Some Studies On Overdetermined Problems And Convexity Estimates In Elliptic Partial Differential Equations

This thesis mainly studies the overdetermined problem of k-curvature equation,the convexity estimates of Green's function,the first eigenfunction of Laplace operator and solutions of torsion problems.Overdetermined problem can be used to describe some physical problems and phe-nomena,such as how to ensure that the magnitude of the resulting traction which occurs at the surface of the bar is independent of position when a solid straight bar is subject to torsion.In addition,the overdetermined problem of the equation can be used to describe the geometric properties of the corresponding region.In chapter 2,we study the overde-termined problem of k-curvature equations satisfying Dirichlet and Neumann boundary value conditions on a bounded open domain.On the assumption of the existence of so-lutions to the overdetermined problems of the curvature equations,the maximum prin-ciple and the Pohozaev identity are used to prove that the solutions of these equations are radially symmetric and the region is a ball up to a translation.As an important geometric feature,convexity has attracted extensive attention from scholars at home and abroad.From chapter 3 to chapter 5,we study the convexity es-timation of the solution of the torsion problem,Green's function and the first eigen-function of Laplace operator,respectively.Under the assumption that the deformation function of the solution is strictly convex,we prove that the corresponding auxiliary function constructed is a superhamonic function.Then we use the deformation method to verify that the above assumption is true.