Power Convexity Of Solutions To A Special Lagrangian Equation In Dimension Two

Given an elliptic boundary value problem on a bounded convex domain in Rn,does the solution inherit some convexity properties from the boundary?In this paper,we prove power convexity result of solution to Dirichlet problem of special Lagrangian equation in dimension two.This provides new example of fully nonlinear elliptic boundary value problem whose solution shares power convexity property previously only knew for 2-Hessian equation in dimension three.The key ingredients consist of microscopic convexity principles and deformation methods.More precisely,the main results of the paper are as follows.Let Ω be a bounded,smooth,strictly convex domain in R2.Let θ∈(0,π/2)be a constant.Assume that u∈C∞(Ω)be the unique solution of Dirichlet problem where λ=(λ1,λ2)are the eigenvalues of D2u.Then the function v=(?)is strictly convex in Ω.