Advanced Search
Article Contents
Article Contents

The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem

  • * Corresponding author

    * Corresponding author 
Abstract Full Text(HTML) Related Papers Cited by
  • We consider the $ p $-Laplacian equation $ -\Delta_p u = 1 $ for $ 1<p<2 $, on a regular bounded domain $ \Omega\subset\mathbb R^N $, with $ N\ge2 $, under homogeneous Dirichlet boundary conditions. In the spirit of Alexandrov's Soap Bubble Theorem and of Serrin's symmetry result for the overdetermined problems, we prove that if the mean curvature $ H $ of $ \partial\Omega $ is constant, then $ \Omega $ is a ball and the unique solution of the Dirichlet $ p $-Laplacian problem is radial. The main tools used are integral identities, the $ P $-function, and the maximum principle.

    Mathematics Subject Classification: Primary: 35J92, 35B06, 35N25, 35A23; Secondary: 53A10.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] A. AftalionJ. Busca and W. Reichel, Approximate radial symmetry for overdetermined boundary value problems, Adv. Differential Equations, 4 (1999), 907-932. 
    [2] A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl., 58 (1962), 303-315.  doi: 10.1007/BF02413056.
    [3] C. Bianchini and G. Ciraolo, Wulff shape characterizations in overdetermined anisotropic elliptic problems, Comm. Partial Differential Equations, 43 (2018), 790-820.  doi: 10.1080/03605302.2018.1475488.
    [4] B. BrandoliniC. NitschP. Salani and C. Trombetti, On the stability of the Serrin problem, J. Differential Equations, 245 (2008), 1566-1583.  doi: 10.1016/j.jde.2008.06.010.
    [5] F. Brock, Symmetry for a general class of overdetermined elliptic problems, Nonlinear Differential Equations and Applications NoDEA, (2016) 23: 36. doi: 10.1007/s00030-016-0390-1.
    [6] F. Brock and A. Henrot, A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative, Rend. Circ. Mat. Palermo, 51 (2002), 375-390.  doi: 10.1007/BF02871848.
    [7] G. Ciraolo and F. Maggi, On the shape of compact hypersurfaces with almost-constant mean curvature, Comm. Pure Appl. Math., 70 (2017), 665-716.  doi: 10.1002/cpa.21683.
    [8] G. CiraoloR. Magnanini and V. Vespri, Hölder stability for Serrin's overdetermined problem, Ann. Mat. Pura Appl., 195 (2016), 1333-1345.  doi: 10.1007/s10231-015-0518-7.
    [9] G. Ciraolo and A. Roncoroni, Serrin's type overdetermined problems in convex cones, preprint, arXiv: 1806.08553.
    [10] G. Ciraolo and L. Vezzoni, A sharp quantitative version of Alexandrov's theorem via the method of moving planes, J. Eur. Math. Soc., 20 (2018), 261-299. doi: 10.4171/JEMS/766.
    [11] L. Damascelli and F. Pacella, Monotonicity and symmetry results for $p$-Laplace equations and applications, Adv. Differential Equations, 5 (2000), 1179-1200. 
    [12] A. Farina and B. Kawohl, Remarks on an overdetermined boundary value problem, Calculus of Variations and Partial Differential Equations, 31 (2008), 351-357.  doi: 10.1007/s00526-007-0115-8.
    [13] I. FragalàF. Gazzola and B. Kawohl, Overdetermined problems with possibly degenerate ellipticity, a geometric approach, Mathematische Zeitschrift, 254 (2006), 117-132.  doi: 10.1007/s00209-006-0937-7.
    [14] N. Garofalo and J. L. Lewis, A symmetry result related to some overdetermined boundary value problems, American Journal of Mathematics, 111 (1989), 9-33.  doi: 10.2307/2374477.
    [15] B. Kawohl, Symmetrization–or how to prove symmetry of solutions to a pde, Partial differential equations (Praha, 1998), 406 (1999), 214-229. 
    [16] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.
    [17] R. Magnanini and G. Poggesi, On the stability for Alexandrov's soap bubble theorem, to appear in J. Anal. Math, (2016).
    [18] R. Magnanini and G. Poggesi, Serrin's problem and Alexandrov's soap bubble theorem: enhanced stability via integral identities, preprint, arXiv: 1708.07392.
    [19] R. Magnanini and G. Poggesi, Nearly optimal stability for Serrin's problem and the soap bubble theorem, preprint, arXiv: 1903.04823.
    [20] F. Pacella and G. Tralli, Overdetermined problems and constant mean curvature surfaces in cones, preprint, arXiv: 1802.03197.
    [21] L. E. Payne and P. W. Schaefer, Duality theorems in some overdetermined boundary value problems, Math. Methods Appl. Sci., 11 (1989), 805-819.  doi: 10.1002/mma.1670110606.
    [22] G. Poggesi, The Soap Bubble Theorem and Serrin's problem: quantitative symmetry, Ph.D thesis, Università di Firenze, arXiv: 1902.08584, 2019.
    [23] C. Pucci and A. Colesanti, A symmetry result for the p-laplacian equation via the moving planes method, Applicable Analysis, 55 (1994), 207-213.  doi: 10.1080/00036819408840300.
    [24] P. Pucci and J. Serrin, The maximum principle, volume 73 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Verlag, Basel, 2007.
    [25] R. C. Reilly, Mean curvature, the laplacian, and soap bubbles, The American Mathematical Monthly, 89 (1982), 180-198.  doi: 10.2307/2320201.
    [26] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.
    [27] R. P. Sperb, Maximum Principles and Their Applications, Elsevier, 1981.
    [28] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  doi: 10.1007/BF01449041.
    [29] H. F. Weinberger, Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal., 43 (1971), 319-320.  doi: 10.1007/BF00250469.
  • 加载中

Article Metrics

HTML views(1312) PDF downloads(394) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint