Solutions of the Special Lagrangian Equation near Infinity

TL;DR

The paper analyzes global solutions to the special Lagrangian equation in exterior domains, proving that such solutions are asymptotic to a quadratic polynomial and that the remainder is governed by a single function $v$ obtained via a modified Kelvin transform. It shows that, after a linear change of variables, the solution has the form $u(x)=\frac12 x^T A x + b\cdot x + c + |\sqrt{I+A^2}\,x|^{2-n} v\left( \frac{\sqrt{I+A^2}\,x}{|\sqrt{I+A^2}\,x|^2} \right)$, where $v$ is smooth in even dimensions and in odd dimensions $v\in C^{n-1,\alpha}$ for any $\alpha\in(0,1)$; in odd $n$, $v$ admits a decomposition $v(y)=v_1(y)+|y|^{n-2}v_2(y)$ with $v_1,v_2$ smooth. The authors establish these regularity properties by transforming the nonlinear equation into a perturbation of the Laplacian, proving removable singularity results, and then extending the expansion to higher orders; they further extend the framework to a broader class of special-Lagrangian-type equations, yielding analogous asymptotics. These results sharpen the understanding of the global behavior of solutions to fully nonlinear elliptic equations and provide a robust method for obtaining high-order asymptotics in both even and odd dimensions.

Abstract

Solutions to special Lagrangian equations near infinity, with supercritical phases or with semiconvexity on solutions, are known to be asymptotic to quadratic polynomials for dimension $n\ge 3$, with an extra logarithmic term for $n=2$. Via modified Kelvin transforms, we characterize remainders in the asymptotic expansions by a single function near the origin. Such a function is smooth in even dimension, but only $C^{n-1,α}$ in odd dimension $n$, for any $α\in (0,1)$.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof