| Title: |
Asymptotics of the spectrum of the mixed boundary value problem for the Laplace operator in a thin spindle-shaped domain |
| Authors: |
Nazarov, S. A.; Taskinen, J. |
| Contributors: |
Department of Mathematics and Statistics |
| Publisher Information: |
American Mathematical Society |
| Publication Year: |
2023 |
| Collection: |
Helsingfors Universitet: HELDA – Helsingin yliopiston digitaalinen arkisto |
| Subject Terms: |
Mathematics; EIGENVALUES; EQUATION; EXTENSIONS; Spindle-shaped thin domain; asymptotics of eigenvalues; boundary layers; selfadjoint extensions |
| Description: |
The asymptotics is examined for solutions to the spectral problem for the Laplace operator in a d-dimensional thin, of diameter O(h), spindle-shaped domain Omega(h) with the Dirichlet condition on small, of size h +0, an ordinary differential equation on the axis (-1, 1) (sic) z of the spindle arises with a coefficient degenerating at the points z = +/- 1 and moreover, without any boundary condition because the requirement on the boundedness of eigenfunctions makes the limit spectral problem well-posed. Error estimates are derived for the one-dimensional model but in the case of d = 3 it is necessary to construct boundary layers near the sets Gamma(h)(+/-) and in the case of d = 2 it is necessary to deal with selfadjoint extensions of the differential operator. The extension parameters depend linearly on In h so that its eigenvalues are analytic functions in the variable 1/vertical bar ln h vertical bar As a result, in all dimensions the one-dimensional model gets the power-law accuracy O(h(delta)d ) with an exponent delta(d) > 0. First (the smallest) eigenvalues, positive in Omega(h) and null in (-1, 1), require individual treatment. Also, infinite asymptotic series are discussed, as well as the static problem (without the spectral parameter) and related shapes of thin domains. ; Peer reviewed |
| Document Type: |
article in journal/newspaper |
| File Description: |
application/pdf |
| Language: |
English |
| ISBN: |
978-0-00-768881-4; 0-00-768881-4 |
| Relation: |
https://hdl.handle.net/10138/353582; 000768881400011 |
| Availability: |
https://hdl.handle.net/10138/353582 |
| Rights: |
cc_by ; info:eu-repo/semantics/openAccess ; openAccess |
| Accession Number: |
edsbas.1635D652 |
| Database: |
BASE |