Ground state energy of the magnetic Laplacian on corner domains

Virginie Bonnaillie-Noël, Monique Dauge, Nicolas Popoff.

The asymptotic behavior of the first eigenvalues of magnetic Laplacian operators with large magnetic fields and Neumann realization in smooth three-dimensional domains is characterized by model problems inside the domain or on its boundary. In two-dimensional polygonal domains, a new set of model problems on sectors has to be taken into account. In this paper, we consider the class of general corner domains. In dimension 3, they include as particular cases polyhedra and axisymmetric cones. We attach model problems not only to each point of the closure of the domain, but also to a hierarchy of tangent substructures'' associated with singular chains. We investigate properties of these model problems, namely continuity, semicontinuity, existence of generalized eigenfunctions satisfying exponential decay. We prove estimates for the remainders of our asymptotic formula. Lower bounds are obtained with the help of an IMS partition based on adequate two-scale coverings of the corner domain, whereas upper bounds are established by a novel construction of quasimodes, qualified as sitting or sliding according to spectral properties of local model problems. A part of our analysis extends to any dimension.

Our main result can be summarized as $$- c\, B^{3/4} \le \lambda(B\,\mathbf{B}_1,\Omega) - B\,\mathcal{E}(\mathbf{B}_1,\Omega) \le c\, B^{2/3}, \quad\mbox{ as } B\to+\infty, \quad \mbox{when \Omega is polyhedral}$$ and $$- c\, B^{9/10} \le \lambda(B\,\mathbf{B}_1,\Omega) - B\,\mathcal{E}(\mathbf{B}_1,\Omega) \le c\, B^{7/8}, \quad\mbox{ as } B\to+\infty, \quad \mbox{when \Omega has conical points}$$ in its large magnetic field'' version (here $\mathbf{B}_1$ is a reference magnetic field).

10 décembre 2015

To appear in Mémoires de la SMF (2016).