Generalized quasi-linear fractional Wentzell problems

TL;DR

This paper studies a complex fractional elliptic boundary value problem involving nonlocal operators and generalized boundary conditions, proving existence, uniqueness, boundedness, and comparison principles for solutions.

Contribution

It introduces a novel fractional quasi-linear elliptic model with generalized Wentzell boundary conditions and establishes fundamental analytical properties for its solutions.

Findings

Existence and uniqueness of weak solutions.

Solutions are globally bounded in the domain.

A priori estimates and comparison principles are established.

Abstract

Given a bounded $(\epsilon,\delta)$-domain $\Omega\subseteq\mathbb{R\!}^N$ ($N\geq2$) whose boundary $\Gamma:=\partial\Omega$ 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 $(-\Delta)^s_{_{p,\Omega}}$ in $\Omega$, and generalized fractional Wentzell boundary conditions of type $$C'_{p,s}\mathcal{N}^{p'(1-s)}u+\beta|u|^{q-2} u+\Theta^{\eta}_qu\,=\,g\indent\indent\indent\textrm{on}\,\,\Gamma,$$ where $\Theta^{\eta}_q$ stands as a nonlocal fractional-type $q$-operator on $\Gamma$ (also refered as a Besov $q$-map), $C'_{p,s}\mathcal{N}^{p'(1-s)}$ denotes the fractional $p$-normal derivative operator in $\Gamma$, 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 $\overline{\Omega}$. 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.