On a gradient term for a class of second-order PDEs and applications to the infinity Laplace equation

TL;DR

This paper introduces a new gradient term for a class of second-order PDEs, providing a unifying framework that extends previous results to broader nonlinear equations like the infinity Laplace equation, including solutions that change sign.

Contribution

It establishes conditions for transforming complex PDEs with quadratic gradient terms into simpler forms, unifying and extending prior results for various operators including the infinity Laplacian.

Findings

Unified framework for PDE transformations involving gradient terms

Extension of previous results to broader classes including infinity Laplace

Analysis of solutions that may change sign and include viscosity solutions

Abstract

We propose a natural gradient term for a class of second-order partial differential equations of the form \begin{equation}\nonumber M(x,Du,D^2u)+g(u)N(x,Du, D^2u)+f(x,u)=0 \;\;\mbox{in}\;\; \Omega, \end{equation} where $\Omega\subset\mathbb{R}^n$ is an open set, $f\in C(\Omega\times \mathbb{R}, \mathbb{R})$, $M$ defines the partial differential operator, $N$ is a quadratic term driven by the gradient $Du$ and $M$ itself, and $g\in C(\mathbb{R},\mathbb{R})$. We establish conditions on the class of operators $M$ for the existence of a change of variables $v = \Phi(u)$ that transforms the previous equation into another one of the form \begin{equation}\nonumber M(x,Dv, D^2v) + h(x,v)=0 \quad \text{in} \;\; \Omega \end{equation} which does not depend on the quadratic term $N$. The results presented here unify previous findings for the Laplacian, $m$-Laplacian, and $k$-Hessian operators, which were derived separately by different authors and are restricted to $C^2$ solutions with fixed sign. Our work provides a more general framework, extending these findings to a broader class of nonlinear partial differential equations, including the infinity-Laplacian o\-pe\-ra\-tor. In addition, we also include both $C^2$ and viscosity solutions that may change sign. As an application, we also obtain an Aronsson-type result and investigate viscosity solutions for the Dirichlet problem associated with the infinity Laplace equation with its natural gradient term.