A note on the $L^{p}$-solvability of a strongly-coupled nonlocal system of equations
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| Format: | Preprint |
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2025
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| _version_ | 1866918524384968704 |
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| author | Mengesha, Tadele Abbate, Miriam |
| author_facet | Mengesha, Tadele Abbate, Miriam |
| contents | The goal of this paper is to study the $L^p$-solvability of the strongly-coupled nonlocal system \[
\mathbb{L} \mathbf{u} (\mathbf{x}) + λ\mathbf{u}(\mathbf{x})= \mathbf{f}(\mathbf{x}) \quad \text{in $\mathbb{R}^{d}$ } \] where $\mathbb{L}$ is a linear nonlocal coupled vector-valued operator associated with a kernel $K$ comparable to $|\mathbf{y}|^{-(d+2s)}$ for $s \in (0,1)$, satisfying certain ellipticity and cancellation conditions. For any $\mathbf{f} \in [L^p(\mathbb{R}^d)]^d$, $1< p < \infty$, the existence of a unique strong solution $\mathbf{u} \in [H^{2s,p}(\mathbb{R}^d)]^d$ is proved via the method of continuity. To apply this method, we establish the continuity of the operator $\mathbb{L}$ and the necessary \textit{a priori} estimates. These are obtained through the study of the corresponding parabolic system. The proof strategy follows and extends recent ideas developed for the scalar setting, combining commutator estimates, Sobolev embeddings, a level set estimates and a bootstrap argument. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_20772 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the $L^{p}$-solvability of a strongly-coupled nonlocal system of equations Mengesha, Tadele Abbate, Miriam Analysis of PDEs The goal of this paper is to study the $L^p$-solvability of the strongly-coupled nonlocal system \[ \mathbb{L} \mathbf{u} (\mathbf{x}) + λ\mathbf{u}(\mathbf{x})= \mathbf{f}(\mathbf{x}) \quad \text{in $\mathbb{R}^{d}$ } \] where $\mathbb{L}$ is a linear nonlocal coupled vector-valued operator associated with a kernel $K$ comparable to $|\mathbf{y}|^{-(d+2s)}$ for $s \in (0,1)$, satisfying certain ellipticity and cancellation conditions. For any $\mathbf{f} \in [L^p(\mathbb{R}^d)]^d$, $1< p < \infty$, the existence of a unique strong solution $\mathbf{u} \in [H^{2s,p}(\mathbb{R}^d)]^d$ is proved via the method of continuity. To apply this method, we establish the continuity of the operator $\mathbb{L}$ and the necessary \textit{a priori} estimates. These are obtained through the study of the corresponding parabolic system. The proof strategy follows and extends recent ideas developed for the scalar setting, combining commutator estimates, Sobolev embeddings, a level set estimates and a bootstrap argument. |
| title | A note on the $L^{p}$-solvability of a strongly-coupled nonlocal system of equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.20772 |