This paper studies the optimization of the lowest eigenvalue of the magnetic Steklov problem on planar domains. In the bounded domain setting and for magnetic fields of moderate strengths, we prove that among all simply-connected smooth domains of given area, the disk maximises the lowest magnetic Steklov eigenvalue. For exterior domains, we establish a similar isoperimetric inequality for magnetic fields of moderate strength under fixed perimeter constraint and additional geometric and symmetry assumptions. The proofs rely on the method of torsion-type trial functions in the bounded domain case and on the method of trial functions dependent only on the distance to the boundary in the exterior domain case.
The Steklov eigenvalue problem, where the spectral parameter appears in the boundary condition, has been extensively studied in spectral geometry (see the review paper [16] and also [32,Chapter 7]). Recently, its magnetic counterpart where the Laplacian is replaced by the magnetic Laplacian has attracted considerable attention (see [9,13,18,19,20]).
For bounded domains, optimization of spectral quantities under geometric constraints is a classical topic. The celebrated Faber-Krahn inequality states that among domains of fixed volume, the first Dirichlet Laplacian eigenvalue is minimized by the ball. Analogous isoperimetric inequalities have been established for the first non-trivial Steklov eigenvalue without magnetic field [5,6,38]. In the magnetic setting, however, the interplay between geometry and the magnetic potential introduces new challenges. For the lowest eigenvalue of the magnetic Dirichlet Laplacian with homogeneous magnetic field on a planar domain an analogue of the Faber-Krahn inequality is proved in [14]. Recent works such as [10,11] and [23,24] have developed techniques for handling shape optimization in magnetic spectral problems with Neumann and also Robin boundary conditions, including the technique of torsion-type trial functions.
Spectral isoperimetric inequalities in exterior domains have been first studied in [28,29,30] and in [7] for the Robin Laplacian. More recent contributions address the magnetic Neumann eigenvalue problem [25] and the (non-magnetic) Steklov problem [8] on exterior domains. In the magnetic setting, the Steklov problem on exterior domains has been studied recently in [18,20], primarily in the context of spectral asymptotics with respect to the magnetic field strength.
Unlike in the non-magnetic case, the presence of the magnetic field makes the lowest Steklov eigenvalue to be non-zero and dependent on the shape of the domain. Optimization of the lowest magnetic Steklov eigenvalue in the case of the Aharonov-Bohm magnetic field was addressed in [10]. In this paper, we establish isoperimetric inequalities for the lowest magnetic Steklov eigenvalue with homogeneous magnetic field in two geometric settings, showing that under moderate magnetic fields, the disk emerges as the optimal shape for maximizing the lowest magnetic Steklov eigenvalue in both bounded and exterior settings.
Firstly, for planar bounded simply-connected smooth domains of given area, we prove that the disk maximizes the eigenvalue provided the magnetic field strength is below a certain threshold that guarantees radial ground states for the disk. This result is stated as Theorem 2.10. It takes the form of a quantitative isoperimetric inequality and, as a corollary, implies that among all domains of fixed perimeter, the disk also maximizes the eigenvalue. This result is proved using the technique of torsion-type trial functions, that is, trial functions constant on the level lines of the torsion function. We employ in the proof some of the ideas developed in [11] and also in [24] for the optimization of the lowest eigenvalue of the magnetic Neumann Laplacian.
Secondly, for exterior domains (the complement of a planar bounded simply connected smooth region), we establish that among domains with a given perimeter and satisfying certain symmetry and geometric conditions (such as central symmetry and a connectedness condition on outer parallel curves), the exterior of the disk maximizes the lowest magnetic Steklov eigenvalue. This holds when the magnetic field strength is below another critical constant guaranteeing a radial and real-valued ground state for the exterior of the disk. The assumption on the outer parallel curves is automatically satisfied in the exterior of a bounded convex domain. This result is stated as Theorem 3.6. In the proof of this result we use trial functions, which depend only on the distance to the boundary. This technique was pioneered in [35] and was employed later in many papers; see e.g. [1,15] for the Robin Laplacian with negative boundary parameter and [23] for the magnetic Robin Laplacian.
The paper is organized as follows. Section 2 treats bounded domains: after introducing the magnetic Steklov eigenvalue and analyzing the case of the disk, we develop the method of torsion-type trial functions and prove the isoperimetric inequality for fixed area. Section 3 is devoted to exterior domains, where we establish a corresponding inequality under fixed perimeter and symmetry assumptions. For completeness, we also include appendices on the existence of ground states and the variational characterization of the lowest magnetic Dirichletto-Neumann eigenvalue.
The aim of this section is to treat optimization of the lowest magnetic Steklov eigenvalue on a bounded planar domain. A central tool is the method of torsion-type trial functions, which reduces the problem to an auxiliary one-dimensional spectral problem. The behavior of this auxiliary problem under perturbations of a
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