[Paper Review] A Bernstein problem for special Lagrangian equations in exterior domains

Dong‐Sheng Li, Zhisu Li+1 more|arXiv (Cornell University)|Sep 14, 2017

Geometry and complex manifolds16 references3 citations

TL;DR

This paper establishes quadratic asymptotics for solutions to special Lagrangian equations with supercritical phases in exterior domains, proving that solutions approach a unique quadratic polynomial at infinity. The method combines an exterior Liouville theorem for fully nonlinear elliptic equations with a rotation-based Hessian bound technique, extending results to Monge-Ampère, quadratic Hessian, and inverse harmonic Hessian equations in exterior domains.

ABSTRACT

We establish quadratic asymptotics for solutions to special Lagrangian equations with supercritical phases in exterior domains. The method is based on an exterior Liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded Hessian, and also certain rotation arguments toward Hessian bound. Our unified approach also leads to quadratic asymptotics for convex solutions to Monge-Ampère equations (previously known), quadratic Hessian equations, and inverse harmonic Hessian equations over exterior domains.

Motivation & Objective

• To establish an exterior Bernstein-type result for special Lagrangian equations with supercritical phases in exterior domains.
• To prove that solutions with supercritical phases are asymptotic to a unique quadratic polynomial at infinity.
• To extend the asymptotic analysis to convex solutions of Monge-Ampère, quadratic Hessian, and inverse harmonic Hessian equations over exterior domains.
• To develop a unified framework using exterior Liouville theorems and rotation arguments to control Hessian growth and establish boundedness.
• To demonstrate that the critical phase condition is necessary, as solutions with critical phase may not exhibit quadratic asymptotics.

Proposed method

• Derives an exterior Liouville theorem for general fully nonlinear uniformly elliptic concave equations with bounded Hessian, ensuring the Hessian of solutions converges at infinity.
• Applies Krylov-Safonov weak Harnack inequality and Evans-Krylov Hessian estimates to control pure second derivatives of solutions.
• Employs a rotation device from previous work (Y02, Y06) to ensure uniform ellipticity and Hessian boundedness in the absence of gradient bounds.
• Uses Legendre transformation to convert the special Lagrangian equation into a Laplace equation with bounded Hessian, enabling asymptotic analysis.
• Applies Schauder estimates and divergence arguments in the transformed space to derive precise asymptotic expansions in both n ≥ 3 and n = 2 cases.
• For n = 2, incorporates complex analysis and logarithmic correction terms via a π/2 rotation and δ-function argument to derive the logarithmic asymptotic term.

Experimental results

Research questions

• RQ1 Can solutions to special Lagrangian equations with supercritical phases in exterior domains be shown to asymptotically approach a quadratic polynomial?
• RQ2 What conditions ensure the Hessian of a solution remains bounded and converges at infinity in exterior domains?
• RQ3 How can the asymptotic behavior of solutions to fully nonlinear elliptic equations be controlled when gradient bounds are not assumed?
• RQ4 To what extent do the asymptotic results extend to other equations such as Monge-Ampère, quadratic Hessian, and inverse harmonic Hessian equations?
• RQ5 Is the critical phase condition necessary for quadratic asymptotics, and what happens when it is violated?

Key findings

• For n ≥ 3, every smooth solution to the special Lagrangian equation with |Θ| > (n−2)π/2 satisfies u(x) = Q(x) + O_k(|x|^{2−n}) as |x| → ∞, for all k ∈ ℕ.
• For n = 2, the solution satisfies u(x) = Q(x) + (d/2)log(x^T(D²Q)²x) + O_k(|x|^{−1}) as |x| → ∞, for all k ∈ ℕ, with d explicitly given by a boundary integral.
• The Hessian of the solution converges to a constant matrix A at infinity, with D²u(x) = A + O_k(|x|^{−n}) as |x| → ∞.
• The asymptotic quadratic polynomial Q(x) is unique, and its Hessian is determined by the limit of D²u at infinity.
• The method applies uniformly to convex solutions of Monge-Ampère equations, quadratic Hessian equations, and inverse harmonic Hessian equations over exterior domains.
• The critical phase condition |Θ| ≥ (n−2)π/2 is necessary, as counterexamples exist at the critical phase, confirming the sharpness of the result.