Year: 2026
Type: preprint
Abstract: We develop a Fredholm alternative for a fractional elliptic operator~$\mathcal{L}$ of mixed order built on the notion of fractional gradient. This operator constitutes the nonlocal extension of the classical second order elliptic operators with measurable coefficients treated by Neil Trudinger in~\cite{trudinger}. We build~$\mathcal{L}$ by weighing the order~$s$ of the fractional gradient over a measure (which can be either continuous, or discrete, or of mixed type). The coefficients of~$\mathcal{L}$ may also depend on~$s$, giving this operator a possibly non-homogeneous structure with variable exponent. These coefficients can also be either unbounded, or discontinuous, or both. A suitable functional analytic framework is introduced and investigated and our main results strongly rely on some custom analysis of appropriate functional spaces.
Source: arXiv (Cornell University)
Authors: De Pas, Francesco, Dipierro, Serena, Valdinoci, Enrico
Institution: The University of Western Australia
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7111907384
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