The \overline\partial-Robin Laplacian and its connection with graphene

This talk addresses the Robin-type problem $-\Delta u = f$ in $\Omega$, $2 \bar \nu \partial_{\bar z} u + au = 0$ on $\partial\Omega$, where $\Omega$ is a bounded C^2 domain in R^2, $f\in L^2(\Omega,\mathbb C)$ is prescribed, and a>0 is a parameter. The boundary condition involves an inherent complex structure, which prevents the use of typical results for real-valued problems (such as maximum principles, rearrangement techniques, or moving plane methods). As we shall see in the talk, this problem is connected with the behavior of electrons conducting electricity in graphene quantum dots. Exploring such a connection, we estimate the energies of these electrons.