| In convex geometry and geometric analysis,the Minkowski problem of torsional rigidity functional is an important research topic.The variational formula of functional is an important prerequisite and key step to solve Minkowski problem,and has been widely used in some classical geometric problems.The k-Hessian equation is an important class of completely nonlinear elliptic equations,which appears in a series of important mathematical problems such as Alexandrov problem for predetermined Weingarter curvature measures,Minkowski problem,optimal transport problem and Special Lagrangian several.The k-Hessian refers to the k order elementary symmetric polynomial of the eigenvalues of the Hessian matrix.When k 2,the k-Hessian equation is completely nonlinear.In this paper,we study the variational formulas of k-Hessian equation torsional rigidity functional by using the properties of k-Hessian operator.The main research achievements and innovations are divided into two parts,as follows:In the first part,we study the properties of torsional rigidity functional corresponding to k-Hessian equation.We use the definition of torsional rigidity functional to prove the equivalence between the extreme value function of torsional rigidity functional and the corresponding solution of k-Hessian equation.Then,with the help of the properties of k-Hessian operator,we derive the expressions of torsional rigidity functional when it takes an extreme value function.In the second part,we study the variational formula of the k-Hessian equation torsional rigidity.First of all,we use the properties of k-Hessian operator to get two identities of the k-Hessian equation.Then,we establish the first order variational formula of torsional rigidity in smooth domain.Finally,we obtain the secondary variational formula of torsional rigidity for the k-Hessian equation on the unit ball. |
