  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, semi-continuity, 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.
