A variable diffusivity fractional Laplacian
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909189083758592 |
|---|---|
| author | Ervin, V. J. |
| author_facet | Ervin, V. J. |
| contents | In this paper we analyze the existence, uniqueness and regularity of the solution to the generalized, variable diffusivity, fractional Laplace equation on the unit disk in $\mathbb{R}^{2}$. For $α$ the order of the differential operator, our results show that for the symmetric, positive definite, diffusivity matrix, $K(\mathbf{x})$, satisfying $λ_{m} \mathbf{v}^{T} \mathbf{v} \le \mathbf{v}^{T} K(\mathbf{x}) \mathbf{v} \le λ_{M} \mathbf{v}^{T} \mathbf{v}$, for all $\mathbf{v} \in \mathbb{R}^{2}$, $\mathbf{x} \in Ω$, with $λ_{M} < \frac{\sqrt{α(2 + α)}}{(2 - α)} λ_{m}$, the problem has a unique solution. The regularity of the solution is given in an appropriately weighted Sobolev space in terms of the regularity of the right hand side function and $K(\mathbf{x})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02457 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A variable diffusivity fractional Laplacian Ervin, V. J. Analysis of PDEs 35S15, 42C10, 35B65, 33C55, 60K50 In this paper we analyze the existence, uniqueness and regularity of the solution to the generalized, variable diffusivity, fractional Laplace equation on the unit disk in $\mathbb{R}^{2}$. For $α$ the order of the differential operator, our results show that for the symmetric, positive definite, diffusivity matrix, $K(\mathbf{x})$, satisfying $λ_{m} \mathbf{v}^{T} \mathbf{v} \le \mathbf{v}^{T} K(\mathbf{x}) \mathbf{v} \le λ_{M} \mathbf{v}^{T} \mathbf{v}$, for all $\mathbf{v} \in \mathbb{R}^{2}$, $\mathbf{x} \in Ω$, with $λ_{M} < \frac{\sqrt{α(2 + α)}}{(2 - α)} λ_{m}$, the problem has a unique solution. The regularity of the solution is given in an appropriately weighted Sobolev space in terms of the regularity of the right hand side function and $K(\mathbf{x})$. |
| title | A variable diffusivity fractional Laplacian |
| topic | Analysis of PDEs 35S15, 42C10, 35B65, 33C55, 60K50 |
| url | https://arxiv.org/abs/2405.02457 |