Optimal stability of complement value problems for p-Lévy operators
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911680376602624 |
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| author | Foghem, Guy |
| author_facet | Foghem, Guy |
| contents | We establish the optimal convergence of solutions to integro-differential equations (IDEs) governed by symmetric integrodifferential $p$-Lévy operators, $1 < p < \infty$, in the presence of nonlocal Dirichlet or Neumann boundary conditions. For illustrative purposes, consider the particular case of the (fractional) $p$-Laplacian $(-Δ)^s_p$ with $0 < s \le 1$. If $(-Δ)^s_p u_s = f_s $ in $Ω\subset \mathbb{R}^d,$ augmented with a Dirichlet or Neumann data $g_s$ then under suitable assumptions on $Ω$, $f_s$ and $g_s$, we show that $(u_s)_s$ strongly converges as $s \to 1^-$ in the the optimal, that is, $\|u_s - u_1\|_{W^{s,p}(Ω)} \to 0$. \smallskip Another subsequent goal underpinning our approach is the robustness of the nonlocal trace spaces; specifically, we also show that the nonlocal trace spaces converge, in an appropriate sense, to the local trace space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13389 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal stability of complement value problems for p-Lévy operators Foghem, Guy Analysis of PDEs 35D30, 35J20, 34B15, 35J60, 35J66, 35B35, 46E35 We establish the optimal convergence of solutions to integro-differential equations (IDEs) governed by symmetric integrodifferential $p$-Lévy operators, $1 < p < \infty$, in the presence of nonlocal Dirichlet or Neumann boundary conditions. For illustrative purposes, consider the particular case of the (fractional) $p$-Laplacian $(-Δ)^s_p$ with $0 < s \le 1$. If $(-Δ)^s_p u_s = f_s $ in $Ω\subset \mathbb{R}^d,$ augmented with a Dirichlet or Neumann data $g_s$ then under suitable assumptions on $Ω$, $f_s$ and $g_s$, we show that $(u_s)_s$ strongly converges as $s \to 1^-$ in the the optimal, that is, $\|u_s - u_1\|_{W^{s,p}(Ω)} \to 0$. \smallskip Another subsequent goal underpinning our approach is the robustness of the nonlocal trace spaces; specifically, we also show that the nonlocal trace spaces converge, in an appropriate sense, to the local trace space. |
| title | Optimal stability of complement value problems for p-Lévy operators |
| topic | Analysis of PDEs 35D30, 35J20, 34B15, 35J60, 35J66, 35B35, 46E35 |
| url | https://arxiv.org/abs/2605.13389 |