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Magnetic spectral bounds on starlike plane domains. (English) Zbl 1319.35130

Let \(\Omega\subset {\mathbb R}^2\) be starlike domain with area \(A\) defined as \[ \Omega = \{ re^{i\theta} : 0 \leq r < R(\theta) \}, \] where the radius function \(R(\cdot)\) is positive, \(2\pi\)-periodic, and Lipschitz continuous. The authors consider the magnetic Laplacian \[ H=\big( i\nabla+ \frac{\beta}{2A}(-x_2,x_1) \big)^2 \] on \(\Omega\) under either Dirichlet or Neumann boundary conditions. The constant \(\beta\) represents the magnetic flux through the domain. Let \(\lambda_j\) denote the \(j\)th eigenvalue of the operator \(H\). The main result of the paper under review says that \[ \sum\limits_{j=1}^n \Phi(\lambda_j A/G) \] is maximal when \(\Omega\) is a disk centered at the origin, where \(\Phi:{\mathbb R}_+\to {\mathbb R}\) is concave and increasing; \(G\) is the following scale-invariant geometric factor measuring the deviation of the domain from roundness: \[ G=\max\left\{1+\frac{1}{2\pi} \int\limits_0^{2\pi} (\log R)'(\theta)^2 d\theta; \frac{2\pi}{A^2}\int\limits_\Omega |x|^2 dx \right\} \] with \(G \geq 1\) for all domains and \(G=1\) for disks.
A similar result is proved for the magnetic Pauli operator with Dirichlet boundary conditions.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J20 Variational methods for second-order elliptic equations

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References:

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