The $\overline\partial$-Robin Laplacian

TL;DR

This paper investigates the properties and boundary parameter dependence of a family of Robin-type Laplacian operators in a bounded domain, focusing on their eigenvalues, convergence, and relation to quantum dot Dirac operators.

Contribution

It provides a detailed analysis of the convergence, eigenvalues, and properties of the $ar{ ext{d}}$-Robin Laplacian operators as the boundary parameter varies.

Findings

Operators converge in the resolvent sense as boundary parameter varies.

Eigenvalues are characterized and their properties described.

Connection established between eigenvalues and quantum dot Dirac operators.

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.