Symmetry results for Serrin-type problems in doubly connected domains

Stefano Borghini; Stefano Borghini

doi:10.3934/mine.2023027

Mathematics in Engineering

2023, Volume 5, Issue 2: 1-16. doi: 10.3934/mine.2023027

Previous Article Next Article

Research article Special Issues

Symmetry results for Serrin-type problems in doubly connected domains

Stefano Borghini ^,

Dipartimento di matematica e applicazioni, Università degli Studi di Milano-Bicocca, Via Roberto Cozzi, 55, 20125 Milano (MI), Italia

Received: 29 October 2021 Revised: 15 March 2022 Accepted: 28 March 2022 Published: 26 April 2022

In this work, we employ the technique developed in ^[2] to prove rotational symmetry for a class of Serrin-type problems for the standard Laplacian. We also discuss in some length how our strategy compares with the classical moving plane method.
- overdetermined boundary value problems,
- doubly-connected domains
Citation: Stefano Borghini. Symmetry results for Serrin-type problems in doubly connected domains[J]. Mathematics in Engineering, 2023, 5(2): 1-16. doi: 10.3934/mine.2023027

Related Papers:

Abstract

In this work, we employ the technique developed in ^[2] to prove rotational symmetry for a class of Serrin-type problems for the standard Laplacian. We also discuss in some length how our strategy compares with the classical moving plane method.

References

[1]	A. Aftalion, J. Busca, Radial symmetry of overdetermined boundary-value problems in exterior domains, Arch. Rational Mech. Anal., 143 (1998), 195–206. http://dx.doi.org/10.1007/s002050050103 doi: 10.1007/s002050050103
[2]	V. Agostiniani, S. Borghini, L. Mazzieri, On the Serrin problem for ring-shaped domains, 2021, arXiv: 2109.11255.
[3]	V. Agostiniani, M. Fogagnolo, L. Mazzieri, Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature, Invent. Math., 222 (2020), 1033–1101. http://dx.doi.org/10.1007/s00222-020-00985-4 doi: 10.1007/s00222-020-00985-4
[4]	V. Agostiniani, L. Mazzieri, F. Oronzio, A Green's function proof of the positive mass theorem, 2021, arXiv: 2108.08402.
[5]	G. Alessandrini, A symmetry theorem for condensers, Math. Method. Appl. Sci., 15 (1992), 315–320. http://dx.doi.org/10.1002/mma.1670150503 doi: 10.1002/mma.1670150503
[6]	A. D. Alexandrov, A characteristic property of spheres, Annali di Matematica, 58 (1962), 303–315. http://dx.doi.org/10.1007/BF02413056 doi: 10.1007/BF02413056
[7]	L. Barbu, C. Enache, A free boundary problem with multiple boundaries for a general class of anisotropic equations, Appl. Math. Comput., 362 (2019), 124551. http://dx.doi.org/10.1016/j.amc.2019.06.065 doi: 10.1016/j.amc.2019.06.065
[8]	B. Barrios, L. Montoro, B. Sciunzi, On the moving plane method for nonlocal problems in bounded domains, JAMA, 135 (2018), 37–57. http://dx.doi.org/10.1007/s11854-018-0031-1 doi: 10.1007/s11854-018-0031-1
[9]	R. Beig, W. Simon, On the uniqueness of static perfect-fluid solutions in general relativity, Commun. Math. Phys., 144 (1992), 373–390. https://dx.doi.org/10.1007/BF02101098 doi: 10.1007/BF02101098
[10]	S. Borghini, Static Black Hole uniqueness for nonpositive masses, Nonlinear Anal., 220 (2022), 112843. http://dx.doi.org/10.1016/j.na.2022.112843 doi: 10.1016/j.na.2022.112843
[11]	S. Borghini, P. T. Chruściel, L. Mazzieri, On the uniqueness of Schwarzschild-de Sitter spacetime, 2019, arXiv: 1909.05941.
[12]	S. Borghini, L. Mazzieri, On the mass of static metrics with positive cosmological constant: Ⅰ, Class. Quantum Grav., 35 (2018), 125001. http://dx.doi.org/10.1088/1361-6382/aac081 doi: 10.1088/1361-6382/aac081
[13]	S. Borghini, L. Mazzieri, On the mass of static metrics with positive cosmological constant: Ⅱ, Commun. Math. Phys., 377 (2020), 2079–2158. http://dx.doi.org/10.1007/s00220-020-03739-8 doi: 10.1007/s00220-020-03739-8
[14]	B. Brandolini, C. Nitsch, P. Salani, C. Trombetti, On the stability of the Serrin problem, J. Differ. Equations, 245 (2008), 1566–1583. http://dx.doi.org/10.1016/j.jde.2008.06.010 doi: 10.1016/j.jde.2008.06.010
[15]	P. T. Chruściel, W. Simon, Towards the classification of static vacuum spacetimes with negative cosmological constant, J. Math. Phys., 42 (2001), 1779–1817. http://dx.doi.org/10.1063/1.1340869 doi: 10.1063/1.1340869
[16]	G. Ciraolo, L. Vezzoni, On Serrin's overdetermined problem in space forms, Manuscripta Math., 159 (2019), 445–452. http://dx.doi.org/10.1007/s00229-018-1079-z doi: 10.1007/s00229-018-1079-z
[17]	A. Enciso, D. Peralta-Salas, Symmetry for an overdetermined boundary problem in a punctured domain, Nonlinear Anal. Theor., 70 (2009), 1080–1086. http://dx.doi.org/10.1016/j.na.2008.01.034 doi: 10.1016/j.na.2008.01.034
[18]	A. Farina, B. Kawohl, Remarks on an overdetermined boundary value problem, Calc. Var., 31 (2008), 351–357. http://dx.doi.org/10.1007/s00526-007-0115-8 doi: 10.1007/s00526-007-0115-8
[19]	A. Farina, E. Valdinoci, A pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature, Adv. Math., 225 (2010), 2808–2827. http://dx.doi.org/10.1016/j.aim.2010.05.008 doi: 10.1016/j.aim.2010.05.008
[20]	M. Fogagnolo, L. Mazzieri, A. Pinamonti, Geometric aspects of $p$-capacitary potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1151–1179. http://dx.doi.org/10.1016/j.anihpc.2018.11.005 doi: 10.1016/j.anihpc.2018.11.005
[21]	M. Fogagnolo, A. Pinamonti, New integral estimates in substatic Riemannian manifolds and the Alexandrov theorem, 2021, arXiv: 2105.04672.
[22]	N. Garofalo, J. L. Lewis, A symmetry result related to some overdetermined boundary value problems, Amer. J. Math., 111 (1989), 9–33. http://dx.doi.org/10.2307/2374477 doi: 10.2307/2374477
[23]	B. Gidas, W. M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209–243. https://dx.doi.org/10.1007/BF01221125 doi: 10.1007/BF01221125
[24]	D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, 2 Eds., Berlin, Heidelberg: Springer, 2001. https://dx.doi.org/10.1007/978-3-642-61798-0
[25]	N. Kamburov, L. Sciaraffia, Nontrivial solutions to Serrin's problem in annular domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 38 (2021), 1–22. http://dx.doi.org/10.1016/j.anihpc.2020.05.001 doi: 10.1016/j.anihpc.2020.05.001
[26]	D. A. Lee, A. Neves, The Penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass, Commun. Math. Phys., 339 (2015), 327–352. http://dx.doi.org/10.1007/s00220-015-2421-x doi: 10.1007/s00220-015-2421-x
[27]	L. Ma, B. Liu, Symmetry results for decay solutions of elliptic systems in the whole space, Adv. Math., 225 (2010), 3052–3063. http://dx.doi.org/10.1016/j.aim.2010.05.022 doi: 10.1016/j.aim.2010.05.022
[28]	R. Magnanini, Alexandrov, Serrin, Weinberger, Reilly: simmetry and stability by integral identities, Bruno Pini Mathematical Analysis Seminar, Italy: University of Bologna, 2017,121–141. http://dx.doi.org/10.6092/issn.2240-2829/7800
[29]	L. E. Payne, G. A. Philippin, On two free boundary problems in potential theory, J. Math. Anal. Appl., 161 (1991), 332–342. http://dx.doi.org/10.1016/0022-247X(91)90333-U doi: 10.1016/0022-247X(91)90333-U
[30]	S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u) = 0$, (Russian), Dokl. Akad. Nauk SSSR, 165 (1965), 36–39.
[31]	W. Reichel, Radial symmetry by moving planes for semilinear elliptic BVPs on annuli and other non-convex domains, In: Progress in partial differential equations: elliptic and parabolic problems, Math. Inst. I, 1995,164–182.
[32]	W. Reichel, Radial symmetry for an electrostatic, a capillarity and some fully nonlinear overdetermined problems on exterior domains, Z. Anal. Anwend., 15 (1996), 619–635. http://dx.doi.org/10.4171/ZAA/719 doi: 10.4171/ZAA/719
[33]	W. Reichel, Radial symmetry for elliptic boundary-value problems on exterior domains, Arch. Rational Mech. Anal., 137 (1997), 381–394. http://dx.doi.org/10.1007/s002050050034 doi: 10.1007/s002050050034
[34]	A. Roncoroni, A Serrin-type symmetry result on model manifolds: an extension of the Weinberger argument, C. R. Math., 356 (2018), 648–656. http://dx.doi.org/10.1016/j.crma.2018.04.012 doi: 10.1016/j.crma.2018.04.012
[35]	J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304–318. http://dx.doi.org/10.1007/BF00250468 doi: 10.1007/BF00250468
[36]	B. Sirakov, Symmetry for exterior elliptic problems and two conjectures in potential theory, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 135–156. http://dx.doi.org/10.1016/S0294-1449(00)00052-4 doi: 10.1016/S0294-1449(00)00052-4
[37]	N. Soave, E. Valdinoci, Overdetermined problems for the fractional Laplacian in exterior and annular sets, J. Anal. Math., 137 (2019), 101–134. http://dx.doi.org/10.1007/s11854-018-0067-2 doi: 10.1007/s11854-018-0067-2
[38]	H. F. Weinberger, Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal., 43 (1971), 319–320. http://dx.doi.org/10.1007/BF00250469 doi: 10.1007/BF00250469
[39]	H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63–89. http://dx.doi.org/10.2307/1989708 doi: 10.2307/1989708
[40]	N. B. Willms, G. M. L. Gladwell, D. Siegel, Symmetry theorems for some overdetermined boundary value problems on ring domains, Z. Angew. Math. Phys., 45 (1994), 556–579. http://dx.doi.org/10.1007/BF00991897 doi: 10.1007/BF00991897

Reader Comments

Your name:*

Email:*
© 2023 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)