Inequalities between Dirichlet and Neumann eigenvalues of the magnetic Laplacian

We consider the magnetic Laplacian with the homogeneous magnetic field in two and three dimensions. We prove that the $(k+1)$-th magnetic Neumann eigenvalue of a bounded convex planar domain is not larger than its $k$-th magnetic Dirichlet eigenvalue. In three dimensions, we restrict our attention to convex domains, which are invariant under rotation by an angle of $\pi$ around an axis parallel to the magnetic field. For such domains, we prove that the $(k+2)$-th magnetic Neumann eigenvalue is not larger than the $k$-th magnetic Dirichlet eigenvalue provided that this Dirichlet eigenvalue is simple. The proofs rely on a modification of the strategy due to Levine and Weinberger.