Research article Special Issues

Strict starshapedness of solutions to the horizontal p-Laplacian in the Heisenberg group

  • Received: 03 July 2020 Accepted: 06 November 2020 Published: 13 November 2020
  • We examine the geometry of the level sets of particular horizontally p-harmonic functions in the Heisenberg group. We find sharp, natural geometric conditions ensuring that the level sets of the p-capacitary potential of a bounded annulus in the Heisenberg group are strictly starshaped.

    Citation: Mattia Fogagnolo, Andrea Pinamonti. Strict starshapedness of solutions to the horizontal p-Laplacian in the Heisenberg group[J]. Mathematics in Engineering, 2021, 3(6): 1-15. doi: 10.3934/mine.2021046

    Related Papers:

  • We examine the geometry of the level sets of particular horizontally p-harmonic functions in the Heisenberg group. We find sharp, natural geometric conditions ensuring that the level sets of the p-capacitary potential of a bounded annulus in the Heisenberg group are strictly starshaped.
    加载中


    [1] V. Agostiniani, M. Fogagnolo, L. Mazzieri, Minkowski inequalities via nonlinear potential theory, 2019. Available from: https://arXiv.org/abs/1906.00322.
    [2] C. Bianchini, G. Ciraolo, Wulff shape characterizations in overdetermined anisotropic elliptic problems, Commun. Part. Diff. Eq., 43 (2018), 790-820.
    [3] A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer, 2007.
    [4] S. Brendle, P. K. Hung, M. T. Wang, A Minkowski inequality for hypersurfaces in the Antide Sitter-Schwarzschild manifold, Commun. Pure Appl. Math., 69 (2016), 124-144.
    [5] L. Capogna, Regularity of quasi-linear equations in the heisenberg group, Commun. Pure Appl. Math., 50 (1997), 867-889.
    [6] L. Capogna, D. Danielli, N. Garofalo, An embedding theorem and the harnack inequality for nonlinear subelliptic equations, Commun. Part. Diff. Eq., 18 (1993), 1765-1794.
    [7] L. Capogna, D. Danielli, N. Garofalo, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Am. J. Math., 118 (1996), 1153-1196.
    [8] L. Capogna, N. Garofalo, D. M. Nhieu, A version of a theorem of Dahlberg for the subelliptic Dirichlet problem, Math. Res. Lett., 5 (1998), 541-549.
    [9] D. Danielli, Regularity at the boundary for solutions of nonlinear subelliptic equations, Indiana U. Math. J., 44 (1995), 269-286.
    [10] D. Danielli, N. Garofalo, Geometric properties of solutions to subelliptic equations in nilpotent lie groups, In: Reaction diffusion sytems, New York: Dekker, 1998, 89-105.
    [11] E. DiBenedetto, C1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.
    [12] A. Domokos, J. Manfredi, c1, α-regularity for p-harmonic functions in the heisenberg group for p near 2, Contemp. Math., 370 (2005), 2126699.
    [13] F. Dragoni, D. Filali, Starshaped and convex sets in carnot groups and in the geometries of vector fields, J. Convex Anal., 26 (2019), 1349-1372.
    [14] F. Dragoni, N. Garofalo, P. Salani, Starshapedeness for fully non-linear equations in Carnot groups, J. Lond. Math. Soc., 99 (2019), 901-918.
    [15] M. Fogagnolo, L. Mazzieri, A. Pinamonti, Geometric aspects of p-capacitary potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1151-1179.
    [16] R. M. Gabriel, An extended principle of the maximum for harmonic functions in 3- dimensions, J. London Math. Soc., 30 (1955), 388-401.
    [17] R. M. Gabriel, Further results concerning the level surfaces of the Green's function for a 3-dimensional convex domain (I), J. London Math. Soc., 32 (1957), 295-302.
    [18] R. M. Gabriel, A result concerning convex level surfaces of 3-dimensional harmonic functions, J. London Math. Soc., 32 (1957), 286-294.
    [19] N. Garofalo, N. C. Phuc, Boundary behavior of p-harmonic functions in the heisenberg group, Math. Ann., 351 (2011), 587-632.
    [20] J. J. Gergen, Note on the green function of a star-shaped three dimensional region, Am. J. Math., 53 (1931), 746-752.
    [21] C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differ. Geom., 32 (1990), 299-314.
    [22] J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear potential theory of degenerate elliptic equations, Mineola, NY: Dover Publications, Inc., 2006.
    [23] G. Huisken, T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differ. Geom., 59 (2001), 353-437.
    [24] D. S. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. I, J. Funct. Anal., 43 (1981), 97-142.
    [25] D. S. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. Ⅱ, J. Funct. Anal., 43 (1981), 224-257.
    [26] J. L. Lewis, Capacitary functions in convex rings, Arch. Ration. Mech. Anal., 66 (1977), 201-224.
    [27] J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana U. Math. J., 32 (1983), 849-858.
    [28] J. J. Manfredi, G. Mingione, Regularity results for quasilinear elliptic equations in the heisenberg group, Math. Ann., 339 (2007), 485-544.
    [29] V. Martino, G. Tralli, On the hopf-oleinik lemma for degenerate-elliptic equations at characteristic points, Calc. Var., 55 (2016), 115.
    [30] G. Mingione, A. Z. Goldstein, X. Zhong, On the regularity of p-harmonic functions in the heisenberg group, Bollettino dell'Unione Matematica Italiana, 1 (2008), 243-253.
    [31] R. Monti, Isoperimetric problem and minimal surfaces in the heisenberg group, In: Geometric measure theory and real analysis, Pisa: Scuola Normale Superiore, 2014, 57-129.
    [32] R. Moser, The inverse mean curvature flow and p-harmonic functions, J. Eur. Math. Soc., 9 (2007), 77-83.
    [33] S. Mukherjee, X. Zhong, c1, α-regularity for variational problems in the heisenberg group, arXiv: 1711.04671, 2017.
    [34] J. A. Pfaltzgraff, Radial symmetrization and capacities in space, Duke Math. J., 34 (1967), 747-756.
    [35] G. Pipoli, Inverse mean curvature flow in complex hyperbolic space, Annales scientifiques de l'ENS, 52 (2019), 1107-1135.
    [36] D. Ricciotti, p-Laplace equation in the Heisenberg group, Cham: Springer, 2015.
    [37] D. Ricciotti, On the c1, α regularity of p-harmonic functions in the heisenberg group, P. Am. Math. Soc., 146 (2016), 2937-2952.
    [38] P. Salani, Starshapedness of level sets of solutions to elliptic PDEs, Appl. Anal., 84 (2005), 1185-1197.
    [39] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equations, 51 (1984), 126-150.
    [40] F. Uguzzoni, E. Lanconelli, On the Poisson kernel for the Kohn Laplacian, Rend. Mat. Appl., 17 (1998), 659-677.
    [41] J. I. E. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z., 205 (1990), 355-372.

    © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
  • Reader Comments
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(40) PDF downloads(10) Cited by()

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog