Weak Harnack inequalities for eigenvalues and constant rank theorem

• 1、 School of Mathematics, Hunan University , ChangSha 410082

Abstract：In the study of partial differential equations, the convexity of the solution is a classical topic. It describes an important geometric property of the solution. The constant rank theorem is a powerful tool to study the convexity of solutions of equations. The constant rank theorem is usually studied by using the strong maximum principle, recently Székelyhidi and Weinkove have introduced a new way to prove it. In this paper, based on this approach, we study the quasi-convex solutions of a class of quasi-linear elliptic equations. First, we slightly improve the first-order derivative estimates of the eigenvalues, and then use the estimates of the first- and second-order derivatives to establish a key differential inequality. Finally, we prove the weak Harnack inequality for the eigenvalues of the level set by applying the weak Harnack inequality for the semi-concave solution. As a direct corollary a quantitative constant rank theorem for the level set of solutions of this class of proposed linear elliptic equations is obtained.

Keywords： partial differential equations constant rank theorem weak harnack inequality level sets.