The Research To Chord Length Of Convex Domain And The Number Of Solutions To Neumann Boundary Value Problem

This paper mainly studies two contents: the first is chord length distribution and averagechord length of plane convex domain;the second is the number of solutions to a second-orderNeumann boundary value problem with singular nonlinearity.The chord length distribution and the average chord length of convex domain is valuableissue in geometry, it is also the focus of many practical problems need to be solved in chemistry,physics, biology and architecture.Therefore,for half a century, it received wide attention andresearch.Many research methods and meaningful results about this problems are obtained.Forexample, explicit expressions for the chord length distribution functions have been known in thecase when the domain is a disc,a regular triangle, a rectangle and a general regular polygon.Butthese results limited to research about some of the simpler and symmetry good convexdomain.Because of the tool restrictions,the calculated very difficult.This paper using generalizedsupport function, restricted chord function and maximum chord length function and making ofintegral geometry theory and methods,and through the deformation of kinematic formulas.Thecalculation method on chord length distribution and the average chord length of a plane convexdomain is obtained.And this method is applied to the triangle,the chord distribution function andthe average chord length of arbitrary triangle is obtained.The methods about these problems in thispaper is relatively simple,and has some applicability.The Neumann boundary value problem has important applications in mathematic physics, forexample,equilibrium problems concerning beams,columns,or strings; fluid flow problems andheat transfer problems and so on. It has attracted considerable attention over the last twodecades.Under suitable conditions,some existence results or methods for the Neumann boundaryvalue problem have been established.In this paper,the number of solutions to a second-orderNeumann boundary value problem with singular nonlinearity are concerned. Comparisontheorem,maximum principle and upper solutions method are employed to come to a conclusionthat,in some condition,the number of solutions for this problem is odd.