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

Author

Listed:
  • Chengwei Yu

    (China Fire and Rescue Institue, 4 Nanyan Road, Changping District, Beijing 102202, China
    China Fire and Rescue Institue, Beihang University, Haidian District, Beijing 100191, 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)

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.

Suggested Citation

  • Chengwei Yu & Huiying Wang & Kunpeng Cui & Zijing Zhao, 2024. "Second-Order Regularity for Degenerate Parabolic Quasi-Linear Equations in the Heisenberg Group," Mathematics, MDPI, vol. 12(22), pages 1-12, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:22:p:3494-:d:1517030
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