On a gradient term for a class of second-order PDEs and applications to the infinity Laplace equation
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| Format: | Preprint |
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2025
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| _version_ | 1866914401370505216 |
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| author | de Oliveira, José Francisco |
| author_facet | de Oliveira, José Francisco |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_00931 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a gradient term for a class of second-order PDEs and applications to the infinity Laplace equation de Oliveira, José Francisco Analysis of PDEs Primary 35J60, Secondary 35J25, 35J65, 35J66 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. |
| title | On a gradient term for a class of second-order PDEs and applications to the infinity Laplace equation |
| topic | Analysis of PDEs Primary 35J60, Secondary 35J25, 35J65, 35J66 |
| url | https://arxiv.org/abs/2511.00931 |