| Convex geometry is a branch of geometry that studies the geometric structure and invariants of convex bodies and star bodies by using the methods of geometry and analysis.It is closely related to partial differential equations,functional analysis,variational methods and other related fields.The Minkowski problem is one of the key problems of convex geometry.It is of great significance to study the existence and regularity of solutions to the Minkowski type problem.This paper draws the following conclusions:First,The existence results of smooth solutions of Lp dual Minkowski problem.Secondly,the existence result of the solution of the dual Orlicz-Minkowski problem.Then the existence result of smooth solution of Gauss image problem is obtained.Finally,regularity results of degenerate solutions of Lp dual Minkowski problem is obtained.The following is the content arrangement of this paper:In the first chapter,we introduce the background and conclusions of the Minkowski type problem,and then we give the structure and main results of this paper.In the second chapter,we first introduce the relevant concepts in convex geometry,especially the properties of support functions and radial functions,and then introduce some important theorems and conclusions.Finally,the related theory of second order parabolic equation is given.In the third chapter,we discuss the existence of smooth solutions for a class of very important Minkowski type problem—Lp dual Minkowski problem.We consider a class of anisotropic inverse Gauss curvature flows,assuming that the initial surface is a closed smooth strictly convex hypersurface and some conditions are satisfied,then the long-time existence of the flows can be obtained,and the solution of inverse Gauss curvature flow will converge to the smooth solution of Lp dual Minkowski problem.In the fourth chapter,we discuss the existence of solutions for a class of dual Orlicz-Minkowski problem by using Gauss curvature flow.The dual Orlicz-Minkowski problem has a nonhomogeneous term,which makes it very difficult to study the problem.After obtaining the long time existence of flows,we successfully obtain the existence of smooth solutions for this class of dual Orlicz-Minkowski problems.In the fifth chapter,we discuss the existence of smooth solution of Gauss image problem,which is also obtained by considering the long time existence and asymptotic behavior of a class of Gauss curvature flow.Compared with the classical Minkowski type problem,the Gauss image problem contains two given measures,so the estimation is more precise.In the sixth chapter,we discuss the C1,1 regularity of degenerate solutions of the Lp dual Minkowski problem.The degenerate problem is transformed into a non degenerate problem by perturbation method.The key is to obtain a priori estimate independent of (?). |
