This paper concerns the spectral properties of the Neumann-Poincaré operator on $m$-fold rotationally symmetric planar domains. An $m$-fold rotationally symmetric simply connected domain $D$ is realized as the $m$th-root transform of a certain domain, say $Ω$. We prove that the domain of definition of the Neumann-Poincaré operator on $D$ is decomposed into invariant subspaces and the spectrum on one of them is the exact copy of the spectrum on $Ω$. It implies in particular that the spectrum on the transformed domain $D$ contains the spectrum on the original domain $Ω$ counting multiplicities.
We consider the spectral structure of the Neumann--Poincaré operators defined on the boundaries of thin domains of rectangle shape in two dimensions. We prove that as the aspect ratio of the domains tends to $\infty$, or equivalently, as the domains get thinner, the spectra of the Neumann--Poincaré operators are densely distributed in the interval $[-1/2,1/2]$.
The Neumann-Poincaré operator on the sphere has $\frac{1}{2(2k+1)}$, $k=0,1,2,\ldots$, as its eigenvalues and the corresponding multiplicity is $2k+1$. We consider the bifurcation of eigenvalues under deformation of domains, and show that Frechét derivative of the sum of the bifurcations is zero. We then discuss the connection of this result with some conjectures regarding the Neumann-Poincaré operator.
It is proved that if a bounded domain in three dimensions satisfies a certain concavity condition, then the Neumann-Poincaré operator on the boundary of the domain or its inversion in a sphere has at least one negative eigenvalue. The concavity condition is quite simple, and is satisfied if there is a point on the boundary at which the Gaussian curvature is negative.
We investigate the decay property of the eigenvalues of the Neumann-Poincaré operator in two dimensions. As is well-known, this operator admits only a sequence of eigenvalues that accumulates to zero as its spectrum for a bounded domain having $C^{1,α}$ boundary with $α\in (0,1)$. In this paper, we show that the eigenvalue $λ_k$'s of the Neumann-Poincaré operator ordered by size satisfy that $|λ_k| = O(k^{-p-α+1/2})$ for an arbitrary simply connected domain having $C^{1+p,α}$ boundary with $p\geq 0,~ α\in(0,1)$ and $p+α>\frac{1}{2}$.
We investigate the spectral structure of the Neumann-Poincaré operator on thin ellipsoids. Two types of thin ellipsoids are considered: long prolate ellipsoids and flat oblate ellipsoids. We show that the totality of eigenvalues of the Neumann-Poincaré operators on a sequence of the prolate spheroids is densely distributed in the interval [0,1/2] as their eccentricities tend to 1, namely, as they become longer. We then prove that eigenvalues of the Neumann-Poincaré operators on the oblate ellipsoids are densely distributed in the interval [-1/2, 1/2] as the ellipsoids become flatter. In partic…
We deduce eigenvalue asymptotics of the Neumann--Poincaré operators in three dimensions. The region $Ω$ is $C^{2, α}$ ($α>0$) bounded in ${\mathbf R}^3$ and the Neumann--Poincaré operator ${\mathcal K}_{\partialΩ} : L^2(\partial Ω) \rightarrow L^2(\partial Ω) $ is defined by $$ {\mathcal K}_{\partialΩ}[ψ]({\bf x}) := \frac{1}{4π} \int_{\partial Ω} \frac{\langle {\bf y}-{\bf x}, {\bf n}({\bf y}) \rangle}{|{\bf x}-{\bf y}|^3} ψ({\bf y})\; dS_{\bf y} $$ where $dS_{\bf y}$ is the surface element and ${\bf n}({\bf y})$ is the outer normal vector on $\partial Ω$. Then the ordering eigenvalues of the…
Sums of Laplace eigenvalues - rotationally symmetric maximizers in the plane
拉普拉斯算子的第一个$n \geq 1$特征值的总和是最大的三角形之间的等边三角形，最大的平行四边形之间的正方形，最大的椭圆之间的磁盘，提供的比例$\text{(area)}^3/\text{(moment of inertia)}$的域是固定的。这一结果对狄利克雷和诺依曼本征值都成立，对于罗宾边界条件和无穷远处增长的势的薛定谔本征值也得到了类似的结论。该方法的一个关键要素是单位根的紧框架属性。对于一般的凸平面区域，圆盘被构造成最大化Neumann特征值之和。
This is a survey of accumulated spectral analysis observations spanning more than a century, referring to the double layer potential integral equation, also known as Neumann-Poincaré operator. The very notion of spectral analysis has evolved along this path. Indeed, the quest for solving this specific singular integral equation, originally aimed at elucidating classical potential theory problems, has inspired and shaped the development of theoretical spectral analysis of linear transforms in XX-th century. We briefly touch some marking discoveries into the subject, with ample bibliographical r…
The elastic Neumann--Poincaré operator is a boundary integral operator associated with the Lamé system of linear elasticity. It is known that if the boundary of a planar domain is smooth enough, it has eigenvalues converging to two different points determined by Lamé parameters. We show that eigenvalues converge at a polynomial rate on smooth boundaries and the convergence rate is determined by smoothness of the boundary. We also show that they converge at an exponential rate if the boundary of the domain is real analytic.
The Neumann-Poincaré operator is a boundary-integral operator associated with harmonic layer potentials. This article proves the existence of eigenvalues within the essential spectrum for the Neumann-Poincaré operator for certain Lipschitz curves in the plane with reflectional symmetry, when considered in the functional space in which it is self-adjoint. The proof combines the compactness of the Neumann-Poincaré operator for curves of class $C^{2,α}$ with the essential spectrum generated by a corner. Eigenvalues corresponding to even (odd) eigenfunctions are proved to lie within the essential …
We consider the double layer potential (Neumann-Poincaré) operator appearing in 3-dimensional elasticity. We show that the recent result about the polynomial compactness of this operator for the case of a homogeneous media follows without additional calculations from previous considerations by Agranovich et.al., based upon pseudodifferential operators. Further on, we define the NP operator for the case of a nonhomogeneous isotropic media and show that its properties depend crucially on the character of non-homogeneity. If the Lamé parameters are constant along the boundary, the NP operator is …
This article constructs a surface whose Neumann-Poincaré (NP) integral operator has infinitely many eigenvalues embedded in its essential spectrum. The surface is a sphere perturbed by smoothly attaching a conical singularity, which imparts essential spectrum. Rotational symmetry allows a decomposition of the operator into Fourier components. Eigenvalues of infinitely many Fourier components are constructed so that they lie within the essential spectrum of other Fourier components and thus within the essential spectrum of the full NP operator. The proof requires the perturbation to be sufficie…
We consider the Neumann-Poincar'e (double layer potential) operator in 3D elasicity on a smooth closed surface. Its essential spectrum consists of 3 points. We find the asymptotics of sequences of eigenvalues converging to these three pounts. They are expressed via the eigenvalue distribution for a matrix eigenvalue problem depending on the points of the cotangent bundle of the surface. For the two-sided asymptotics for the distribution of the the union of sequences of eigenvalues converging to the point of the essential spectrum from both sides, an expression is found via the Euler characteri…
We consider the asymptotic properties of the eigenvalues of the Neumann-Poincare (NP) operator in three dimensions. The region $Ω\subset \mathbb{R}^3$ is bounded by a compact surface $Γ=\partial Ω$, with certain smoothness conditions imposed. The NP operator is defined by $ \mathcal{K}_Γ[ψ](\mathbf{x}):=\frac{1}{4π}\int_Γ\frac{\langle \mathbf{y}-\mathbf{x},\mathbf{n}(\mathbf{y})\rangle}{|\mathbf{x}-\mathbf{y}|^3}ψ(\mathbf{y})dS_{\mathbf{y}},$ where $dS_{\mathbf{y}}$ is the surface element and $\mathbf{n}(\mathbf{y})$ is the outer unit normal vector on $Γ$. The first-named author established ea…
We discuss isoperimetric inequalities for the magnetic Laplacian on bounded domains of $\mathbb R^2$ endowed with an Aharonov-Bohm potential. When the flux of the potential around the pole is not an integer, the lowest eigenvalue for the Neumann and the Steklov problems is positive. We establish isoperimetric inequalitites for the lowest eigenvalue in the spirit of the classical inequalities of Szegö-Weinberger, Brock and Weinstock, the model domain being a disk with the pole at its center. We consider more generally domains in the plane endowed with a rotationally invariant metric, which incl…
It is known that the Neumann--Poincaré operator for the Lamé system of linear elasticity is polynomially compact and, as a consequence, that its spectrum consists of three non-empty sequences of eigenvalues accumulating to certain numbers determined by Lamé parameters, if the boundary of the domain where the operator is defined is $C^\infty$-smooth. We extend this result to less smooth boundaries, namely, $C^{1, α}$-smooth boundaries for some $α> 0$. The results are obtained by proving certain identities for surface Riesz transforms, which are singular integral operators of nonconvolution type…
Let $γ$ be a smooth, non-closed, simple curve whose image is symmetric with respect to the $y$-axis, and let $D$ be a planar domain consisting of the points on one side of $γ$, within a suitable distance $δ$ of $γ$. Denote by $μ_1^{odd}(D)$ the smallest nontrivial Neumann eigenvalue having a corresponding eigenfunction that is odd with respect to the $y$-axis. If $γ$ satisfies some simple geometric conditions, then $μ_1^{odd}(D)$ can be sharply estimated from below in terms of the length of $γ$, its curvature, and $δ$. Moreover, we give explicit conditions on $δ$ that ensure $μ_1^{odd}(D)=μ_1(…
We consider the plasmonic eigenvalue problem for a general 2D domain with a curvilinear corner, studying the spectral theory of the Neumann--Poincaré operator of the boundary. A limiting absorption principle is proved, valid when the spectral parameter approaches the essential spectrum. Putting the principle into use, it is proved that the corner produces absolutely continuous spectrum of multiplicity 1. The embedded eigenvalues are discrete. In particular, there is no singular continuous spectrum.
We characterize the essential spectrum of the plasmonic problem for polyhedra in $\mathbb{R}^3$. The description is particularly simple for convex polyhedra and permittivities $ε< - 1$. The plasmonic problem is interpreted as a spectral problem through a boundary integral operator, the direct value of the double layer potential, also known as the Neumann--Poincaré operator. We therefore study the spectral structure of the the double layer potential for polyhedral cones and polyhedra.
Sharp $L^2$-stability and gradient stability hierarchies of the $L^2$-Poincaré inequalities on Euclidean balls and Gaussian Poincaré inequality
School of Science and the Environment, Grenfell Campus, Memorial University of Newfoundland（纽芬兰纪念大学格伦菲尔夫学院科学与环境学院）；Department of Mathematics, University of Connecticut（康涅狄格大学数学系）
