Solutions of the minimal surface equation and of the Monge–Ampère equation near infinity

Abstract

Abstract Classical results assert that, under appropriate assumptions, solutions near infinity are asymptotic to linear functions for the minimal surface equation and to quadratic polynomials for the Monge–Ampère equation for dimension n ≥ 3 n\geq 3 , with an extra logarithmic term for n = 2 n=2 . Via Kelvin transforms, we characterize remainders in the asymptotic expansions by a single function near the origin. Such a function is smooth in the entire neighborhood of the origin for the minimal surface equation in every dimension and for the Monge–Ampère equation in even dimension, but only C n − 1 , α C^{n-1,\alpha} for the Monge–Ampère equation in odd dimension 𝑛, for any α ∈ ( 0 , 1 ) \alpha\in(0,1) .