Eigenvalues of the fractional Laplace operator in the unit ball
| Autor: | Bartłomiej Dyda, Mateusz Kwaśnicki, Alexey Kuznetsov |
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| Rok vydání: | 2017 |
| Předmět: |
Unit sphere
Dense set Antisymmetric relation General Mathematics 010102 general mathematics Mathematical analysis 01 natural sciences 010101 applied mathematics symbols.namesake Variational method Dirichlet boundary condition Orthogonal polynomials symbols 0101 mathematics Laplace operator Eigenvalues and eigenvectors Mathematics |
| Zdroj: | Journal of the London Mathematical Society. 95:500-518 |
| ISSN: | 0024-6107 |
| DOI: | 10.1112/jlms.12024 |
| Popis: | We describe a highly efficient numerical scheme for finding two-sided bounds for the eigenvalues of the fractional Laplace operator (−Δ)α/2 in the unit ball D⊂Rd, with a Dirichlet condition in the complement of D. The standard Rayleigh–Ritz variational method is used for the upper bounds, while the lower bounds involve the lesser known Aronszajn method of intermediate problems. Both require explicit expressions for the fractional Laplace operator applied to a linearly dense set of functions in L2(D). We use appropriate Jacobi-type orthogonal polynomials, which were studied in a companion paper (B. Dyda, A. Kuznetsov and M. Kwaśnicki, ‘Fractional Laplace operator and Meijer G-function’, Constr. Approx., to appear, doi:10.1007/s00365-016-9336-4). Our numerical scheme can be applied analytically when polynomials of degree two are involved. This is used to partially resolve the conjecture of Kulczycki, which claims that the second smallest eigenvalue corresponds to an antisymmetric function: we prove that this is the case when either d⩽2 and α∈(0,2], or d⩽9 and α=1, and we provide strong numerical evidence for d⩽9 and general α∈(0,2]. |
| Databáze: | OpenAIRE |
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