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arXiv · 2310.00767

Sobolev spaces for singular perturbation of Laplace operato

Vladimir Georgiev·Mario Rastrelli
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Abstract

We study the perturbed Sobolev space $H^{1,r}_\alpha$, $r \in (1,\infty),$ associated with singular perturbation $\Delta_\alpha$ of Laplace operator in Euclidean space of dimension $2.$ The main results give the possibility to extend the $L^2$ theory of perturbed Sobolev space to the $L^r$ case. When $r \in (2,\infty)$ we have appropriate representation of the functions in $H^{1,r}_\alpha$ in regular and singular part. An application to local well - posedness of the NLS associated with this singular perturbation in the mass critical and mass supercritical cases is established too.

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Vladimir Georgiev, Mario Rastrelli. 2023-10-01. Sobolev spaces for singular perturbation of Laplace operato. https://arxiv.org/abs/2310.00767

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