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Register a new userIsoperimetric and Universal Inequalities for Eigenvalues
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Mark S. Ashbaugh
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This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with ``Dirichlet boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to present some new ones. Some of the names associated with these inequalities are Rayleigh, Faber-Krahn, Szego-Weinberger, Payne-Polya-Weinberger, Sperner, Hile-Protter, and H. C. Yang. Occasionally, we will also comment on extensions of some of our inequalities to bounded domains in other spaces, specifically, S^n or H^n.
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We consider the problem of geometric optimization of the lowest eigenvalue for the Laplacian on a compact, simply-connected two-dimensional manifold with boundary subject to an attractive Robin boundary condition. We prove that in the sub-class of manifolds with the Gauss curvature bounded from above by a constant $K_circ ge 0$ and under the constraint of fixed perimeter, the geodesic disk of constant curvature $K_circ$ maximizes the lowest Robin eigenvalue. In the same geometric setting, it is proved that the spectral isoperimetric inequality holds for the lowest eigenvalue of the Dirichlet-to-Neumann operator. Finally, we adapt our methods to Robin Laplacians acting on unbounded three-dimensional cones to show that, under a constraint of fixed perimeter of the cross-section, the lowest Robin eigenvalue is maximized by the circular cone.
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Let $H_{0, D}$ (resp., $H_{0,N}$) be the Schroedinger operator in constant magnetic field on the half-plane with Dirichlet (resp., Neumann) boundary conditions, and let $H_ell : = H_{0, ell} - V$, $ell =D,N$, where the scalar potential $V$ is non negative, bounded, does not vanish identically, and decays at infinity. We compare the distribution of the eigenvalues of $H_D$ and $H_N$ below the respective infima of the essential spectra. To this end, we construct effective Hamiltonians which govern the asymptotic behaviour of the discrete spectrum of $H_ell$ near $inf sigma_{ess}(H_ell) = inf sigma(H_{0,ell})$, $ell = D,N$. Applying these Hamiltonians, we show that $sigma_{disc}(H_D)$ is infinite even if $V$ has a compact support, while $sigma_{disc}(H_N)$ could be finite or infinite depending on the decay rate of $V$.
In this article, we consider the semiclassical Schrodinger operator $P = - h^{2} Delta + V$ in $mathbb{R}^{d}$ with confining non-negative potential $V$ which vanishes, and study its low-lying eigenvalues $lambda_{k} ( P )$ as $h to 0$. First, we give a necessary and sufficient criterion upon $V^{-1} ( 0 )$ for $lambda_{1} ( P ) h^{- 2}$ to be bounded. When $d = 1$ and $V^{-1} ( 0 ) = { 0 }$, we are able to control the eigenvalues $lambda_{k} ( P )$ for monotonous potentials by a quantity linked to an interval $I_{h}$, determined by an implicit relation involving $V$ and $h$. Next, we consider the case where $V$ has a flat minimum, in the sense that it vanishes to infinite order. We give the asymptotic of the eigenvalues: they behave as the eigenvalues of the Dirichlet Laplacian on $I_{h}$. Our analysis includes an asymptotic of the associated eigenvectors and extends in particular cases to higher dimensions.
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