Optimal Asymptotic Behavior at Infinity for Solutions of the Supercritical Lagrangian Mean Curvature Equation in Exterior Domains

Abstract

We study the asymptotic behavior at infinity of solutions to the supercritical Lagrangian mean curvature equation \[ \sum_{i=1}^n \arctan \lambda_i(D^2u)=\theta+f(x) \] on exterior domains in \(\mathbb R^n\), \(n\ge 2\), where \(|\theta|>((n-2)\pi)/2\). The perturbation \(f\) is assumed to be locally Lipschitz near infinity and to satisfy a decay condition with rate \(\beta>0\). The main new ingredient is a scale-dependent difference quotient argument, combined with a nonlocal potential method, which avoids differentiating \(f\) twice and yields quantitative Hessian convergence under only Lipschitz regularity. We establish optimal asymptotic expansions in all dimensions and for all decay rates \(\beta>0\), including the critical logarithmic cases. This improves previous results requiring higher regularity of \(f\) and faster decay in \cite{BJ2026}.