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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.04328 |
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| _version_ | 1866910007033856000 |
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| author | Dweik, Samer Sabra, Ahmad |
| author_facet | Dweik, Samer Sabra, Ahmad |
| contents | This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term $f(u)$, where $f:\mathbb{R}^+ \mapsto \mathbb{R}^+$: $$\begin{cases} L[u]=f(u) &\qquad in \{u>0\}\\ u \geq 0 &\qquad in\, Ω\\ u=g &\qquad on\, \partial Ω\end{cases},$$ with $$L[u](x)=\sup_{y\in Ω,\,y\neq x}\dfrac{u(y)-u(x)}{|y-x|^α}+\inf_{y\in Ω,\,y\neq x} \dfrac{u(y)-u(x)}{|y-x|^α},\qquad 0<α<1.$$ Under the assumptions that $f$ is a continuous and monotone function and that the boundary datum $g$ is in $C^{0,β}(\partialΩ)$ for some $0<β<α$, we prove existence of a solution $u$ to this problem. Moreover, this solution $u$ is $β-$Hölderian on $\overlineΩ$. Our proof is based on an approximation of $f$ by an appropriate sequence of functions $f_\varepsilon$ where we prove using Perron's method the existence of solutions $u_\varepsilon$, for every $\varepsilon>0$. Then, we show some uniform Hölder estimates on $u_\varepsilon$ that guarantee that $u_\varepsilon \rightarrow u$ where this limit function $u$ turns out to be a solution to our obstacle problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04328 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractional Infinity Laplacian with Obstacle Dweik, Samer Sabra, Ahmad Analysis of PDEs 5D40, 35J60, 35J65 This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term $f(u)$, where $f:\mathbb{R}^+ \mapsto \mathbb{R}^+$: $$\begin{cases} L[u]=f(u) &\qquad in \{u>0\}\\ u \geq 0 &\qquad in\, Ω\\ u=g &\qquad on\, \partial Ω\end{cases},$$ with $$L[u](x)=\sup_{y\in Ω,\,y\neq x}\dfrac{u(y)-u(x)}{|y-x|^α}+\inf_{y\in Ω,\,y\neq x} \dfrac{u(y)-u(x)}{|y-x|^α},\qquad 0<α<1.$$ Under the assumptions that $f$ is a continuous and monotone function and that the boundary datum $g$ is in $C^{0,β}(\partialΩ)$ for some $0<β<α$, we prove existence of a solution $u$ to this problem. Moreover, this solution $u$ is $β-$Hölderian on $\overlineΩ$. Our proof is based on an approximation of $f$ by an appropriate sequence of functions $f_\varepsilon$ where we prove using Perron's method the existence of solutions $u_\varepsilon$, for every $\varepsilon>0$. Then, we show some uniform Hölder estimates on $u_\varepsilon$ that guarantee that $u_\varepsilon \rightarrow u$ where this limit function $u$ turns out to be a solution to our obstacle problem. |
| title | Fractional Infinity Laplacian with Obstacle |
| topic | Analysis of PDEs 5D40, 35J60, 35J65 |
| url | https://arxiv.org/abs/2507.04328 |