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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.02803 |
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| _version_ | 1866909753801703424 |
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| author | Biswas, Nirjan Prasad, Harsh |
| author_facet | Biswas, Nirjan Prasad, Harsh |
| contents | For $p \in (1, \infty)$ and $s \in (0,1)$, we consider the following mixed local-nonlocal equation
$$ - Δ_p u + (-Δ_p)^s u = f \; \text{in} \; Ω,$$ where $Ω\subset \mathbb{R}^d$ is a bounded domain and the function $f \in L_{loc}^1(Ω)$. Depending on the dimension $d$, we prove gradient potential estimates of weak solutions for the entire ranges of $p$ and $s$. As a byproduct, we recover the corresponding estimates in the purely diffusive setup, providing connections between the local and nonlocal aspects of the equation. Our results are new, even for the linear case $p=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_02803 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lipschitz potential estimates for diffusion with jumps Biswas, Nirjan Prasad, Harsh Analysis of PDEs 35B65, 35J60, 31C15, 35D30 For $p \in (1, \infty)$ and $s \in (0,1)$, we consider the following mixed local-nonlocal equation $$ - Δ_p u + (-Δ_p)^s u = f \; \text{in} \; Ω,$$ where $Ω\subset \mathbb{R}^d$ is a bounded domain and the function $f \in L_{loc}^1(Ω)$. Depending on the dimension $d$, we prove gradient potential estimates of weak solutions for the entire ranges of $p$ and $s$. As a byproduct, we recover the corresponding estimates in the purely diffusive setup, providing connections between the local and nonlocal aspects of the equation. Our results are new, even for the linear case $p=2$. |
| title | Lipschitz potential estimates for diffusion with jumps |
| topic | Analysis of PDEs 35B65, 35J60, 31C15, 35D30 |
| url | https://arxiv.org/abs/2307.02803 |