$L^p$ continuity of eigenprojections for 2-d Dirichlet Laplacians under perturbations of the domain
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913249640841216 |
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| author | Babb, Ryan L. Acosta Robinson, James C. |
| author_facet | Babb, Ryan L. Acosta Robinson, James C. |
| contents | We generalise results by Lamberti and Lanza de Cristoforis (2005) concerning the continuity of projections onto eigenspaces of self-adjoint differential operators with compact inverses as the (spatial) domain of the functions is perturbed in $\mathbb{R}^2$. Our main case of interest is the Dirichlet Laplacian. We extend these results from bounds from $H_0^1$ to $H_0^1$ to bounds from $L^p$ to $L^p$, under the assumption that $(-Δ^{-1}-z)^{-1}$ is $L^p$ bounded when $z$ lies outside of the spectrum of $-Δ^{-1}$. We show that this assumption is met if the initial domain is a square or a rectangle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10066 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^p$ continuity of eigenprojections for 2-d Dirichlet Laplacians under perturbations of the domain Babb, Ryan L. Acosta Robinson, James C. Analysis of PDEs 47A70 (Primary), 35P10, 42B08 (Secondary) We generalise results by Lamberti and Lanza de Cristoforis (2005) concerning the continuity of projections onto eigenspaces of self-adjoint differential operators with compact inverses as the (spatial) domain of the functions is perturbed in $\mathbb{R}^2$. Our main case of interest is the Dirichlet Laplacian. We extend these results from bounds from $H_0^1$ to $H_0^1$ to bounds from $L^p$ to $L^p$, under the assumption that $(-Δ^{-1}-z)^{-1}$ is $L^p$ bounded when $z$ lies outside of the spectrum of $-Δ^{-1}$. We show that this assumption is met if the initial domain is a square or a rectangle. |
| title | $L^p$ continuity of eigenprojections for 2-d Dirichlet Laplacians under perturbations of the domain |
| topic | Analysis of PDEs 47A70 (Primary), 35P10, 42B08 (Secondary) |
| url | https://arxiv.org/abs/2401.10066 |