Convexity Estimates And Superharmonicity For The Curvature Function To A Class Of Elliptic Partial Differential Equations

As one of the most important geometric features in mathematics,convexity has been studied for a long time.For the convexity of the solutions or the level sets of solutions to the most partial differential equations,it can be solved by means of differentiation.The P-function method is one of the classical microscopic methods.The advantage of this method is that it can obtain some kind of convexity estimate of the solution for the elliptic partial differential equation through auxiliary functions.In this paper,we use the P-function method to study the convexity estimate of the solution of the Dirichlet problem of semilinear elliptic equation:where 0≤p<1,λ>0,Ω is a smooth bounded convex domain in Rn.In this paper,we first find a curvature function related to the solution,and then prove that it is a superharmonic function.Therefore,we can use the minumum principle to show that the auxiliary function attains its minimum on the boundary,and then give the relevant convexity estimate of the solution.