Spectral optimization for Robin Laplacian on domains admitting parallel coordinates
Pavel Exner, Vladimir Lotoreichik
备注：
14 pages, no figures, the title and the assumptions of the main theorem slightly modified in comparison with the first version. To appear in Mathematische Nachrichten
In this paper we deal with spectral optimization for the Robin Laplacian on a family of planar domains admitting parallel coordinates, namely a fixed-width strip built over a smooth closed curve and the exterior of a convex set with a smooth boundary. We show that if the curve length is kept fixed, the first eigenvalue referring to the fixed-width strip is for any value of the Robin parameter maximized by a circular annulus. Furthermore, we prove that the second eigenvalue in the exterior of a convex domain $Ω$ corresponding to a negative Robin parameter does not exceed the analogous quantity for a disk whose boundary has a curvature larger than or equal to the maximum of that for $\partialΩ$.
This paper is devoted to establish semiclassical Weyl formulae for the Robin Laplacian on smooth domains in any dimension. Theirs proofs are reminiscent of the Born-Oppenheimer method.
We determine a counterexample to strong diamagnetism for the Laplace operator in the unit disc with a uniform magnetic field and Robin boundary condition. The example follows from the accurate asymptotics of the lowest eigenvalue when the Robin parameter tends to $-\infty$.
We discuss several geometric conditions guaranteeing the finiteness or the infiniteness of the discrete spectrum for Robin Laplacians on conical domains.
We determine accurate asymptotics of the lowest eigenvalue for the Laplace operator with a smooth magnetic field and Robin boundary conditions in a smooth 3D domain, when the Robin parameter tends to $+\infty$. Our results identify a critical regime where the contribution of the magnetic field and the Robin condition are of the same order. In this critical regime, we derive an effective operator defined on the boundary of the domain.
We determine accurate asymptotics for the low-lying eigenvalues of the Robin Laplacian when the Robin parameter goes to $-\infty$. The two first terms in the expansion have been obtained by K. Pankrashkin in the $2D$-case and by K. Pankrashkin and N. Popoff in higher dimensions. The asymptotics display the influence of the scalar curvature and the splitting between every two consecutive eigenvalues. The analysis is based on the approach developed by Fournais-Helffer for the semi-classical magnetic Laplacian. We also propose a WKB construction as a candidate for the ground state energy.
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 Dirichle…
This paper is devoted to the asymptotic analysis of the eigenvalues of the Laplace operator with a strong magnetic field and Robin boundary condition on a smooth planar domain and with a negative boundary parameter. We study the singular limit when the Robin parameter tends to infinity which is equivalent to a semi-classical limit involving a small positive semi-classical parameter. The main result is a comparison between the spectrum of the Robin Laplacian with an effective operator defined on the boundary of the domain via the Born-Oppenheimer approximation. More precisely, the low-lying eig…
By developing the method of multipliers, we establish sufficient conditions which guarantee the total absence of eigenvalues of the Laplacian in the half-space, subject to variable complex Robin boundary conditions. As a further application of this technique, uniform resolvent estimates are derived under the same assumptions on the potential. Some of the results are new even in the self-adjoint setting, where we obtain quantum-mechanically natural conditions.
We consider the Laplacian in a curved two-dimensional strip of constant width squeezed between two curves, subject to Dirichlet boundary conditions on one of the curves and variable Robin boundary conditions on the other. We prove that, for certain types of Robin boundary conditions, the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Laplacian in a Dirichlet-Robin annulus determined by the geometry of the strip. Moreover, we show that an appropriate combination of the geometric setting and boundary conditions leads to a Hardy-type inequality in infi…
Motivated by the study of high energy Steklov eigenfunctions, we examine the semi-classical Robin Laplacian. In the two dimensional situation, we determine an effective operator describing the asymptotic distribution of the negative eigenvalues, and we prove that the corresponding eigenfunctions decay away from the boundary, for all dimensions.
The Laplacian in an unbounded tubular neighbourhood of a hyperplane with non-Hermitian complex-symmetric Robin-type boundary conditions is investigated in the limit when the width of the neighbourhood diminishes. We show that the Laplacian converges in a norm resolvent sense to a self-adjoint Schroedinger operator in the hyperplane whose potential is expressed solely in terms of the boundary coupling function. As a consequence, we are able to explain some peculiar spectral properties of the non-Hermitian Laplacian by known results for Schroedinger operators.
Previous works provided several counterexamples to monotonicity of the lowest eigenvalue for the magnetic Laplacian in the two-dimensional case. However, the three-dimensional case is less studied. We use the results obtained by Helffer, Kachmar and Raymond to provide one of the first counterexamples in 3D. Considering the Robin magnetic Laplacian on the unit ball with a constant magnetic field, we show the non-monotonicity of the lowest eigenvalue asymptotics when the Robin parameter tends to $+\infty$.
We prove that the optimal cluster problem for the sum of the first Robin eigenvalue of the Laplacian, in the limit of a large number of convex cells, is asymptotically solved by (the Cheeger sets of) the honeycomb of regular hexagons. The same result is established for the Robin torsional rigidity.
We study the Robin Laplacian in a domain with two corners of the same opening, and we calculate the asymptotics of the two lowest eigenvalues as the distance between the corners increases to infinity.
In this paper we establish a Hardy inequality for Laplace operators with Robin boundary conditions. For convex domains, in particular, we show explicitly how the corresponding Hardy weight depends on the coefficient of the Robin boundary conditions. We also study several extensions to non-convex and unbounded domains.
A generalization of Robin boundary conditions leading to self-adjoint operators is developed for the second derivative operator on metric graphs with compact completion and totally disconnected boundary. Harmonic functions and their properties play an essential role.
In the present paper we introduce the perturbed two-dimensional Robin bi-Laplacian in the exterior of a bounded simply-connected $C^2$-smooth open set. The considered perturbation is of lower order and corresponds to tension. We prove that the essential spectrum of this operator coincides with the positive semi-axis and that the negative discrete spectrum is non-empty if, and only if, the boundary parameter is negative. As the main result, we obtain an isoperimetric inequality for the lowest eigenvalue of such a perturbed Robin bi-Laplacian with a negative boundary parameter in the exterior of…
We consider the Laplace operator in a planar waveguide, i.e., an infinite two-dimensional straight strip of constant width, with particular types of Robin boundary conditions. We study the essential spectrum of the corresponding Laplacian when the boundary coupling function has a limit at infinity. Furthermore, we derive sufficient conditions for the existence of discrete spectrum.
We study the Robin eigenvalue problem for the Laplace-Beltrami operator on Riemannian manifolds. Our first result is a comparison theorem for the second Robin eigenvalue on geodesic balls in manifolds whose sectional curvatures are bounded from above. Our second result asserts that geodesic balls in nonpositively curved space forms maximize the second Robin eigenvalue among bounded domains of the same volume.
The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $Θ$ which give rise to self-adjoint Laplacians $-Δ_{Θ, Ω}$ in $L^2(Ω; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $Ω\subset\bbR^n$, $n\in\bbN$, $n\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $Ω$, following an approach introduced by Filonov for this type of problems.
In this note self-adjoint extensions of symmetric operators are investigated by using the abstract technique of quasi boundary triples and their Weyl functions. The main result is an extension of Theorem 2.6 in [5] which provides sufficient conditions on the parameter in the boundary space to induce self-adjoint realizations. As an example self-adjoint Robin Laplacians on the half-space with boundary conditions involving an unbounded coefficient are considered.
In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals $((p-1)/p)^p$ whenever Dirichlet boundary conditions are imposed on a subset of the boundary of non-zero measure. We also discuss some generalizations to non-convex domains.
The difference of the resolvents of two Laplacians on a half-space subject to Robin boundary conditions is studied. In general this difference is not compact, but it will be shown that it is compact and even belongs to some Schatten-von-Neumann class, if the coefficients in the boundary condition are sufficiently close to each other in a proper sense. In certain cases the resolvent difference is shown to belong even to the same Schatten-von Neumann class as it is known for the resolvent difference of two Robin Laplacians on a domain with a compact boundary.
We show that eigenvalues of the Robin Laplacian with a positive boundary parameter $α$ on rectangles and unions of rectangtes satisfy Pólya-type inequalities, albeit with an exponent smaller than that of the corresponding Weyl asympotics for a fixed domain. We determine the optimal exponents in either case, showing that they are different in the two situations. Our approach to proving these results includes a characterisation of the corresponding extremal domains for the $k$th eigenvalue in regions of the $(k,α)$-plane.
In this note we investigate the asymptotic behaviour of the $s$-numbers of the resolvent difference of two generalized self-adjoint, maximal dissipative or maximal accumulative Robin Laplacians on a bounded domain $Ω$ with smooth boundary $\partialΩ$. For this we apply the recently introduced abstract notion of quasi boundary triples and Weyl functions from extension theory of symmetric operators together with Krein type resolvent formulae and well-known eigenvalue asymptotics of the Laplace-Beltrami operator on $\partialΩ$. It will be shown that the resolvent difference of two generalized Rob…
We consider the first Robin eigenvalue $ł_p(M,\a)$ for the $p$-Laplacian on a compact Riemannian manifold $M$ with nonempty smooth boundary, with $\a \in \R$ being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of Cheng type for $ł_p(M,\a)$. Secondly, when $\a>0$ we establish sharp lower bound of $ł_p(M,\a)$ in terms of dimension, inradius, Ricci curvature lower bound and boundary mean curvature lower bound, via comparison with an associated one-dimensional eigenvalue problem. The lower bound becomes an upper bound when $\a<0$. Our results cover corresponding comparison …
The first two eigenvalues of the Robin Laplacian are investigated along with their gap and ratio. Conjectures by various authors for arbitrary domains are supported here by new results for rectangular boxes. Results for rectangular domains include that: the square minimizes the first eigenvalue among rectangles under area normalization, when the Robin parameter $α\in \mathbb{R}$ is scaled by perimeter; that the square maximizes the second eigenvalue for a sharp range of $α$-values; that the line segment minimizes the Robin spectral gap under diameter normalization for each $α\in \mathbb{R}$; a…
For a bounded corner domain $Ω$, we consider the Robin Laplacian in $Ω$ with large Robin parameter. Exploiting multiscale analysis and a recursive procedure, we have a precise description of the mechanism giving the ground state of the spectrum. It allows also the study of the bottom of the essential spectrum on the associated tangent structures given by cones. Then we obtain the asymptotic behavior of the principal eigenvalue for this singular limit in any dimension, with remainder estimates. The same method works for the Schrödinger operator in $\mathbb{R}^n$ with a strong attractive delta-i…
We consider the magnetic Robin Laplacian with a negative boundary parameter. Among a certain class of domains, we prove that the disk maximizes the ground state energy under the fixed perimeter constraint provided that the magnetic field is of moderate strength. This class of domains includes, in particular, all domains that are contained upon translations in the disk of the same perimeter and all convex centrally symmetric domains.
Let $Ω$ be a curvilinear polygon and $Q^γ_Ω$ be the Laplacian in $L^2(Ω)$, $Q^γ_Ωψ=-Δψ$, with the Robin boundary condition $\partial_νψ=γψ$, where $\partial_ν$ is the outer normal derivative and $γ>0$. We are interested in the behavior of the eigenvalues of $Q^γ_Ω$ as $γ$ becomes large. We prove that the asymptotics of the first eigenvalues of $Q^γ_Ω$ is determined at the leading order by those of model operators associated with the vertices: the Robin Laplacians acting on the tangent sectors associated with $\partial Ω$. In the particular case of a polygon with straight edges the first eigenp…
We study the Laplacian in a smooth bounded domain, with a varying Robin boundary condition singular at one point. The associated quadratic form is not semi-bounded from below, and the corresponding Laplacian is not self-adjoint, it has the residual spectrum covering the whole complex plane. We describe its self-adjoint extensions and exhibit a physically relevant skew-symmetric one. We approximate the boundary condition, giving rise to a family of self-adjoint operators, and we describe their eigenvalues by the method of matched asymptotic expansions. These eigenvalues acquire a strange behavi…
We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain in euclidean space. Functions in the domain of the operator are subject to a boundary condition of the third type (a magnetic Robin condition). In addition to the Landau levels, we obtain that the spectrum of this operator consists of clusters of eigenvalues around the Landau levels and that they do accumulate to the Landau levels from below. We give a precise asymptotic formula for the rate of accumulation of eigenvalues in these clusters, which is independent of the boundary condition.
Let $Ω\subset \mathbb{R}^2$ be the exterior of a convex polygon whose side lengths are $\ell_1,...,\ell_M$. For $α>0$, let $H^Ω_α$ denote the Laplacian in $Ω$, $u\mapsto -Δu$, with the Robin boundary conditions $\partial u/\partialν=αu$, where $ν$ is the exterior unit normal at the boundary of $Ω$. We show that, for any fixed $m\in\mathbb{N}$, the $m$th eigenvalue $E^Ω_m(α)$ of $H^Ω_α$ behaves as \[ E^Ω_m(α)=-α^2+μ^D_m +\mathcal{O}\Big(\dfrac{1}{\sqrtα}\Big) \quad {as $α$ tends to $+\infty$}, \] where $μ^D_m$ stands for the $m$th eigenvalue of the operator $D_1\oplus...\oplus D_M$ and $D_n$ de…
We consider the problem of minimising the $n^{th}-$eigenvalue of the Robin Laplacian in $\mathbb{R}^{N}$. Although for $n=1,2$ and a positive boundary parameter $α$ it is known that the minimisers do not depend on $α$, we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on $α$. We derive a Wolf-Keller type result for this problem and show that optimal eigenvalues grow at most with $n^{1/N}$, which is in sharp contrast with the Weyl asymptotics for a fixed domain. We further show that the gap between consecutive eigenvalues does go to ze…
We consider the Laplacian in a tubular neighbourhood of a hyperplane subjected to non-self-adjoint $\mathcal{PT}$-symmetric Robin boundary conditions. Its spectrum is found to be purely essential and real for constant boundary conditions. The influence of the perturbation in the boundary conditions on the threshold of the essential spectrum is studied using the Birman-Schwinger principle. Our aim is to derive a sufficient condition for existence, uniqueness and reality of discrete eigenvalues. We show that discrete spectrum exists when the perturbation acts in the mean against the unperturbed …
We consider the Robin Laplacian in the domains $Ω$ and $Ω^\varepsilon$, $\varepsilon >0$, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient $a$ is large enough, the spectrum of the problem in $Ω$ is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain $Ω^\varepsilon$ is discrete. However, our results reveal the strange behavior of the discrete spectrum as the blunting parameter $\varepsilon$ tends to 0: we construct asymptotic forms of the eigenvalues and detect families of "hardly movable" and "plumm…
Let $Ω\subset \mathbb{R}^n$ be a bounded $C^1$ domain and $p>1$. For $α>0$, define the quantity \[ Λ(α)=\inf_{u\in W^{1,p}(Ω),\, u\not\equiv 0} \Big(\int_Ω|\nabla u|^p\,\mathrm{d}x - α\int_{\partialΩ} |u|^p \,\mathrm{d} s\Big)\Big/ \int_Ω|u|^p\,\mathrm{d} x \] with $\mathrm{d} s$ being the hypersurface measure, which is the lowest eigenvalue of the $p$-laplacian in $Ω$ with a non-linear $α$-dependent Robin boundary condition. We show the asymptotics $Λ(α) =(1-p)α^{p/(p-1)}+o(α^{p/(p-1)})$ as $α$ tends to $+\infty$. The result was only known for the linear case $p=2$ or under stronger smoothnes…
Let $\varepsilon>0$ be a small parameter. We consider the domain $Ω_\varepsilon:=Ω\setminus D_\varepsilon$, where $Ω$ is an open domain in $\mathbb{R}^n$, and $D_\varepsilon$ is a family of small balls of the radius $d_\varepsilon=o(\varepsilon)$ distributed periodically with period $\varepsilon$. Let $Δ_\varepsilon$ be the Laplace operator in $Ω_\varepsilon$ subject to the Robin condition ${\partial u\over \partial n}+γ_\varepsilon u = 0$ with $γ_\varepsilon\ge 0$ on the boundary of the holes and the Dirichlet condition on the exterior boundary. Kaizu (1985, 1989) and Brillard (1988) have sho…
We study the spectrum of the Robin Laplacian with a complex Robin parameter $α$ on a bounded Lipschitz domain $Ω$. We start by establishing a number of properties of the corresponding operator, such as generation properties, local analytic dependence of the eigenvalues and eigenspaces on $α\in \mathbb C$, and basis properties of the eigenfunctions. Our focus, however, is on bounds and asymptotics for the eigenvalues as functions of $α$: we start by providing estimates on the numerical range of the associated operator, which lead to new eigenvalue bounds even in the case $α\in \mathbb R$. For t…
We study the discrete spectrum of the Robin Laplacian $Q^Ω_α$ in $L^2(Ω)$, \[ u\mapsto -Δu, \quad \dfrac{\partial u}{\partial n}=αu \text{ on }\partialΩ, \] where $Ω\subset \mathbb{R}^{3}$ is a conical domain with a regular cross-section $Θ\subset \mathbb{S}^2$, $n$ is the outer unit normal, and $α>0$ is a fixed constant. It is known from previous papers that the bottom of the essential spectrum of $Q^Ω_α$ is $-α^2$ and that the finiteness of the discrete spectrum depends on the geometry of the cross-section. We show that the accumulation of the discrete spectrum of $Q^Ω_α$ is determined by th…
Robin problem for the Laplacian in a bounded planar domain with a smooth boundary and a large parameter in the boundary condition is considered. We prove a two-sided three-term asymptotic estimate for the negative eigenvalues. Furthermore, improving the upper bound we get a two term asymptotics in terms of the coupling constant and the maximum of the boundary curvature.
The second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $α/L(Ω)$, and $α$ lies between $-2π$ and $2π$. Corollaries include Szegő's sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock's inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter. The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization o…
For $α\in(0,π)$, let $U_α$ denote the infinite planar sector of opening $2α$, \[ U_α=\big\{ (x_1,x_2)\in\mathbb R^2: \big|\arg(x_1+ix_2) \big|<α\big\}, \] and $T^γ_α$ be the Laplacian in $L^2(U_α)$, $T^γ_αu= -Δu$, with the Robin boundary condition $\partial_νu=γu$, where $\partial_ν$ stands for the outer normal derivative and $γ>0$. The essential spectrum of $T^γ_α$ does not depend on the angle $α$ and equals $[-γ^2,+\infty)$, and the discrete spectrum is non-empty iff $α<\fracπ2$. In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a contino…
