Solutions of the Special Lagrangian Equation near Infinity

TL;DR

This paper investigates the asymptotic behavior of solutions to the special Lagrangian equation near infinity, revealing dimension-dependent regularity of the remainders through Kelvin transforms.

Contribution

It characterizes the asymptotic remainders of solutions using a single function via Kelvin transforms, highlighting regularity differences between even and odd dimensions.

Findings

Solutions are asymptotic to quadratic polynomials with logarithmic terms in 2D.

Remainders are characterized by a smooth function in even dimensions.

In odd dimensions, the remainder function is $C^{n-1,eta}$ for any $eta ext{ in }(0,1)$.

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 $C^{n-1,\alpha}$ in odd dimension $n$, for any $\alpha\in (0,1)$.