A sharp upper bound on the spectral gap for graphene quantum dots (Q2418706)

For planar domains \(\Omega\subset\mathbb{R}^{2}\) containing the origin, with \(C^{3}\) boundary, which are either convex or nearly circular, the spectrum of the Dirac operator with 'infinite mass boundary conditions' is considered; see Definition 4. The main result is a Faber-Krahn type inequality \(\mathcal{F}\left(\Omega\right)\mu_{1}\left(\Omega\right)\leq\mathcal{F}\left(\mathbb{D}_{r}\right)\mu_{1}\left(\mathbb{D}_{r}\right)\) for \(\mu_{1}\) the first positive eigenvalue; here \(\mathbb{D}_{r}\) is the disk of radius of \(r\), and \(\mathcal{F}\) is an explicit functional given in terms of the total area as well as the minimum/maximum radii and curvature of points on the boundary.