A Microscopic Convexity Principle for Nonlinear Partial Differential Equations

Caffarelli-Friedman [7] proved a constant rank theorem for convex solutions of semilinear elliptic equations in R2, a similar result was also discovered by Yau [28] at the same time. The result in [7] was generalized to R by Korevaar-Lewis [27] shortly after. This type of constant rank theorem is called microscopic convexity principle. It is a powerful tool in the study of geometric properties of solutions of nonlinear differential equations, it is particularly useful in producing convex solutions of differential equations via homotopic deformations. The great advantage of the microscopic convexity principle is that it can treat geometric nonlinear differential equations involving tensors on general manifolds. The proof of such microscopic convexity principle for σk-equation on the unit sphere