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

TL;DR

The paper develops a general framework to remove a natural gradient-type term from a broad class of second-order PDEs of the form $M(x,Du,D^2u) + g(u) N(x,Du,D^2u) + f(x,u)=0$ by introducing a change of variables $v=\Phi_g(u)$, where $\Phi_g$ is built from $g$ via $G(s)=\int_0^s g(\tau)d\tau$. Under $\alpha$-homogeneity of $M$ in $p$ and a compatibility condition with $N$ (parameters $\alpha,\beta$), the transformed equation $M(x,Dv,D^2v) + h(x,v)=0$ emerges with $h(x,s)=e^{(\alpha+\beta+1)G(\Phi_g^{-1}(s))} f(x,\Phi_g^{-1}(s))$, and the original problem is recovered by $u=\Phi_g^{-1}(v)$. This invariance is established for $C^2$-solutions and extended to viscosity solutions (Theorem thm-VS), unifying results for the Laplacian, $m$-Laplacian, $k$-Hessian, and infinity-Laplacian, and enabling applications to the infinity Laplace equation, including Aronsson-type results and Dirichlet problems. The approach yields a versatile tool for transforming gradient-influenced PDEs into gradient-free forms, facilitating analysis of existence, uniqueness, and nonexistence in both classical and viscosity frameworks, with potential implications for absolute minimizers and game-theoretic operators.

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}\;\; Ω, \end{equation} where $Ω\subset\mathbb{R}^n$ is an open set, $f\in C(Ω\times \mathbb{R}, \mathbb{R})$, $M$ 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 $v = Φ(u)$ 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} \;\; Ω\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.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof
• Theorem 6
• proof