Gardado en:
| Autor Principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| Subjects: | |
| Acceso en liña: | https://arxiv.org/abs/2405.02457 |
| Tags: |
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Table of 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 $K(\mathbf{x})$.