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Virginie Bonnaillie-Noël

Ground state energy of the magnetic Laplacian on corner domains

Abstract: 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 talk, 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 spectral properties of these model problems, namely semicontinuity and existence of bounded generalized eigenfunctions. We prove estimates for the remainders of our asymptotic formula. Lower bounds are obtained with the help of an IMS type 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.
This is a joint work with M. Dauge and N. Popoff.