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.
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 $[-1/2,1/2]$.
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.
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 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…
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 introduce higher-order Poincar'e constants for compact weighted manifolds and estimate them from above in terms of subsets. These estimates imply upper bounds for eigenvalues of the weighted Laplacian and the first nontrivial eigenvalue of the $p$-Laplacian. In the case of the closed eigenvalue problem and the Neumann eigenvalue problem these are related with the estimates obtained by Chung-Grigor'yan-Yau and Gozlan-Herry. We also obtain similar upper bounds for Dirichlet eigenvalues and multi-way isoperimetric constants. As an application, for manifolds with boundary of non-negative dimens…
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…
We fnd the asymptotics of eigenvalues of polynomially compact zero order pseudodiferential operators, the motivating example being the Neumann- Poincare operator in linear elasticity.
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）
