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Mathematics in Engineering

2021, Volume 3, Issue 1: 1-31. doi: 10.3934/mine.2021008
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Regularity for a class of quasilinear degenerate parabolic equations in the Heisenberg group

  • 1.
    Department of Mathematical Sciences, Worcester Polytechnic Institute, MA 01609
  • 2.
    Dipartimento di Matematica, Piazza Porta S. Donato 5, Università di Bologna, 40126 Bologna, Italy
  • 3.
    Dipartimento di Ingegneria Civile e Ambientale (DICEA), Università di Padova, 35131 Padova, Italy
  • Received: 23 January 2020 Accepted: 31 August 2020 Published: 22 September 2020

  • We extend to the parabolic setting some of the ideas originated with Xiao Zhong's proof in [31] of the Hölder regularity of $p-$harmonic functions in the Heisenberg group $\mathbb{H}^n$. Given a number $p\ge 2$, in this paper we establish the $C^{\infty}$ smoothness of weak solutions of a class of quasilinear PDE in $\mathbb{H}^n$ modeled on the equation $$?_t u = \sum_{i = 1}^{2n} X_i \bigg((1+|\nabla_0 u|^2)^{\frac{p-2}{2}} X_i u\bigg).$$

    Citation: Luca Capogna, Giovanna Citti, Nicola Garofalo. Regularity for a class of quasilinear degenerate parabolic equations in the Heisenberg group[J]. Mathematics in Engineering, 2021, 3(1): 1-31. doi: 10.3934/mine.2021008

    Related Papers:

  • We extend to the parabolic setting some of the ideas originated with Xiao Zhong's proof in [31] of the Hölder regularity of $p-$harmonic functions in the Heisenberg group $\mathbb{H}^n$. Given a number $p\ge 2$, in this paper we establish the $C^{\infty}$ smoothness of weak solutions of a class of quasilinear PDE in $\mathbb{H}^n$ modeled on the equation $$?_t u = \sum_{i = 1}^{2n} X_i \bigg((1+|\nabla_0 u|^2)^{\frac{p-2}{2}} X_i u\bigg).$$


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