxiliary function and new subtle skills,Hardy type inequality on Heisenberg group in nonconvex domain has been generalized to Hardy type inequality corresponding to Greiner vector field in nonconvex domain.(3)Hardy type inequality corresponding to anisotropic Greiner vector field.In the second part,on the basis of the research,through subtle analysis and calculation,this paper further obtained Hardy type inequalities corresponding to anisotropic Greiner vector field in nonconvex domain,thus further promote the existing results.This paper adopted some new methods and techniques to the research of Hardy type inequality in nonconvex domain,obtained a type of Hardy inequality on the anisotropic Heisenberg group in nonconvex domain,a type of Hardy inequality corresponding to Greiner vector field and Hardy type inequality corresponding to anisotropic Greiner vector field.There are a series of discussion on Hardy type inequality innonconvex domain,this is supplement in the field of research on Hardy inequality,and will help the study of the regularity of elliptic equation.