Local C0,1-Regularity for the Parabolic p-Laplacian Equation on the Group SU(3)

He, Yongming; Yu, Chengwei; Wang, Hongqing

doi:10.3390/math12091288

Open AccessArticle

Local C^0,1-Regularity for the Parabolic p-Laplacian Equation on the Group SU(3)

by

Yongming He

¹,

Chengwei Yu

^1,2,*

and

Hongqing Wang

¹

Department of Basic, China Fire and Rescue Institue, 4 Nanyan Road, Changping District, Beijing 102202, China

²

School of Mathematical Sciences, Beihang University, Haidian District, Beijing 100191, China

^*

Author to whom correspondence should be addressed.

Mathematics 2024, 12(9), 1288; https://doi.org/10.3390/math12091288

Submission received: 2 April 2024 / Revised: 20 April 2024 / Accepted: 23 April 2024 / Published: 24 April 2024

(This article belongs to the Special Issue Research on Dynamical Systems and Differential Equations)

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Abstract

In this article, when

2 \leq p \leq 4

, we establish the

C_{loc}^{0, 1}

-regularity of weak solutions to the degenerate parabolic p-Laplacian equation

\partial_{t} u = - \sum_{i = 1}^{6} X_{i}^{*} (| \nabla_{H} u |^{p - 2} X_{i} u)

on the group SU(3) granted with horizontal vector fields

X_{1}, \dots, X_{6}

. Compared to the Heisenberg group,

H^{n}

, we obtained the optimal range of p; that is,

2 \leq p \leq 4

.

Keywords: p-Laplacian type;

C loc 0 , 1

-regularity; parabolic p-Laplacian; the group SU(3); range of p; Caccioppoli-type inequality

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MDPI and ACS Style

He, Y.; Yu, C.; Wang, H. Local C^0,1-Regularity for the Parabolic p-Laplacian Equation on the Group SU(3). Mathematics 2024, 12, 1288. https://doi.org/10.3390/math12091288

AMA Style

He Y, Yu C, Wang H. Local C^0,1-Regularity for the Parabolic p-Laplacian Equation on the Group SU(3). Mathematics. 2024; 12(9):1288. https://doi.org/10.3390/math12091288

Chicago/Turabian Style

He, Yongming, Chengwei Yu, and Hongqing Wang. 2024. "Local C^0,1-Regularity for the Parabolic p-Laplacian Equation on the Group SU(3)" Mathematics 12, no. 9: 1288. https://doi.org/10.3390/math12091288

APA Style

He, Y., Yu, C., & Wang, H. (2024). Local C^0,1-Regularity for the Parabolic p-Laplacian Equation on the Group SU(3). Mathematics, 12(9), 1288. https://doi.org/10.3390/math12091288

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Local C^0,1-Regularity for the Parabolic p-Laplacian Equation on the Group SU(3)

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