Removing singularities for fully nonlinear PDEs

TL;DR

This paper introduces a new method to remove certain singularities in viscosity solutions of fully nonlinear elliptic PDEs, including Monge-Ampère and minimal surface equations, using a quick doubling technique and the Jacobi inequality.

Contribution

It presents a novel quick doubling proof that extends singularity removability to a broader class of fully nonlinear PDEs with classical density and Jacobi inequality.

Findings

Removability of half-line singularities in viscosity solutions.

Application to Monge-Ampère, minimal surface, and special Lagrangian equations.

The method applies to singularities satisfying a single side condition.

Abstract

We show removability of half-line singularities for viscosity solutions of fully nonlinear elliptic PDEs which have classical density and a Jacobi inequality. An example of such a PDE is the Monge-Amp\`ere equation, and the original proof follows from Caffarelli 1990. Other examples are the minimal surface and special Lagrangian equations. The present paper's quick doubling proof combines Savin's small perturbation theorem with the Jacobi inequality. The method more generally removes singularities satisfying the single side condition.