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

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.