A Bernstein problem for special Lagrangian 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.

Asymptotic behavior of solutions of fully nonlinear equations over exterior domains

“…2.1. D 2 u tends to some symmetry positive definite matrix at infinity with (5) The following Lemma 7 shows that the Hessian matrix of u in Theorem 4 tends to some symmetric positive definite matrix at infinity by making use of the method given by Li et al in [14,Lemma 2.1]. Then, the following auxiliary lemma together with Theorem 1 implies Theorem 4, immediately.…”

“…Then there exists a constant v ∞ such that (12) holds if t < n, (13) if t ≥ n, and (15) if t > n with (14).…”

“…Essentially, [5] provided a incisive observation to analyse the asymptotic behavior for fully nonlinear equations. Recently, many researchers studied the Liouville theorem and asymptotic behavior for various types of fully nonlinear equations such as k-Hessian equations [2,7], parabolic k-Hessian equations [19], Hessian quotient equations [13], special Lagrangian equations [14,18], Lagrangian mean curvature equations [1], parabolic Monge-Ampère equations [21][22][23], some fully nonlinear degenerate equations [15][16][17], and the references therein. Especially, Li et al [14] investigated the asymptotic behavior at infinity for general fully nonlinear elliptic equations…”

On Symmetry Reduction of the (1 + 3)-Dimensional Inhomogeneous Monge-Amp&amp;#232;re Equation to the First-Order ODEs

“…is reduced to the investigation of the Monge-Ampère equations in the spaces of different dimensions and different types. Some details on this theme can be found in [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] (see, also, the references therein).…”

On the exterior Dirichlet problem for special Lagrangian equations

“…Recently, we proved in [LLY17] that any smooth solution u of (1.1) with supercritical phase in the exterior domain must tend to a quadratic polynomial Q at infinity, and satisfy u(x) = Q(x) + O 1 |x| n−2 (|x| → +∞).…”