Generalized quasi-linear fractional Wentzell problems

2025/08/12 by Efren Mesino-Espinosa, Mesino-Espinosa, Efren, Alejandro Vélez-Santiago +1
Mathematics · #35B45 #35D30 #35J62 #35J92 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2508.08813

openalex publication_date 2025/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a bounded (ε,δ)-domain Ω⊆\mathbbR ^N (N≥2) whose boundary Γ:=∂Ω is a d-set for d∈(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,sN^p'(1-s)u+β|u|^q-2 u+Θ^η_qu = g\indent\indent\indent\textrmon Γ, where Θ^η_q stands as a nonlocal fractional-type q-operator on Γ (also refered as a Besov q-map), C'_p,sN^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 Ω. Furthermore, given two distinct weak solution related to this boundary value problem with different data values, we establish a priori L^∞-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.

Citations

• Diffusion over ramified domains: solvability and fine regularity
• The generalized anisotropic dynamical Wentzell heat equation with nonstandard growth conditions
• 3D Koch-type crystals
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Related

• The Dirichlet elliptic problem involving regional fractional Laplacian
• Existence of solutions for a class of fractional elliptic problems on exterior domains
• Existence of weak solutions for fractional p(x, .)-Laplacian Dirichlet problems with nonhomogeneous boundary conditions
• Regularity results for quasilinear elliptic problems driven by the fractional Φ-Laplacian operator
• Singular boundary behaviour and large solutions for fractional elliptic equations