Fractional Sobolev Spaces for the Singular-perturbed Laplace Operator in the $L^p$ setting
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917467137245184 |
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| author | Georgiev, Vladimir Rastrelli, Mario |
| author_facet | Georgiev, Vladimir Rastrelli, Mario |
| contents | We study the perturbed Sobolev spaces ${H^{s,p}_α(\mathbb{R}^d)}$, associated with singular perturbation $Δ_α$ of Laplace operator in Euclidean space of dimensions 2 and 3. We extend the $L^2$ theory of perturbed Sobolev space to the $L^p$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19732 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractional Sobolev Spaces for the Singular-perturbed Laplace Operator in the $L^p$ setting Georgiev, Vladimir Rastrelli, Mario Analysis of PDEs 46E35, 47A60, 81Q15, 35Q41 We study the perturbed Sobolev spaces ${H^{s,p}_α(\mathbb{R}^d)}$, associated with singular perturbation $Δ_α$ of Laplace operator in Euclidean space of dimensions 2 and 3. We extend the $L^2$ theory of perturbed Sobolev space to the $L^p$ 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. |
| title | Fractional Sobolev Spaces for the Singular-perturbed Laplace Operator in the $L^p$ setting |
| topic | Analysis of PDEs 46E35, 47A60, 81Q15, 35Q41 |
| url | https://arxiv.org/abs/2504.19732 |