Existence and uniqueness of viscosity solutions to the infinity Laplacian relative to a class of Grushin-type vector fields

Yıl 2023, Cilt: 6 Sayı: 2, 77 - 89, 15.06.2023

Thomas Bieske Zachary Forrest

https://doi.org/10.33205/cma.1245581

Cited By: 1

Öz

In this paper we pose the $\infty$-Laplace Equation as a Dirichlet Problem in a class of Grushin-type spaces whose vector fields are of the form
\begin{equation*}
X_k(p):=\sigma_k(p)\frac{\partial}{\partial x_k}
\end{equation*}
and $\sigma_k$ is not a polynomial for indices $m+1 \leq k \leq n$. Solutions to the $\infty$-Laplacian in the viscosity sense have been shown to exist and be unique in [3], when $\sigma_k$ is a polynomial; we extend these results by exploiting the relationship between Grushin-type and Euclidean second-order jets and utilizing estimates on the viscosity derivatives of sub- and supersolutions in order to produce a comparison principle for semicontinuous functions.

Anahtar Kelimeler

Infinite-Laplace equation, viscosity solution, Grushin-type spaces

Kaynakça

• F. Beatrous, T. Bieske and J. Manfredi: The Maximum Principle for Vector Fields, Contemp. Math., 370 (2005), Amer. Math. Soc. Providence, RI, 1–9.
• T. Bieske: On Infinite Harmonic Functions on the Heisenberg Group, Comm. in PDE, 27 (3 & 4) (2002), 727–762.
• T. Bieske: Lipschitz Extensions on Generalized Grushin Spaces, Michigan Math. J., 53 (1) (2005), 3–31.
• T. Bieske: A Sub-Riemannian Maximum Principle and its Application to the p-Laplacian in Carnot Groups, Ann. Acad. Sci. Fenn., 37 (2012), 119–134 .
• A. Bellaïche: The Tangent Space in Sub-Riemannian Geometry, In Sub-Riemannian Geometry; Bellaïche, André., Risler, Jean-Jacques., Eds.; Progress in Mathematics; Birkhäuser: Basel, Switzerland. 144, 1–78 (1996).
• M. Crandall, H. Ishii, P.-L. Lions: User’s Guide to Viscosity Solutions of Second Order Partial Differential Equations, Bull. Amer. Math. Soc., 27 (1) (1992), 1–67.
• R. Jensen: Uniqueness of Lipschitz Extensions: Minimizing the Sup Norm of the Gradient, Arch. Rational. Mech. Anal., 123 (1993), 51–74.
• P. Juutinen: Minimization Problems for Lipschitz Functions via Viscosity Solutions, Ann. Acad. Sci. Fenn. Math. Diss., 115 (1998).