Linear non-divergence elliptic equations in a bounded, infinitely winding planar domain
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910024866988032 |
|---|---|
| author | Hoang, Luan Ibragimov, Akif |
| author_facet | Hoang, Luan Ibragimov, Akif |
| contents | We study the second order elliptic equations of non-divergence form in a planar domain with complicated geometry. In this case the domain winds around a fixed circle infinitely many times and converges to it when the rotating angle goes to infinity. For the homogeneous equation and the homogeneous Dirichlet boundary condition, in the case of bounded drifts, we prove that the maximum of the solution on the cross-section corresponding to a given rotating angle either grows or decays exponentially as the angle goes to infinity. Results for the oscillation and its asymptotic estimates are also obtained for inhomogeneous Dirichlet data. If the drift is unbounded but does not grow to infinity too fast, then the above maximum also goes to either zero or infinity. For the inhomogeneous equation, we obtain the estimates in the case of bounded forcing functions. Moreover, we establish the uniqueness of the solution and its continuous dependence on the boundary data and the forcing function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14352 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear non-divergence elliptic equations in a bounded, infinitely winding planar domain Hoang, Luan Ibragimov, Akif Analysis of PDEs 35A09, 35B30, 35B40, 35B50 We study the second order elliptic equations of non-divergence form in a planar domain with complicated geometry. In this case the domain winds around a fixed circle infinitely many times and converges to it when the rotating angle goes to infinity. For the homogeneous equation and the homogeneous Dirichlet boundary condition, in the case of bounded drifts, we prove that the maximum of the solution on the cross-section corresponding to a given rotating angle either grows or decays exponentially as the angle goes to infinity. Results for the oscillation and its asymptotic estimates are also obtained for inhomogeneous Dirichlet data. If the drift is unbounded but does not grow to infinity too fast, then the above maximum also goes to either zero or infinity. For the inhomogeneous equation, we obtain the estimates in the case of bounded forcing functions. Moreover, we establish the uniqueness of the solution and its continuous dependence on the boundary data and the forcing function. |
| title | Linear non-divergence elliptic equations in a bounded, infinitely winding planar domain |
| topic | Analysis of PDEs 35A09, 35B30, 35B40, 35B50 |
| url | https://arxiv.org/abs/2509.14352 |