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Dissipative nonlinear schrödinger equations for large data in one space dimension
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 |
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.
References:
[1] |
A. Aftalion, J. 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. 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. |
[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. 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. |
[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. 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. |
[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. |
show all references
References:
[1] |
A. Aftalion, J. 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. 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. |
[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. 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. |
[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. 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. |
[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. |
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