Research On The Minkowski Type Problems And The Brunn-Minkowski Type Inequalities For Several Types Of Functionals

The main contents of this article come from convex geometry,the core of which is the Brunn-Minkowski theory.The Brunn-Minkowski inequality and the Minkowski prob-lem play important roles in the Brunn-Minkowski theory.We study the Minkowski type problems and the Brunn-Minkowski type inequalities for several types of functionals.The specific contents are as follows.In Chapter 2,we study the Orlicz Brunn-Minkowski theory for the first eigenvalue of the p-Laplacian operator.We firstly establish the Orlicz Hadamard variational formula for the first eigenvalue of the p-Laplacian operator when p>1,and then obtain the Orlicz type Minkowski inequality by defining the Orlicz mixed first eigenvalue of the p-Laplacian operator reasonably.Furthermore,we obtain the Orlicz Brunn-Minkowski inequality for the first eigenvalue of the p-Laplacian operator when p>1,and obtain the equality con-ditions.Moreover,the Brunn-Minkowski inequality is a key step to solve the uniqueness of solutions to Minkowski type problem.So we prove the uniqueness of solutions to the Orlicz Minkowski type problem in the smooth case with p>1 by using the equality case of the Orlicz Brunn-Minkowski inequality.These results are generalizations of the classical Brunn-Minkowski theory.In Chapter 3,we continue to study the extension of Brunn-Minkowski theory.We propose the L_qMinkowski type problem for the eigenvalue of the k-th Hessian operator.Unlike Chapter 2,we use the concavity of the eigenvalue of the k-th Hessian operator to prove its L_qBrunn-Minkowski inequality.Meanwhile,we prove the Minkowski inequality for the eigenvalue of the k-th Hessian operator.Subsequently,the L_qHadamard variational formula is further established.So we define the L_qmixed eigenvalues of the k-th Hessian operator,and then prove the uniqueness of solutions to the L_qMinkowski type problem when q>1 in the smooth case.In Chapter 4,we study the L_qdual Minkowski type problem of the mixed Hessian operators.Solving this problem is equivalent to find the geometrically convex solution of the following mixed Hessian equations.σ_k（η）=u^q-1f（x）（u²+|u|²）^k+12^-p,?x∈Sⁿ,（0.2）where σ_k（η）is mixed Hessian operator and k=1,...,n.We firstly obtain a prior estimate of the positiveΓ_kadmissible solutions to the equation（0.2）.The existence of the admissible solutions to the equation（0.2）are obtained by using the continuity method when q>p and p≤1.Moreover,the existence of the geometric convex solutions to the equation（0.2）has been proved by the Full Rank Theorem when q>p and p≤1.In Chapter 5,we study the shape optimization problems of p-capacity under the L_qsum when p∈（1,2）and q>1,which belongs to the isoperimetric problem in Brunn-Minkowski theory.In fact,the kind of shape optimization problems we study are the new isoperimetric inequalities equivalent to the Brunn-Minkowski inequalities.We introduce the definition of q-perimeter and given symmetry constraints,and consider that in a class of convex polygons with fixed number of sides.We use the classical Brunn-Minkowski inequality of p-capacity to prove that the regular polygons are the optimal solutions for a variety of shape optimization problems with p-capacity under the L_q sum.