Fractional Sobolev spaces for the singular-perturbed Laplace operator in the $$L^p$$ L p setting

Vladimir Georgiev () and Mario Rastrelli ()
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Vladimir Georgiev: University of Pisa
Mario Rastrelli: University of Pisa

Partial Differential Equations and Applications, 2025, vol. 6, issue 4, 1-40

Abstract: Abstract We study the perturbed Sobolev spaces $${H^{s,p}_\alpha (\mathbb {R}^d)}$$ H α s , p ( R d ) , associated with singular perturbation $$\Delta _\alpha $$ Δ α of Laplace operator in Euclidean space of dimensions 2 and 3. We extend the $$L^2$$ L 2 theory of perturbed Sobolev space to the $$L^p$$ L p case, finding an analogue description in terms of standard Sobolev spaces. This enables us to extend the Strichartz estimates to the energy space and to treat the local well-posedness of the Nonlinear Schrödinger equation associated with this singular perturbation, with the contraction method.

Keywords: Singular perturbation of Laplace operator; Sobolev spaces; Nonlinear Schrödinger equation; Fractional operators; 46E35; 47A60; 81Q15; 35Q41 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s42985-025-00336-z

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