Nonradiality of second eigenfunctions of the fractional Laplacian in a ball

Benedikt, Jiří; Bobkov, Vladimir; Dhara, Raj; Girg, Petr

doi:10.1090/proc/16062

Nonradiality of second eigenfunctions of the fractional Laplacian in a ball
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by Jiří Benedikt, Vladimir Bobkov, Raj Narayan Dhara and Petr Girg PDF

Proc. Amer. Math. Soc. 150 (2022), 5335-5348 Request permission

Abstract:

Using symmetrization techniques, we show that, for every $N \geq 2$, any second eigenfunction of the fractional Laplacian in the $N$-dimensional unit ball with homogeneous Dirichlet conditions is nonradial, and hence its nodal set is an equatorial section of the ball.

References

T. V. Anoop, P. Drábek, and Sarath Sasi, On the structure of the second eigenfunctions of the $p$-Laplacian on a ball, Proc. Amer. Math. Soc. 144 (2016), no. 6, 2503–2512. MR 3477066, DOI 10.1090/proc/12902
S. Axler, P. Bourdon, and R. Wade, Harmonic function theory, vol. 137, Springer Science & Business Media, 2013.
Rodrigo Bañuelos and Tadeusz Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal. 211 (2004), no. 2, 355–423. MR 2056835, DOI 10.1016/j.jfa.2004.02.005
William Beckner, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $S^n$, Proc. Nat. Acad. Sci. U.S.A. 89 (1992), no. 11, 4816–4819. MR 1164616, DOI 10.1073/pnas.89.11.4816
Jiří Benedikt, Pavel Drábek, and Petr Girg, The second eigenfunction of the $p$-Laplacian on the disk is not radial, Nonlinear Anal. 75 (2012), no. 12, 4422–4435. MR 2927111, DOI 10.1016/j.na.2011.06.012
Umberto Biccari and Víctor Hernández-Santamaría, The Poisson equation from non-local to local, Electron. J. Differential Equations (2018), Paper No. 145, 13. MR 3856421
Vladimir Bobkov and Sergey Kolonitskii, On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 149 (2019), no. 5, 1163–1173. MR 4010517, DOI 10.1017/prm.2018.88
Lorenzo Brasco and Enea Parini, The second eigenvalue of the fractional $p$-Laplacian, Adv. Calc. Var. 9 (2016), no. 4, 323–355. MR 3552458, DOI 10.1515/acv-2015-0007
Friedemann Brock and Alexander Yu. Solynin, An approach to symmetrization via polarization, Trans. Amer. Math. Soc. 352 (2000), no. 4, 1759–1796. MR 1695019, DOI 10.1090/S0002-9947-99-02558-1
Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. MR 2944369, DOI 10.1016/j.bulsci.2011.12.004
Bartłomiej Dyda, Alexey Kuznetsov, and Mateusz Kwaśnicki, Eigenvalues of the fractional Laplace operator in the unit ball, J. Lond. Math. Soc. (2) 95 (2017), no. 2, 500–518. MR 3656279, DOI 10.1112/jlms.12024
Mouhamed Moustapha Fall, Pierre Aime Feulefack, Remi Yvant Temgoua, and Tobias Weth, Morse index versus radial symmetry for fractional Dirichlet problems, Adv. Math. 384 (2021), Paper No. 107728, 22. MR 4246096, DOI 10.1016/j.aim.2021.107728
Rui A. C. Ferreira, Anti-symmetry of the second eigenfunction of the fractional Laplace operator in a 3-D ball, NoDEA Nonlinear Differential Equations Appl. 26 (2019), no. 1, Paper No. 6, 8. MR 3905370, DOI 10.1007/s00030-019-0554-x
Rupert L. Frank, Eigenvalue bounds for the fractional Laplacian: a review, Recent developments in nonlocal theory, De Gruyter, Berlin, 2018, pp. 210–235. MR 3824213, DOI 10.1515/9783110571561-007
Nicola Garofalo, Fractional thoughts, New developments in the analysis of nonlocal operators, Contemp. Math., vol. 723, Amer. Math. Soc., [Providence], RI, [2019] ©2019, pp. 1–135. MR 3916700, DOI 10.1090/conm/723/14569
Erik Lindgren and Peter Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 795–826. MR 3148135, DOI 10.1007/s00526-013-0600-1
Xavier Ros-Oton and Joaquim Serra, Fractional Laplacian: Pohozaev identity and nonexistence results, C. R. Math. Acad. Sci. Paris 350 (2012), no. 9-10, 505–508 (English, with English and French summaries). MR 2929057, DOI 10.1016/j.crma.2012.05.011
Xavier Ros-Oton and Joaquim Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl. (9) 101 (2014), no. 3, 275–302 (English, with English and French summaries). MR 3168912, DOI 10.1016/j.matpur.2013.06.003
Raffaella Servadei and Enrico Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 2105–2137. MR 3002745, DOI 10.3934/dcds.2013.33.2105
Raffaella Servadei and Enrico Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal. 12 (2013), no. 6, 2445–2464. MR 3060890, DOI 10.3934/cpaa.2013.12.2445
C. L. Siegel (1929), Uber einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss., Phys.-math. Kl., Reprinted in Gesammelte Abhandlungen I, Berlin-Heidelberg-New York: Springer-Verlag, 1966.
Pablo Raúl Stinga and José Luis Torrea, Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations 35 (2010), no. 11, 2092–2122. MR 2754080, DOI 10.1080/03605301003735680
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746