The $\overline\partial$-Robin Laplacian
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912910082572288 |
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| author | Duran, Joaquim |
| author_facet | Duran, Joaquim |
| contents | We study the family of operators $\{\mathcal{R}_a\}_{a\in [0,+\infty)}$ associated to the Robin-type problems in a bounded domain $Ω\subset\mathbb{R}^2$ $$
\begin{cases}
-Δu = f & \text{in } Ω, \\
2 \bar ν\partial_{\bar z} u + au = 0 & \text{on } \partialΩ,
\end{cases} $$ and their dependency on the boundary parameter $a$ as it moves along $[0,+\infty)$. In this regard, we study the convergence of such operators in a resolvent sense. We also describe the eigenvalues of such operators and show some of their properties, both for all fixed $a$ and as functions of the parameter $a$. As shall be seen in more detail in arXiv:2507.18698, the eigenvalues of these operators characterize the positive eigenvalues of quantum dot Dirac operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16895 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $\overline\partial$-Robin Laplacian Duran, Joaquim Analysis of PDEs 35P05, 47A10 We study the family of operators $\{\mathcal{R}_a\}_{a\in [0,+\infty)}$ associated to the Robin-type problems in a bounded domain $Ω\subset\mathbb{R}^2$ $$ \begin{cases} -Δu = f & \text{in } Ω, \\ 2 \bar ν\partial_{\bar z} u + au = 0 & \text{on } \partialΩ, \end{cases} $$ and their dependency on the boundary parameter $a$ as it moves along $[0,+\infty)$. In this regard, we study the convergence of such operators in a resolvent sense. We also describe the eigenvalues of such operators and show some of their properties, both for all fixed $a$ and as functions of the parameter $a$. As shall be seen in more detail in arXiv:2507.18698, the eigenvalues of these operators characterize the positive eigenvalues of quantum dot Dirac operators. |
| title | The $\overline\partial$-Robin Laplacian |
| topic | Analysis of PDEs 35P05, 47A10 |
| url | https://arxiv.org/abs/2507.16895 |