# American Institute of Mathematical Sciences

February  2020, 19(2): 983-1000. doi: 10.3934/cpaa.2020045

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

 1 Dipartimento di Matematica "Giuseppe Peano", Università degli Studi di Torino, via Carlo Alberto, 10 - 10123 Torino, Italy 2 Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, piazza di Porta S. Donato, 5 - 40126 Bologna, Italy

* Corresponding author

Received  March 2019 Revised  July 2019 Published  October 2019

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.

Citation: Francesca Colasuonno, Fausto Ferrari. The Soap Bubble Theorem and a $p$-Laplacian overdetermined problem. Communications on Pure & Applied Analysis, 2020, 19 (2) : 983-1000. doi: 10.3934/cpaa.2020045
##### References:
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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.  Google Scholar [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.  Google Scholar [8] G. Ciraolo, R. 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.  Google Scholar [9] G. Ciraolo and A. Roncoroni, Serrin's type overdetermined problems in convex cones, preprint, arXiv: 1806.08553. Google Scholar [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.  Google Scholar [11] L. Damascelli and F. Pacella, Monotonicity and symmetry results for $p$-Laplace equations and applications, Adv. Differential Equations, 5 (2000), 1179-1200.   Google Scholar [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.  Google Scholar [13] I. Fragalà, F. 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.  Google Scholar [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.  Google Scholar [15] B. Kawohl, Symmetrization–or how to prove symmetry of solutions to a pde, Partial differential equations (Praha, 1998), 406 (1999), 214-229.   Google Scholar [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.  Google Scholar [17] R. Magnanini and G. Poggesi, On the stability for Alexandrov's soap bubble theorem, to appear in J. Anal. Math, (2016). Google Scholar [18] R. Magnanini and G. Poggesi, Serrin's problem and Alexandrov's soap bubble theorem: enhanced stability via integral identities, preprint, arXiv: 1708.07392. Google Scholar [19] R. Magnanini and G. Poggesi, Nearly optimal stability for Serrin's problem and the soap bubble theorem, preprint, arXiv: 1903.04823. Google Scholar [20] F. Pacella and G. Tralli, Overdetermined problems and constant mean curvature surfaces in cones, preprint, arXiv: 1802.03197. Google Scholar [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.  Google Scholar [22] G. Poggesi, The Soap Bubble Theorem and Serrin's problem: quantitative symmetry, Ph.D thesis, Università di Firenze, arXiv: 1902.08584, 2019. Google Scholar [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.  Google Scholar [24] P. Pucci and J. Serrin, The maximum principle, volume 73 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Verlag, Basel, 2007.  Google Scholar [25] R. C. Reilly, Mean curvature, the laplacian, and soap bubbles, The American Mathematical Monthly, 89 (1982), 180-198.  doi: 10.2307/2320201.  Google Scholar [26] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.  Google Scholar [27] R. P. Sperb, Maximum Principles and Their Applications, Elsevier, 1981.  Google Scholar [28] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  doi: 10.1007/BF01449041.  Google Scholar [29] H. F. Weinberger, Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal., 43 (1971), 319-320.  doi: 10.1007/BF00250469.  Google Scholar

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##### References:
 [1] A. Aftalion, J. Busca and W. Reichel, Approximate radial symmetry for overdetermined boundary value problems, Adv. Differential Equations, 4 (1999), 907-932.   Google Scholar [2] A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl., 58 (1962), 303-315.  doi: 10.1007/BF02413056.  Google Scholar [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.  Google Scholar [4] B. Brandolini, C. Nitsch, P. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] G. Ciraolo, R. 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.  Google Scholar [9] G. Ciraolo and A. Roncoroni, Serrin's type overdetermined problems in convex cones, preprint, arXiv: 1806.08553. Google Scholar [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.  Google Scholar [11] L. Damascelli and F. Pacella, Monotonicity and symmetry results for $p$-Laplace equations and applications, Adv. Differential Equations, 5 (2000), 1179-1200.   Google Scholar [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.  Google Scholar [13] I. Fragalà, F. 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.  Google Scholar [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.  Google Scholar [15] B. Kawohl, Symmetrization–or how to prove symmetry of solutions to a pde, Partial differential equations (Praha, 1998), 406 (1999), 214-229.   Google Scholar [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.  Google Scholar [17] R. Magnanini and G. Poggesi, On the stability for Alexandrov's soap bubble theorem, to appear in J. Anal. Math, (2016). Google Scholar [18] R. Magnanini and G. Poggesi, Serrin's problem and Alexandrov's soap bubble theorem: enhanced stability via integral identities, preprint, arXiv: 1708.07392. Google Scholar [19] R. Magnanini and G. Poggesi, Nearly optimal stability for Serrin's problem and the soap bubble theorem, preprint, arXiv: 1903.04823. Google Scholar [20] F. Pacella and G. Tralli, Overdetermined problems and constant mean curvature surfaces in cones, preprint, arXiv: 1802.03197. Google Scholar [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.  Google Scholar [22] G. Poggesi, The Soap Bubble Theorem and Serrin's problem: quantitative symmetry, Ph.D thesis, Università di Firenze, arXiv: 1902.08584, 2019. Google Scholar [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.  Google Scholar [24] P. Pucci and J. Serrin, The maximum principle, volume 73 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Verlag, Basel, 2007.  Google Scholar [25] R. C. Reilly, Mean curvature, the laplacian, and soap bubbles, The American Mathematical Monthly, 89 (1982), 180-198.  doi: 10.2307/2320201.  Google Scholar [26] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.  Google Scholar [27] R. P. Sperb, Maximum Principles and Their Applications, Elsevier, 1981.  Google Scholar [28] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  doi: 10.1007/BF01449041.  Google Scholar [29] H. F. Weinberger, Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal., 43 (1971), 319-320.  doi: 10.1007/BF00250469.  Google Scholar
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