A microscopic convexity theorem of level sets for solutions to elliptic equations

摘要

We study the microscopic level-set convexity theorem for elliptic equation Lu = f(x, u, Du), which generalize Korevaars’ result in (Korevaar, Commun Part Diff Eq 15(4):541–556, 1990) by using different expression for the elementary symmetric functions of the principal curvatures of the level surface. We find out that the structure conditions on equation are as same as conditions in macroscopic level-set convexity results (see e.g. (Colesanti and Salani, Math Nachr 258:3–15, 2003; Greco, Bound Value Prob 1–15, 2006)). In a forthcoming paper, we use the same techniques to deal with Hessian type equations.