The Weak Harnack Inequalities For Eigenvalues And Their Applications

In the study of partial differential equations,convexity of solutions is a classical and important topic.The constant rank theorem is a powerful tool for the study of convexity.It can deal not only with equations in Euclidean space,but also with various nonlinear differential equations involving symmetric curvature tensor on general manifolds.Moreover,the constant rank theorem has been widely applied to some classical geometric problems,such as the Christoffel-Minkowski problem and the prescribed Weingarten problem.Constant rank theorem is generally proved by using the strong maximum principle,recently Székelyhidi and Weinkove give a new idea of proof by introducing a linear combination of eigenvalues as an auxiliary function and establishing a weak Harnack inequality for the eigenvalues of the Hessian matrix,which leads to a quantitative version of the constant rank theorem for a class of fully nonlinear elliptic equations.The main work of this paper is to extend the new idea to obtain a new proof of the related constant rank theorem.The main result of this paper is divided into two parts,which we describe in detail below.In the first part,we study the quasi-convex solutions of a class of quasi-linear elliptic equations.We simply improve the first-order derivative estimates of the eigenvalues to obtain more precise results.For elliptic equations satisfying given structural condition,a critical elliptic differential inequality is established by using the first-order derivatives and the second-order derivative estimates.Then using the weak Harnack inequality for nonnegative semi-concave supersolution,we obtain the weak Harnack inequality for the eigenvalues of Weingarten tensor of level set.As a direct corollary,we obtain a constant rank theorem for the level set of solutions of this class of equations.In the second part,we study the convex solutions of a class of fullly nonlinear parabolic equations.First,we prove the parabolic weak Harnack inequalities for nonnegative semi-concave supersolutions by approximation.Since the estimates of the derivative of eigenvalues of the spatial Hessian matrix is consistent with those of the eigenvalues in the elliptic equation,we similarly establish parabolic differential inequalities.The technique of establishing the weak Harnack inequality for eigenvalues introduced by Székelyhidi and Weinkove is then extended to the parabolic equation.We obtain the weak Harnack inequality for the eigenvalues of the spatial Hessian matrix,and this result implies that rank of the spatial Hessian matrix is monotonically increasing with respect to time.