Differentiability of the nonlocal-to-local transition in fractional Poisson problems
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| Format: | Preprint |
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2023
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| _version_ | 1866912001419116544 |
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| author | Saldaña, Alberto Jarohs, Sven Weth, Tobias |
| author_facet | Saldaña, Alberto Jarohs, Sven Weth, Tobias |
| contents | Let $u_s$ denote a solution of the fractional Poisson problem $$
(-Δ)^s u_s = f\quad\text{ in }Ω,\qquad u_s=0\quad \text{ on }\mathbb{R}^N\setminus Ω, $$ where $N\geq 2$ and $Ω\subset \mathbb{R}^N$ is a bounded domain of class $C^2$. We show that the solution mapping $s\mapsto u_s$ is differentiable in $L^\infty(Ω)$ at $s=1$, namely, at the nonlocal-to-local transition. Moreover, using the logarithmic Laplacian, we characterize the derivative $\partial_s u_s$ as the solution to a boundary value problem. This complements the previously known differentiability results for $s$ in the open interval $(0,1)$. Our proofs are based on an asymptotic analysis to describe the collapse of the nonlocality of the fractional Laplacian as $s$ approaches 1. We also provide a new representation of $\partial_s u_s$ for $s \in (0,1)$ which allows us to refine previously obtained Green function estimates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_18476 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Differentiability of the nonlocal-to-local transition in fractional Poisson problems Saldaña, Alberto Jarohs, Sven Weth, Tobias Analysis of PDEs 35S15, 35B40, 35C20, 35B30, 35C15 Let $u_s$ denote a solution of the fractional Poisson problem $$ (-Δ)^s u_s = f\quad\text{ in }Ω,\qquad u_s=0\quad \text{ on }\mathbb{R}^N\setminus Ω, $$ where $N\geq 2$ and $Ω\subset \mathbb{R}^N$ is a bounded domain of class $C^2$. We show that the solution mapping $s\mapsto u_s$ is differentiable in $L^\infty(Ω)$ at $s=1$, namely, at the nonlocal-to-local transition. Moreover, using the logarithmic Laplacian, we characterize the derivative $\partial_s u_s$ as the solution to a boundary value problem. This complements the previously known differentiability results for $s$ in the open interval $(0,1)$. Our proofs are based on an asymptotic analysis to describe the collapse of the nonlocality of the fractional Laplacian as $s$ approaches 1. We also provide a new representation of $\partial_s u_s$ for $s \in (0,1)$ which allows us to refine previously obtained Green function estimates. |
| title | Differentiability of the nonlocal-to-local transition in fractional Poisson problems |
| topic | Analysis of PDEs 35S15, 35B40, 35C20, 35B30, 35C15 |
| url | https://arxiv.org/abs/2311.18476 |