| Abstract
| - We develop sharp upper bounds for energy levels of the magnetic Laplacian on starlike plane domains, under either Dirichlet or Neumann boundary conditions and assuming a constant magnetic field in the transverse direction. Our main result says that ∑ j=1 nΦ( λjA/ G) is maximal for a disk whenever Φ is concave increasing, n ≥ 1 , the domain has area A, and λj is the jth Dirichlet eigenvalue of the magnetic Laplacian ( i∇ + \hbox{$\frac{\beta}{2A}$}(− x2, x1)) 2. Here the flux β is constant, and the scale invariant factor G penalizes deviations from roundness, meaning G ≥ 1 for all domains and G = 1 for disks.
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