The ∂ ‾ -Robin Laplacian
Abstract
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.
Fields of Study & Subjects
Spectral Theory in Mathematical Physics
Nonlinear Partial Differential Equations
Advanced Operator Algebra Research
Publication details
- Journal / source
- Journal of Mathematical Analysis and Applications
- Publisher
- Elsevier BV
- Volume
- 558
- Issue
- 1
- Pages
- 130411–130411
- Publication date
- 14 January 2026
- Language
- EN