Magnetic spectral bounds on starlike plane domains

Laugesen, R.S.; Siudeja, B.A.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 3, pp. 670-689.

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 ${\Sigma }_{j=1}^n\Phi ({\lambda }_{j}A/G)$ is maximal for a disk whenever $\Phi$ is concave increasing, $n\ge 1$, the domain has area $A$, and $\lambda {}_j$ is the $j$th Dirichlet eigenvalue of the magnetic Laplacian ${(i\nabla +\frac{\beta }{2A}(-x_2,x_1))}^2$. Here the flux $\beta$ is constant, and the scale invariant factor $G$ penalizes deviations from roundness, meaning $G\ge 1$ for all domains and $G=1$ for disks.

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