Curvature Estimates For The Level Sets Of The Solutions To A Class Of Fully Nonlinear Equations

As one of the important geometric features,convexity has always been a hot topic in elliptic partial differential equations,so it is necessary to study convexity.Qualitative research is to prove the convexity of the solution itself or prove the convexity for the level sets of the solutions to the elliptic equations.Quantitative research is to further obtain the degree of convexity on the basis of qualitative research.The curvature estimates is one of the common methods in quantitative research.With this method,we can know the degree of convexity for the level sets of the solution and the degree of convexity for the graph of the solution.In this paper,we mainly study Gauss curvature estimates for the level sets of the strictly convex solution u to the following 0 boundary value Dirichlet problem of the Hessian quotient equation:Firstly,we construct the auxiliary function related to the Gauss curvature of the level sets of the solution.Secondly,we prove that the auxiliary function satisfies the differential inequality,that is to say,it is greater than or equal to zero for the elliptic operator acting on the auxiliary function.Finally,by maximum principle,the auxiliary function attains its maximum on the boundary.Therefore,we obtain the Gauss curvature estimates of the level sets for the solution to this equation.