Research On Two Types Of Minkowski Problems

Minkowski problem is a predetermined measure problem,that is,given a non-zero finite Borel measure on the unit sphere,can a convex body be found to be geometric measure of the convex body?Minkowski put forward the problem of the existence and uniqueness of polyhedron with predetermined area and outer normal in the Euclidean space in 1897.This is the classic Minkowski problem.Aleksandrov extended the Minkowski problem on a polyhedron to the general convex body case in 1938,proved the variational formula of volume and derived the surface area measure,thus forming the variational theory system for solving the weak solution of the Minkowski problem.Since then,the Minkowski problem has developed rapidly.Under the condition that the convex body is sufficiently smooth,the solvability of the Minkowski problem is equivalent to the the solvability of corresponding Monge-Ampere equation.Studying this kind of problem promotes the intersection and fusion of convex geometry and partial differential equations.This thesis mainly studies the existence,regularity and uniqueness of solutions to the Lp Gaussian Minkowski problem and function class Minkowski problem of the dual Minkowski problem.The first content of this thesis is to study the Lp Gaussian Minkowski problem.On the one hand,we obtain the uniqueness of the solution of Lp Gaussian Minkowski problem by using Ehrhard inequality and its equal sign characterization.On the other hand,we first make a series of prior estimates,analyze the regularity of the solution,prove the existence of smooth solutions with the help of the degree theory in partial differential equations,and finally obtain the existence of weak solutions of Lp Gaussian Minkowski problem(p≥1)by using the approximation idea.The second content of this thesis is to study the function class Minkowski problem of the dual Minkowski problem.Due to the close connection between convex body sets and log-concave function space,we study the extension of the dual quermassintegrals on the logconcave function set-the(q-n)-th moment of the log-concave function f.In particular,when the function f is the characteristic function of a convex body which contain the origin in interior,the(q-n)-th moment is the dual quermassintegrals.In the dual theory,the reason why it took more than 40 years to find the dual curvature measure is that it is difficult to obtain the variational formula.With the help of the theory of bounded anisotropic weighted variation,we prove the variational formula of the(q-n)-th moment of the log-concave function,and derive two measures defined on Rn and Sn-1-Euclidean q-th dual curvature measure Cqe(f;·)and spherical q-th dual curvature measure Cqs(f;·).We mainly study the related Minkowski problem of q-th dual curvature measure Cqe(f;·),using the Aleksandrov variational method,we obtain the existence of weak solutions of the problem.