Semiclassical states for the curl-curl problem
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929688304156672 |
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| author | Bieganowski, Bartosz Konysz, Adam Mederski, Jarosław |
| author_facet | Bieganowski, Bartosz Konysz, Adam Mederski, Jarosław |
| contents | We show the existence of the so-called semiclassical states $\mathbf{U}:\mathbb{R}^3\to\mathbb{R}^3$ to the following curl-curl problem $$ \varepsilon^2\; \nabla \times (\nabla \times \mathbf{U}) + V(x) \mathbf{U} = g(\mathbf{U}), $$ for sufficiently small $\varepsilon > 0$. We study the asymptotic behaviour of solutions as $\varepsilon\to 0^+$ and we investigate also a related nonlinear Schrödinger equation involving a singular potential. The problem models large permeability nonlinear materials satisfying the system of Maxwell equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_03658 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Semiclassical states for the curl-curl problem Bieganowski, Bartosz Konysz, Adam Mederski, Jarosław Analysis of PDEs We show the existence of the so-called semiclassical states $\mathbf{U}:\mathbb{R}^3\to\mathbb{R}^3$ to the following curl-curl problem $$ \varepsilon^2\; \nabla \times (\nabla \times \mathbf{U}) + V(x) \mathbf{U} = g(\mathbf{U}), $$ for sufficiently small $\varepsilon > 0$. We study the asymptotic behaviour of solutions as $\varepsilon\to 0^+$ and we investigate also a related nonlinear Schrödinger equation involving a singular potential. The problem models large permeability nonlinear materials satisfying the system of Maxwell equations. |
| title | Semiclassical states for the curl-curl problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2312.03658 |