Generalized quasi-linear fractional Wentzell problems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909734002491392 |
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| author | Mesino-Espinosa, Efren Vélez-Santiago, Alejandro |
| author_facet | Mesino-Espinosa, Efren Vélez-Santiago, Alejandro |
| contents | Given a bounded $(ε,δ)$-domain $Ω\subseteq\mathbb{R\!}^N$ ($N\geq2$) whose boundary $Γ:=\partialΩ$ is a $d$-set for $d\in(N-p,N)$, we investigate a generalized quasi-linear elliptic boundary value problem governed by the regional fractional $p$-Laplacian $(-Δ)^s_{_{p,Ω}}$ in $Ω$, and generalized fractional Wentzell boundary conditions of type $$C'_{p,s}\mathcal{N}^{p'(1-s)}u+β|u|^{q-2} u+Θ^η_qu\,=\,g\indent\indent\indent\textrm{on}\,\,Γ,$$ where $Θ^η_q$ stands as a nonlocal fractional-type $q$-operator on $Γ$ (also refered as a Besov $q$-map), $C'_{p,s}\mathcal{N}^{p'(1-s)}$ denotes the fractional $p$-normal derivative operator in $Γ$, and $p,\,q$ are two growth exponents acting on the interior and boundary, respectively (which are in general unrelated between each other). We first show that this model equation admits a unique weak solution, which is globally bounded in $\overlineΩ$. Furthermore, given two distinct weak solution related to this boundary value problem with different data values, we establish a priori $L^{\infty}$-estimates for the difference of weak solutions with upper bound depending in the differences of the respective interior and boundary data functions. Additionally, a Weak Comparison Principle is derived, and we conclude by establishing a sort of nonlinear Fredholm Alternative related to this generalized elliptic fractional model equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_08813 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized quasi-linear fractional Wentzell problems Mesino-Espinosa, Efren Vélez-Santiago, Alejandro Analysis of PDEs 35J92, 35J62, 35D30, 35B45 Given a bounded $(ε,δ)$-domain $Ω\subseteq\mathbb{R\!}^N$ ($N\geq2$) whose boundary $Γ:=\partialΩ$ is a $d$-set for $d\in(N-p,N)$, we investigate a generalized quasi-linear elliptic boundary value problem governed by the regional fractional $p$-Laplacian $(-Δ)^s_{_{p,Ω}}$ in $Ω$, and generalized fractional Wentzell boundary conditions of type $$C'_{p,s}\mathcal{N}^{p'(1-s)}u+β|u|^{q-2} u+Θ^η_qu\,=\,g\indent\indent\indent\textrm{on}\,\,Γ,$$ where $Θ^η_q$ stands as a nonlocal fractional-type $q$-operator on $Γ$ (also refered as a Besov $q$-map), $C'_{p,s}\mathcal{N}^{p'(1-s)}$ denotes the fractional $p$-normal derivative operator in $Γ$, and $p,\,q$ are two growth exponents acting on the interior and boundary, respectively (which are in general unrelated between each other). We first show that this model equation admits a unique weak solution, which is globally bounded in $\overlineΩ$. Furthermore, given two distinct weak solution related to this boundary value problem with different data values, we establish a priori $L^{\infty}$-estimates for the difference of weak solutions with upper bound depending in the differences of the respective interior and boundary data functions. Additionally, a Weak Comparison Principle is derived, and we conclude by establishing a sort of nonlinear Fredholm Alternative related to this generalized elliptic fractional model equation. |
| title | Generalized quasi-linear fractional Wentzell problems |
| topic | Analysis of PDEs 35J92, 35J62, 35D30, 35B45 |
| url | https://arxiv.org/abs/2508.08813 |