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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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    [1] Aronson DG, Serrin J (1967) Local behavior of solutions of quasilinear parabolic equations. Arch Ration Mech Anal 25: 81-122.
    [2] Avelin B, Capogna L, Citti G, et al. (2014) Harnack estimates for degenerate parabolic equations modeled on the subelliptic p-Laplacian. Adv Math 257: 25-65.
    [3] Bieske T (2005) Comparison principle for parabolic equations in the Heisenberg group. Electron J Differ Eq 95: 1-11.
    [4] Bieske T (2006) Equivalence of weak and viscosity solutions to the p-Laplace equation in the Heisenberg group. Ann Acad Sci Fenn Math 31: 363-379.
    [5] Bramanti M, Brandolini L (2007) Schauder estimates for parabolic nondivergence operators of Hörmander type. J Differ Equations 234: 177-245.
    [6] Capogna L (1999) Regularity for quasilinear equations and 1-quasiconformal mappings in Carnot groups. Math Ann 313: 263-295.
    [7] Capogna L, Citti G (2016) Regularity for subelliptic PDE through uniform estimates in multiscale geometries. B Math Sci 6: 173-230.
    [8] Capogna L, Citti G, Ottazzi A, et al. (2019) Conformality and Q-harmonicity in sub-Riemannian manifolds. J Math Pure Appl 122: 67-124.
    [9] Capogna L, Citti G, Rea G (2013) A subelliptic analogue of Aronson-Serrin's Harnack inequality. Math Ann 357: 1175-1198.
    [10] Citti G, Manfredini M (2006) Uniform estimates of the fundamental solution for a family of hypoelliptic operators. Potential Anal 25: 147-164.
    [11] Citti G, Lanconelli E, Montanari A (2002) Smoothness of Lipschitz-continuous graphs with nonvanishing Levi curvature. Acta Math 188: 87-128.
    [12] Da Prato G (1964) Spazi Lp, θ(?, δ) e loro proprietà. Annali di Matematica 69: 383-392.
    [13] Di Benedetto E (1993) Degenerate Parabolic Equations, Springer, Universitext.
    [14] Domokos A (2004) Differentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg group. J Differ Equations 204: 439-470.
    [15] Domokos A, Manfredi JJ (2020) C1, α-subelliptic regularity on S U(3) and compact, semi-simple Lie groups. Anal Math Phys 10: 4.
    [16] Folland GB, Stein EM (1982) Hardy Spaces on Homogeneous Groups, Princeton: Princeton University Press.
    [17] Grigor'yan AA (1992) The heat equation on noncompact Riemannian manifolds. Sb Math 72: 47-77.
    [18] Hörmander L (1967) Hypoelliptic second order differential equations. Acta Math 119: 147-171.
    [19] Koranyi A (1983) Geometric aspects of analysis on the Heisenberg group, In: Topics in Modern Harmonic Analysis, Vol. I, II (Turin/Milan, 1982), Francesco Severi, Rome: Ist. Naz. Alta Mat., 209-258.
    [20] Ladyzhenskaya O, Solonnikov VA, Ural'tseva NN (1967) Linear and Quasilinear Parabolic Equations, Moskow: Nauka.
    [21] Manfredi JJ, Mingione G (2007) Regularity results for quasilinear elliptic equations in the Heisenberg group. Math Ann 339: 485-544.
    [22] Mingione G, Zatorska-Goldstein A, Zhong X (2009) Gradient regularity for elliptic equations in the Heisenberg group. Adv Math 222: 62-129.
    [23] Morrey Jr CB (1959) Second order elliptic equations in several variables and Hölder continuity. Math Z 72: 146-164.
    [24] Moser J (1964) A Harnack inequality for parabolic differential equations. Commun Pure Appl Math 17: 101-134.
    [25] Mukherjee S (2019) On local Lipschitz regularity for quasilinear equations in the Heisenberg group. arXiv: 1804.00751.
    [26] Ricciotti D (2015) p-Laplace Equation in the Heisenberg Group, Cham: Springer.
    [27] Ricciotti D (2018) On the C1, α regularity of p-harmonic functions in the Heisenberg group. P Am Math Soc 146: 2937-2952.
    [28] Saloff-Coste L (1992) A note on Poincaré, Sobolev, and Harnack inequalities. Int Math Res Notices 1992: 27-38.
    [29] Stein EM (1993) Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton: Princeton University Press.
    [30] Xu CJ (1992) Regularity for quasilinear second-order subelliptic equations. Commun Pure Appl Math 45: 77-96.
    [31] Zhong X (2009) Regularity for variational problems in the Heisenberg group. arXiv: 1711.03284.
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