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5. Dirac operators

    1. Faber-Krahn for Dirac operator with infinite mass boundary condition


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      Problem 5.1.

      [T. Ourmières-Bonafos] Prove or disprove that among all planar domains of a given area, the disk is the unique minimizer for the smallest positive eigenvalue $\nu_1(\Omega)$ of the Dirac problem with infinite mass boundary conditions $$ \left\{ \begin{aligned} -\mathsf{i} \big(\partial_1 v - \mathsf{i}\partial_2 v\big) &= \nu u &&\mbox{in} \quad \Omega \,, \\ -\mathsf{i}\big(\partial_1 u + \mathsf{i}\partial_2 u\big) &= \nu v &&\mbox{in} \quad \Omega \,, \\ \mathsf{i}\big(n_1 + \mathsf{i} n_2\big) u &= v &&\mbox{on} \quad \partial\Omega \,, \end{aligned} \right. $$ where $n = (n_1,n_2)^\top$ is the outer unit normal to $\Omega$.
          A geometric lower bound for the first non-negative eigenvalue has been obtained by R. Benguria, S. Fournais, E. Stockmeyer, and H. Van Den Bosch [MR3625007]. Moreover, a numerical evidence supporting that the disk is in fact the minimizer is provided in [arxiv:2003.04061].

          Cite this as: AimPL: Shape optimization with surface interactions, available at http://aimpl.org/shapesurface.