The Steklov problem for exterior domains: asymptotic behavior and applications
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918090783064064 |
|---|---|
| author | Grebenkov, Denis S. Chaigneau, Adrien |
| author_facet | Grebenkov, Denis S. Chaigneau, Adrien |
| contents | We investigate the spectral properties of the Steklov problem for the modified Helmholtz equation $(p-Δ) u = 0$ in the exterior of a compact set, for which the positive parameter $p$ ensures exponential decay of the Steklov eigenfunctions at infinity. We obtain the small-$p$ asymptotic behavior of the eigenvalues and eigenfunctions and discuss their features for different space dimensions. These results find immediate applications to the theory of stochastic processes and unveil the long-time asymptotic behavior of probability densities of various first-passage times in exterior domains. Theoretical results are validated by solving the exterior Steklov problem by a finite-element method with a transparent boundary condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09864 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Steklov problem for exterior domains: asymptotic behavior and applications Grebenkov, Denis S. Chaigneau, Adrien Mathematical Physics Analysis of PDEs Spectral Theory Chemical Physics We investigate the spectral properties of the Steklov problem for the modified Helmholtz equation $(p-Δ) u = 0$ in the exterior of a compact set, for which the positive parameter $p$ ensures exponential decay of the Steklov eigenfunctions at infinity. We obtain the small-$p$ asymptotic behavior of the eigenvalues and eigenfunctions and discuss their features for different space dimensions. These results find immediate applications to the theory of stochastic processes and unveil the long-time asymptotic behavior of probability densities of various first-passage times in exterior domains. Theoretical results are validated by solving the exterior Steklov problem by a finite-element method with a transparent boundary condition. |
| title | The Steklov problem for exterior domains: asymptotic behavior and applications |
| topic | Mathematical Physics Analysis of PDEs Spectral Theory Chemical Physics |
| url | https://arxiv.org/abs/2407.09864 |