On the exterior Dirichlet problem for Hessian type fully nonlinear elliptic equations

TL;DR

This paper establishes existence and uniqueness of viscosity solutions for the exterior Dirichlet problem of fully nonlinear elliptic equations, including Monge-Ampère and Hessian equations, without requiring the concavity of the function f.

Contribution

It introduces a Perron’s method-based approach to solve the exterior Dirichlet problem for a broad class of fully nonlinear elliptic equations without the concavity assumption.

Findings

Proved existence and uniqueness of solutions for the exterior Dirichlet problem.

Extended solvability results to equations like Monge-Ampère and Hessian equations.

Provided a new approach that relaxes previous concavity requirements.

Abstract

We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form $$f(\lambda(D^2u))=g(x),$$ with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli--Nirenberg--Spruck \cite{Caffarelli1985}, Trudinger \cite{Trudinger1995} and many others, and there had been significant discussions on the solvability of the classical Dirichlet problem via the continuity method, under the assumption that $f$ is a concave function. In this paper, based on the Perron's method, we establish an exterior existence and uniqueness result for viscosity solutions of the equations by assuming $f$ to satisfy certain structure conditions as in \cite{Caffarelli1985,Trudinger1995}, which may embrace the well-known Monge--Amp\`ere equations, Hessian equations and Hessian quotient equations as special cases but do not require the concavity.