Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2023, , 77 - 89, 15.06.2023
https://doi.org/10.33205/cma.1245581

Öz

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).

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

Yıl 2023, , 77 - 89, 15.06.2023
https://doi.org/10.33205/cma.1245581

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.

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).
Toplam 8 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Thomas Bieske Bu kişi benim 0000-0003-2029-0562

Zachary Forrest 0000-0002-6636-0047

Erken Görünüm Tarihi 2 Mayıs 2023
Yayımlanma Tarihi 15 Haziran 2023
Yayımlandığı Sayı Yıl 2023

Kaynak Göster

APA Bieske, T., & Forrest, Z. (2023). Existence and uniqueness of viscosity solutions to the infinity Laplacian relative to a class of Grushin-type vector fields. Constructive Mathematical Analysis, 6(2), 77-89. https://doi.org/10.33205/cma.1245581
AMA Bieske T, Forrest Z. Existence and uniqueness of viscosity solutions to the infinity Laplacian relative to a class of Grushin-type vector fields. CMA. Haziran 2023;6(2):77-89. doi:10.33205/cma.1245581
Chicago Bieske, Thomas, ve Zachary Forrest. “Existence and Uniqueness of Viscosity Solutions to the Infinity Laplacian Relative to a Class of Grushin-Type Vector Fields”. Constructive Mathematical Analysis 6, sy. 2 (Haziran 2023): 77-89. https://doi.org/10.33205/cma.1245581.
EndNote Bieske T, Forrest Z (01 Haziran 2023) Existence and uniqueness of viscosity solutions to the infinity Laplacian relative to a class of Grushin-type vector fields. Constructive Mathematical Analysis 6 2 77–89.
IEEE T. Bieske ve Z. Forrest, “Existence and uniqueness of viscosity solutions to the infinity Laplacian relative to a class of Grushin-type vector fields”, CMA, c. 6, sy. 2, ss. 77–89, 2023, doi: 10.33205/cma.1245581.
ISNAD Bieske, Thomas - Forrest, Zachary. “Existence and Uniqueness of Viscosity Solutions to the Infinity Laplacian Relative to a Class of Grushin-Type Vector Fields”. Constructive Mathematical Analysis 6/2 (Haziran 2023), 77-89. https://doi.org/10.33205/cma.1245581.
JAMA Bieske T, Forrest Z. Existence and uniqueness of viscosity solutions to the infinity Laplacian relative to a class of Grushin-type vector fields. CMA. 2023;6:77–89.
MLA Bieske, Thomas ve Zachary Forrest. “Existence and Uniqueness of Viscosity Solutions to the Infinity Laplacian Relative to a Class of Grushin-Type Vector Fields”. Constructive Mathematical Analysis, c. 6, sy. 2, 2023, ss. 77-89, doi:10.33205/cma.1245581.
Vancouver Bieske T, Forrest Z. Existence and uniqueness of viscosity solutions to the infinity Laplacian relative to a class of Grushin-type vector fields. CMA. 2023;6(2):77-89.