Convex Solutions of Elliptic Differential Equations in Classical Differential Geometry

The issue of convexity is fundamental in the theory of partial differential equations. We discuss some recent progress of convexity estimates for solutions of nonlinear elliptic equations arising from some classical problems in differential geometry. We first review some works in the literature on the convexity of solutions of quasilinear elliptic equations in Rn. The study of geometric properties of the harmonic function and solutions of general elliptic partial differential equations was initiated long time ago, such as the location of the critical points and the star-shapeness of the level set, etc. Gabriel [16] obtained the strict convexity of level set for the Green function in thee dimension convex domain in R3. Makar-Limanov [34] studied equation ∆u = −1 in Ω, (1.1) u = 0 on ∂Ω,