Generalizations of the drift Laplace equation over the quaternions in a class of Grushin-type spaces

Yıl 2023, Cilt: 6 Sayı: 3, 164 - 175, 15.09.2023

Thomas Bieske , Keller Blackwell

https://doi.org/10.33205/cma.1324774

Öz

Beals, Gaveau, and Greiner established a formula for the fundamental solution to the Laplace equation with drift term in Grushin-type planes. The first author and Childers expanded these results by invoking a p-Laplace type generalization that encompasses these formulas while the authors explored a different natural generalization of the p-Laplace equation with drift term that also encompasses these formulas. In both, the drift term lies in the complex domain. We extend these results by considering a drift term in the quaternion realm and show our solutions are stable under limits as p tends to infinity.

Anahtar Kelimeler

p-Laplace equation, Grushin-type plane, Grushin-type spaces

Kaynakça

• A. Belläche: The Tangent Space in Sub-Riemannian Geometry, in: Sub-Riemannian Geometry; A. Belläche, J. J. Risler, Eds.; Progress in Mathematics; Birkhäuser: Basel, Switzerland, 144 (1996), 1–78.
• R. Beals, B. Gaveau, P. Greiner: On a Geometric Formula for the Fundamental Solution of Subelliptic Laplacians, Math. Nachr., 181 (1996), 81–163.
• T. Bieske, K. Blackwell: Generalizations of the Drift p-Laplace Equation in the Heisenberg Group and a Class of Grushin-type Planes, (2019), submitted for publication, preprint available at https://arxiv.org/abs/1906.01467.
• T. Bieske, K. Childers: Generalizations of a Laplacian-type Equation in the Heisenberg Group and a Class of Grushin-type Spaces, Proc. Amer. Math. Soc., 142 (3) (2013), 989–1003.
• T. Bieske, J. Gong: The p-Laplacian Equation on a Class of Grushin-Type Spaces, Amer. Math. Society, 134 (2006), 3585–3594.
• 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.