Second-Order Regularity for Degenerate Parabolic Quasi-Linear Equations in the Heisenberg Group

Chengwei Yu (), Huiying Wang (), Kunpeng Cui and Zijing Zhao
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Chengwei Yu: China Fire and Rescue Institue, 4 Nanyan Road, Changping District, Beijing 102202, China
Huiying Wang: China Fire and Rescue Institue, 4 Nanyan Road, Changping District, Beijing 102202, China
Kunpeng Cui: China Fire and Rescue Institue, 4 Nanyan Road, Changping District, Beijing 102202, China
Zijing Zhao: China Fire and Rescue Institue, 4 Nanyan Road, Changping District, Beijing 102202, China

Mathematics, 2024, vol. 12, issue 22, 1-12

Abstract: In the Heisenberg group H n , we obtain the local second-order H W loc 2 , 2 -regularity for the weak solution u to a class of degenerate parabolic quasi-linear equations ∂ t u = ∑ i = 1 2 n X i A i ( X u ) modeled on the parabolic p -Laplacian equation. Specifically, when 2 ≤ p ≤ 4 , we demonstrate the integrability of ( ∂ t u ) 2 , namely, ∂ t u ∈ L loc 2 ; when 2 ≤ p < 3 , we demonstrate the H W loc 2 , 2 -regularity of u , namely, X X u ∈ L loc 2 . For the H W loc 2 , 2 -regularity, when p ≥ 2 , the range of p is optimal compared to the Euclidean case.

Keywords: parabolic quasi-linear equation; second-order regularity; Heisenberg group; parabolic p -Laplacian equation; H W loc 2 , 2 -regularity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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