Scientific seminar on the differential and functional differential equation “Non-radiality of the second eigenfunction of the fractional Laplace in the ball”
28 March at 12.00 (Moscow time)
Speaker: Bobkov Vladimir, Institute of Mathematics with Computing Centre, Ufa Federal Research Centre, Russian Academy of Sciences, Ufa.
Topic: Non-radiality of the second eigenfunction of the fractional Laplace in the ball.
In connection with the work [Banuelos, Kulczycki. J. Funct. Anal., 2004], Kulchitsky put forward the hypothesis that the second eigenfunction of the fractional Laplace operator in a ball is not radial, and moreover, it is antisymmetric with respect to the central section of the ball by a plane. In the case of space dimensions 1 and 2, this conjecture was proved in [Dyda, Kuznetsov, Kwaśnicki. J. Lond. Math. Soc., 2017], and in the case of dimension 3, in [Ferreira. NoDEA, 2019], using the Aronshine and Rayleigh-Ritz methods to find lower and upper bounds for eigenvalues. In a recent work [Fall, Feulefack, Temgoua, Weth. Adv. Math., 2021], Kulchitsky’s conjecture was proved using estimates of the Morse index of eigenfunctions.
The report will be devoted to the work [Benedikt, Bobkov, Dhara, Girg. Proc. amer. Math. Soc., 2022], in which we presented an alternative approach to proving this hypothesis in all dimensions, based on the development of the so-called. moving polarization method introduced earlier in [Bobkov, Kolonitskii. Proc. Roy. soc. Edinburgh Sect. A, 2019], as applied to problems with fractional operators, which is fundamentally different from previous approaches.