Boundary regularity and a priori estimates for fractional equations on unbounded domains
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912805032034304 |
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| author | Guo, Yahong Li, Congming Ouyang, Yugao |
| author_facet | Guo, Yahong Li, Congming Ouyang, Yugao |
| contents | In this paper, we study the boundary Hölder regularity for solutions to the fractional Dirichlet problem in unbounded domains with boundary \begin{equation*}
\begin{cases}
(-Δ)^s u(x) = g(x),&\text{in } Ω,
u(x)=0, &\text{in } Ω^c.
\end{cases}
\end{equation*} Existing results rely on the global $L^{\infty}$ norm of solutions to control their boundary $C^s$ norm, which is insufficient for blow-up and rescaling analysis to obtain a priori estimates in unbounded domains. To overcome this limitation, we first derive a local version of boundary Hölder regularity for nonnegative solutions in which we replace the global $L^{\infty}$ norm by only a local $L^{\infty}$ norm. Then as an important application, we establish a priori estimates for nonnegative solutions to a family of nonlinear equations on unbounded domains with boundaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17325 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundary regularity and a priori estimates for fractional equations on unbounded domains Guo, Yahong Li, Congming Ouyang, Yugao Analysis of PDEs In this paper, we study the boundary Hölder regularity for solutions to the fractional Dirichlet problem in unbounded domains with boundary \begin{equation*} \begin{cases} (-Δ)^s u(x) = g(x),&\text{in } Ω, u(x)=0, &\text{in } Ω^c. \end{cases} \end{equation*} Existing results rely on the global $L^{\infty}$ norm of solutions to control their boundary $C^s$ norm, which is insufficient for blow-up and rescaling analysis to obtain a priori estimates in unbounded domains. To overcome this limitation, we first derive a local version of boundary Hölder regularity for nonnegative solutions in which we replace the global $L^{\infty}$ norm by only a local $L^{\infty}$ norm. Then as an important application, we establish a priori estimates for nonnegative solutions to a family of nonlinear equations on unbounded domains with boundaries. |
| title | Boundary regularity and a priori estimates for fractional equations on unbounded domains |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.17325 |