Second Order Elliptic Equations Convexity

The geometrical property of the solutions of the second order elliptic equations is a classical topic in the research of the partial differential equation. In this paper, we introduce some method to deal with the convexity of the solutions of the second order elliptic equations. Moreover, we use the method of establishing the concave envelope, proving the following theorem:Theorem Let Rn be a bounded, convex domain, A > 0, p > 0, if u is the solution of the problemThen u1/2 is concave in. In particular the level sets of u are convex.Problem (0.1) was proposed by Kawohl [23]. When p = 1, Kawohl [23] proved arcsin(1 - u) is convex in SI, and so it has convex level sets. However, when p > 0 and p=1, the method of Kawohl cann't be applied.