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Interpolation Inequalities in Fuzzy Fractional Sobolev-Slobodeckij Spaces |
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PP: 469-483 |
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doi:10.18576/amis/200213
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Author(s) |
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Asokan Vasudevan,
Sulieman Ibrahim Shelash Mohammad,
Yogeesh Nijalingappa,
Hanan Jadallah,
Raja Natarajan,
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Abstract |
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| Fractional Sobolev-Slobodeckij spaces provide a powerful framework for capturing nonlocal smoothness, while fuzzy set theory offers a systematic way to model uncertainty; however, a unified theory combining both has been lacking. In this work, we define the fuzzy fractional Sobolev space We (s, p) (Ω ) by equipping level-sets of fuzzy-valued functions with the classical Gagliardo seminorm and extend the real-interpolation K - and J-methods to this fuzzy setting. We prove a sharp interpolation inequality
∥u∥ (s ,p ) (s ,p ) ≤C ∥u∥1−θ ∥u∥θ
(We 0 0 ,We 1 1 ) F We(s0,p0) We(s1,p1)
θ,q
and derive corollaries including continuous and compact embeddings, a fuzzy fractional Poincare ́-Wirtinger inequality, and weighted - space extensions. Detailed examples on Ω = [0,1] illustrate how fuzziness amplifies fractional norms, and applications to fuzzy fractional Poisson equations establish well-posedness, uniqueness, and regularity via level-set Lax-Milgram and interpolation estimates. Our framework recovers classical results when uncertainty vanishes and lays the groundwork for future developments such as nonlinear Gagliardo-Nirenberg analogues, manifold generalizations, and stochastic - fuzzy differential equations.
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