Two-step minimization approach to an $L^\infty$-constrained variational problem with a generalized potential
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910944267862016 |
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| author | Apriliani, Vina Kimura, Masato Ohtsuka, Hiroshi |
| author_facet | Apriliani, Vina Kimura, Masato Ohtsuka, Hiroshi |
| contents | We study a variational problem on $H^1({\mathbb R})$ under an $L^\infty$-constraint related to Sobolev-type inequalities for a class of generalized potentials, including $L^p$-potentials, non-positive potentials, and signed Radon measures. We establish various essential tools for this variational problem, including the decomposition principle, the comparison principle, and the perturbation theorem, which are the basis of the two-step minimization method. As for their applications, we present precise results for minimizers of minimization problems, such as the study of potentials of Dirac's delta measure type and the analysis of trapped modes in potential wells. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00730 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two-step minimization approach to an $L^\infty$-constrained variational problem with a generalized potential Apriliani, Vina Kimura, Masato Ohtsuka, Hiroshi Analysis of PDEs 35J20 (Primary), 46E35 (Secondary) We study a variational problem on $H^1({\mathbb R})$ under an $L^\infty$-constraint related to Sobolev-type inequalities for a class of generalized potentials, including $L^p$-potentials, non-positive potentials, and signed Radon measures. We establish various essential tools for this variational problem, including the decomposition principle, the comparison principle, and the perturbation theorem, which are the basis of the two-step minimization method. As for their applications, we present precise results for minimizers of minimization problems, such as the study of potentials of Dirac's delta measure type and the analysis of trapped modes in potential wells. |
| title | Two-step minimization approach to an $L^\infty$-constrained variational problem with a generalized potential |
| topic | Analysis of PDEs 35J20 (Primary), 46E35 (Secondary) |
| url | https://arxiv.org/abs/2405.00730 |