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arXiv · 2507.16895

The $\overline\partial$-Robin Laplacian

Joaquim Duran
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Abstract

We study the family of operators $\{\mathcal{R}_a\}_{a\in [0,+\infty)}$ associated to the Robin-type problems in a bounded domain $\Omega\subset\mathbb{R}^2$ $$ \begin{cases} -\Delta u = f & \text{in } \Omega, \\ 2 \bar \nu \partial_{\bar z} u + au = 0 & \text{on } \partial\Omega, \end{cases} $$ 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.

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Joaquim Duran. 2025-07-22. The $\overline\partial$-Robin Laplacian. https://doi.org/10.1016/j.jmaa.2026.130411

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