A local symmetry result for linear elliptic problems with solutions changing sign

Canuto, B.

doi:10.1016/j.anihpc.2011.03.005

Canuto, B.

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 4, pp. 551-564.

Résumé

We prove that the only domain Ω such that there exists a solution to the following problem $Δ u + ω^{2} u = - 1$ in Ω, $u = 0$ on ∂Ω, and $\frac{1}{| \partial Ω |} \int_{\partial Ω} \partial_{𝐧} u = c$ , for a given constant c, is the unit ball $B_{1}$ , if we assume that Ω lies in an appropriate class of Lipschitz domains.

Zbl

DOI : 10.1016/j.anihpc.2011.03.005

@article{AIHPC_2011__28_4_551_0,
     author = {Canuto, B.},
     title = {A local symmetry result for linear elliptic problems with solutions changing sign},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {551--564},
     publisher = {Elsevier},
     volume = {28},
     number = {4},
     year = {2011},
     doi = {10.1016/j.anihpc.2011.03.005},
     zbl = {1242.35182},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.03.005/}
}

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Canuto, B. A local symmetry result for linear elliptic problems with solutions changing sign. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 4, pp. 551-564. doi : 10.1016/j.anihpc.2011.03.005. http://www.numdam.org/articles/10.1016/j.anihpc.2011.03.005/

Bibliographie
Cité par

[1] A. Aftalion, J. Busca, W. Reichel, Approximate radial symmetry for overdetermined boundary value problems, Adv. Differential Equations 4 no. 6 (1999), 907-932 | Zbl

[2] G. Alessandrini, A symmetry theorem for condensers, Math. Methods Appl. Sci. 15 (1992), 315-320 | Zbl

[3] E. Berchio, F. Gazzola, T. Weth, Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems, J. Reine Angew. Math. 620 (2008), 165-183 | Zbl

[4] F. Brock, A. Henrot, A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative, Rend. Circ. Mat. Palermo 51 (2002), 375-390 | Zbl

[5] B. Canuto, D. Rial, Local overdetermined linear elliptic problems in Lipschitz domains with solutions changing sign, Rend. Istit. Mat. Univ. Trieste XL (2009), 1-27 | Zbl

[6] B. Canuto, D. Rial, Some remarks on solutions to an overdetermined elliptic problem in divergence form in a ball, Ann. Mat. Pura Appl. 186 (2007), 591-602 | Zbl

[7] M. Choulli, A. Henrot, Use of the domain derivative to prove symmetry results in partial differential equations, Math. Nachr. 192 (1998), 91-103 | Zbl

[8] A. Farina, B. Kawohl, Remarks on an overdetermined boundary value problem, Calc. Var. Partial Differential Equations 31 (2008), 351-357 | Zbl

[9] I. Fragalà, F. Gazzola, J. Lamboley, M. Pierre, Counterexamples to symmetry for partially overdetermined elliptic problems, Analysis (Munich) 29 (2009), 85-93 | Zbl

[10] I. Fragalà, F. Gazzola, Partially overdetermined elliptic boundary value problems, J. Differential Equations 245 (2008), 1299-1322 | Zbl

[11] I. Fragalà, F. Gazzola, B. Kawohl, Overdetermined problems with possibly degenerate ellipticity, a geometric approach, Math. Z. 254 (2006), 117-132 | Zbl

[12] N. Garofalo, J.L. Lewis, A symmetry result related to some overdetermined boundary value problems, Amer. J. Math. 111 (1989), 9-33 | Zbl

[13] F. Gazzola, No geometric approach for general overdetermined elliptic problems with nonconstant source, Matematiche (Catania) 60 (2005), 259-268 | Zbl

[14] A. Greco, Radial symmetry and uniqueness for an overdetermined problem, Math. Methods Appl. Sci. 24 (2001), 103-115 | Zbl

[15] L.E. Payne, G.A. Philippin, On two free boundary problems in potential theory, J. Math. Anal. Appl. 161 no. 2 (1991), 332-342 | Zbl

[16] G.A. Philippin, On a free boundary problem in electrostatics, Math. Methods Appl. Sci. 12 (1990), 387-392 | Zbl

[17] G.A. Philippin, L.E. Payne, On the conformal capacity problem, G. Talenti (ed.), Geometry of Solutions to Partial Differential Equations, Academic, London (1989) | Zbl

[18] J. Prajapat, Serrinʼs result for domains with a corner or cusp, Duke Math. J. 91 (1998), 29-31 | Zbl

[19] W. Reichel, Radial symmetry for elliptic boundary value problems on exterior domains, Arch. Rat. Mech. Anal. 137 (1997), 381-394 | Zbl

[20] J. Serrin, A symmetry problem in potential theory, Arch. Rat. Mech. Anal. 43 (1971), 304-318 | Zbl

[21] B. Sirakov, Symmetry for exterior elliptic problems and two conjectures in potential theory, Ann. Inst. H. Poincaré, Anal. Non Linéaire 18 (2001), 135-156 | EuDML | Numdam | Zbl

[22] A.L. Vogel, Symmetry and regularity for general regions having solutions to certain overdetermined boundary value problems, Atti Sem. Mat. Fis. Univ. Modena 40 (1992), 443-484 | Zbl

[23] H. Weinberger, Remark on the preceding paper by Serrin, Arch. Rat. Mech. Anal. 43 (1971), 319-320 | Zbl

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