The Maximum Principles and Symmetry results for Viscosity Solutions of Fully Nonlinear Equations

TL;DR

This paper establishes maximum principles and radial symmetry for viscosity solutions of fully nonlinear PDEs, including nonexistence results and symmetry properties under decay conditions, with applications to elliptic and parabolic equations.

Contribution

It introduces new maximum principles and symmetry results for viscosity solutions of fully nonlinear equations, extending to elliptic and parabolic cases, and includes nonexistence theorems under certain conditions.

Findings

Radial symmetry and monotonicity for solutions of nonlinear PDEs.

A new maximum principle for viscosity solutions of fully nonlinear elliptic equations.

Nonexistence of nontrivial solutions for Pucci extremal operators.

Abstract

This paper is concerned about maximum principles and radial symmetry for viscosity solutions of fully nonlinear partial differential equations. We obtain the radial symmetry and monotonicity properties for nonnegative viscosity solutions of begin{equation}F (D^2 u)+u^p=0 \quad \quad \mbox{in}\ \mathbb R^n \label{abs}\end{equation} under the asymptotic decay rate $u=o(|x|^{-\frac{2}{p-1}})$ at infinity, where $p>1$ (Theorem 1, Corollary 1). As a consequence of our symmetry results, we obtain the nonexistence of any nontrivial and nonnegative solution when $F$ is the Pucci extremal operators (Corollary 2). Our symmetry and monotonicity results also apply to Hamilton-Jacobi-Bellman or Isaccs equations. A new maximum principle for viscosity solutions to fully nonlinear elliptic equations is established (Theorem 2). As a result, different forms of maximum principles on bounded and unbounded domains are obtained. Radial symmetry, monotonicity and the corresponding maximum principle for fully nonlinear elliptic equations in a punctured ball are shown (Theorem 3). We also investigate the radial symmetry for viscosity solutions of fully nonlinear parabolic partial differential equations (Theorem 4).