Global solutions to special Lagrangian equations

Abstract

We show that any global solution to the special Lagrangian equations with the phase larger than a critical value must be quadratic.

A Bernstein problem for special Lagrangian equations in exterior domains

“…Simple solution sin x 1 e x2 and precious one (x 2 1 + x 2 2 )e x3 − e x3 + e −x3 /4 [W16] to (1.1) with Θ = (n − 2)π/2, n = 2 and n = 3 respectively show that the "critical" phase condition in Theorem 1.1 is indeed necessary. The "entire" Bernstein-Liouville type problem has been well-studied, see for instance [Bo92,Fu98,BCGJ03,Y02,Y06,WY08].…”

Hessian estimates for the sigma‐2 equationin dimension 3

“…[15,16]). Another consequence is the regularity (analyticity) of the C 0 viscosity solutions to (1.1) or (1.2) with n D 3 and ‚ D˙ 2 .…”

“…One immediate consequence of the above estimates is a Liouville-type result for global solutions with quadratic growth to (1.1); namely, any such solution must be quadratic (cf. [15,16]). Another consequence is the regularity (analyticity) of the C 0 viscosity solutions to (1.1) or (1.2) with n D 3 and ‚ D˙ 2 .…”

A priori estimate for convex solutions to special Lagrangian equations and its application

“…Actually, theorem 6.2 in [8] on the Dirichlet problem for general fully nonlinear elliptic equations with starlike/convex level set already applies to (1.1) with j‚j .n 1/ 2 , even j‚j .n 2/ 2 . This is because the level set (4.1) † is convex for large phase j‚j .n 2/ 2 ; see lemma 2.1 in [25].…”