Nonlocal Boundary Value Problems Governed by Symmetric Nonlocal Operators
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912853561180160 |
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| author | Frerick, Leonhard Huschens, Julia Vu, Michael |
| author_facet | Frerick, Leonhard Huschens, Julia Vu, Michael |
| contents | Nonlocal boundary value problems with Dirichlet or Neumann boundary are well-studied for nonlocal operators of the type $\mathcal{L}_γu = \operatorname{PV} \int_{\mathbb{R}^d} \big(u(\cdot)-u(y)\big) γ(\cdot,y) \, \mathrm{d}y$ where the underlying kernel function $γ: \mathbb{R}^d \times \mathbb{R}^d \rightarrow [0,\infty)$ is assumed to be measurable and symmetric. In this paper, a theory is introduced for problems whose governing operator is of the more general type \[\mathcal{L}u:= \operatorname{PV} \int_{\mathbb{R}^d}\big(u(\cdot)-u(y)\big) \, K(\cdot, \mathrm{d}y)\] where ${K: \mathbb{R}^d \times \mathcal{B}(\mathbb{R}^d) \rightarrow [0,\infty]}$ is a symmetric transition kernel. Our main focus is on nonlocal Dirichlet and Neumann problems and a classical Hilbert space approach is developed for solving designated weak formulations. As an example, the discrete Poisson problem on $Ω=(0,1)^d$ is discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19872 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nonlocal Boundary Value Problems Governed by Symmetric Nonlocal Operators Frerick, Leonhard Huschens, Julia Vu, Michael Analysis of PDEs 35A23, 47G10, 46E35 Nonlocal boundary value problems with Dirichlet or Neumann boundary are well-studied for nonlocal operators of the type $\mathcal{L}_γu = \operatorname{PV} \int_{\mathbb{R}^d} \big(u(\cdot)-u(y)\big) γ(\cdot,y) \, \mathrm{d}y$ where the underlying kernel function $γ: \mathbb{R}^d \times \mathbb{R}^d \rightarrow [0,\infty)$ is assumed to be measurable and symmetric. In this paper, a theory is introduced for problems whose governing operator is of the more general type \[\mathcal{L}u:= \operatorname{PV} \int_{\mathbb{R}^d}\big(u(\cdot)-u(y)\big) \, K(\cdot, \mathrm{d}y)\] where ${K: \mathbb{R}^d \times \mathcal{B}(\mathbb{R}^d) \rightarrow [0,\infty]}$ is a symmetric transition kernel. Our main focus is on nonlocal Dirichlet and Neumann problems and a classical Hilbert space approach is developed for solving designated weak formulations. As an example, the discrete Poisson problem on $Ω=(0,1)^d$ is discussed. |
| title | Nonlocal Boundary Value Problems Governed by Symmetric Nonlocal Operators |
| topic | Analysis of PDEs 35A23, 47G10, 46E35 |
| url | https://arxiv.org/abs/2601.19872 |