Viscosity solutions to the ∞-Laplace equation in Grushin-type spaces
Keywords:
p-Laplace equation, ∞-Laplace equation, viscosity solution, Grushin-type spaces, sub-Riemannian geometryAbstract
In this paper, we prove the existence and uniqueness of viscosity solutions to the infinite Laplace equation in Grushin-type spaces whose tangent spaces consist of arbitrary triangular vector fields.
References
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: Fundamental solutions to the p-Laplace equation in a class of Grushin vector fields, Electron. J. Diff. Equ., Vol. 2011 84 (2011), 1–10.
T. Bieske: A comparison principle for a class of subparabolic equations in Grushin-type spaces, Electron. J. Diff. Eqns., Vol. 2007 30 (2007), 1–9.
T. Bieske: Properties of Infinite Harmonic Functions on Grushin-Type Spaces, Rocky Mountain J. Math., 39 (3) (2009), 729–756.
T. Bieske: A sub-Riemannian maximum principle and its application to the p-Laplacian in Carnot groups, Ann. Acad. Sci. Fenn. Math., 37 (1) (2012), 119–134.
T. Bieske: The∞(x)-Equation in Grushin-Type Spaces, Electron. J. Diff. Eqns., Vol. 2016 125 (2016), 1–13.
T. Bieske, Z. Forrest: Existence and Uniqueness of Viscosity Solutions to the Infinity Laplacian Relative to a Class of Grushin-Type Vector Fields, Constr. Math. Anal., 6 (2) (2023), 77–89.
T. Bieske, J. Gong: The P-Laplace Equation on a Class of Grushin-Type Spaces, Proc. Amer. Math. Soc., 134 (12) (2006), 3585–3594.
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 (1996), 1–78.
L. Capogna, G. Giovannardi, A. Pinamonti and S. Verzellesi: The Asymptotic p-Poisson Equation as p → ∞ in Carnot-Carathéodory Spaces, Math. Ann., to appear.
M. Crandall, H. Ishii and 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, P. Lindqvist and J. Manfredi: On the Equivalence of Viscosity Solutions and Weak Solutions for a Quasi-Linear Equation, SIAM J. Math. Anal., 33 (3) (2001), 699–717.
P. Juutinen: Minimization Problems for Lipschitz Functions via Viscosity Solutions, Ann. Acad. Sci. Fenn. Math. Diss., 115 (1998).
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Modern Mathematical Methods
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution 4.0 International License.