Spectral analysis on the elastic Neumann–Poincaré operator ∗Daisuke Kawagoe†
AMS subject classifications. 35J47 (primary), 35P05 (secondary)
Key words. Neumann–Poincaré operator, Lamé system, polynomial compactness, essential spectrum, corner
singularity

1Introduction

The elastic Neumann-Poincaré (eNP) operator is a boundary integral operator that appears naturally when we solve classical boundary value problems for the Lamé system using layer potentials.
Recently, there is rapidly growing interest in the spectral properties of the eNP operator in relation
to cloaking by anomalous localized resonance (CALR). Anomalous localized resonance occurs at
the accumulation point of eigenvalues, which motivates us to investigate the spectral structure of
the eNP operator.
The Lamé system, a system of equations of linear elasticity, is described by
Lλ,µ u := µ∆u + (λ + µ)∇∇ · u = f,where u = (u1 , . . . , ud ) (d = 2, 3) is the displacement, (λ, µ) are the Lamé constants, and f is
the force term. In what follows, we assume that the pair of constants (λ, µ) satisfies the strong
convexity condition:µ > 0, dλ + 2µ > 0.
Let C = (Cijkl )di,j,k,l=1 be the isotropic elasticity tensor corresponding to (λ, µ), namely,Cijkl := λδij δkl + µ(δik δjl + δil δjk ).
b be the symmetric gradient of a vector-valued function u, namely,
Also, let ∇ub := 1 ∇u + (∇u)T ,∇u2where (∇u)T is the transpose of the matrix ∇u. Then, the Lamé system is also described asb = f.
Lλ,µ u = ∇(C∇u)
We see what CALR is. Let Ω be a bounded domain in Rd (d = 2, 3) with the Lipschitz
boundary. Let a(x) be a function in Rd \∂Ω such that(k, x ∈ Ω,a(x) =1, x ∈ Rd \Ω,
where k is a complex number. We consider the following transmission problem:(b = f in Rd ,∇(a(x)C∇u)|u(x)| = O(|x|1−d ) as |x| → ∞,∗ This work was supported by NRF grants No. 2016R1A2B4011304 and 2017R1A4A1014735.
† Graduate School of Informatics, Kyoto University (d.kawagoe@acs.i.kyoto-u.ac.jp)

1(1.1)

where f is a function compactly supported in Rd \Ω and satisfies
Zf dx = 0.
RdLetZZb 2 dx.
λ|∇ · u|2 + (d − 1)µ|∇u|b : C∇ub dx =∇uE(u) :=ΩΩPHere, A : B = ij aij bij for two matrices A = (aij ) and B = (bij ). We assume that k = k0 + iδin (1.1) for δ > 0, whereλ + 3µ.
k0 := −λ+µLet uδ be the solution to the problem (1.1). Then, CALR is characterized by two conditions:lim sup δE(uδ ) = ∞,δ↓0

|u (x)| < C,
δ|x| > Rfor some positive pconstants C and R independent of δ.
Let vδ := uδ / δE(uδ ). Then, we have δE(vδ ) = 1 and |vδ (x)| → 0 as δ ↓ 0 for |x| > R. In
other words, we cannot observe the displacement outside a ball.
It is worth mentioning that, in the case that Ω is a ball or an ellipse in R2 , CALR occurs when
the support of f locates in a suitable area, and that we can replace k0 by 1/k0 to see CALR [1].
Let Γ(x) = (Γ(x))di,j=1 be the fundamental solution to the Lamé system associated with theLamé constants (λ, µ), namely,αα2 xi xj
1, d = 2,
 δij log |x| −2π2π |x|2Γij (x) :=|x| =6 0,α δα xx
− 1 ij − 2 i 3j , d = 3,4π |x| 4π |x|
where
α1 :=

12



11+µ λ + 2µ


,

α2 :=

12



11−µ λ + 2µ.

Also, for a vector-valued function u, the conormal derivative ∂ν u corresponding to the Lamé system
is defined bybb∂ν u := (C∇u)n= λ(∇ · u)n + 2µ(∇u)n,
where n is the outward unit normal to ∂Ω. Then, the eNP operator K∗ is defined byZK∗ [f ](x) := p.v.
∂νx Γ(x − y)f (y) dσy , a.e. x ∈ ∂Ω.
∂ΩHere, we consider the conormal derivative ∂νx Γ(x − y) of the matrix columnwise and p.v. stands
for the Cauchy principal value.
Let κ0 := (k0 + 1)/2(k0 − 1). Then, κ0 and −κ0 are accumulation points of eigenvalues of K∗
when ∂Ω is smooth.

2Main Results

So far, we obtain two results on the spectral structure of the eNP operator K∗ on the energy space
H −1/2 (∂Ω)d .
The first one is polynomial compactness of the eNP operator. This result was obtained by ajoint work with Hyeonbae Kang (Inha University).

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Theorem 2.1. Let Ω be a bounded domain in R3 with the C 1,α -smooth boundary for some α > 0.
Let K∗ be the eNP operator on ∂Ω corresponding to the pair of Lamé constants (λ, µ). Let p3 (t) :=t(t + κ0 )(t − κ0 ), where κ0 is given byκ0 :=k0 + 1µ=.
2(k0 − 1)2(λ + 2µ)Then, p3 (K∗ ) is compact on H −1/2 (∂Ω)3 . Moreover, K∗ (K∗ +κ0 I), K∗ (K∗ −κ0 I) and (K∗ )2 −κ20 I
are not compact on H −1/2 (∂Ω)3 .
From Theorem 2.1 and the spectral mapping theorem, we obtain the following result on theasymptotic behavior of eigenvalues.
Corollary 2.1. The spectrum of K∗ on H −1/2 (∂Ω)3 consists of three non-empty sequences ofeigenvalues which converge to 0, κ0 and −κ0 , respectively.
The counterpart of Theorem 2.1 is proved in [1] replacing the polynomial p3 by p2 (t) := (t +κ0 )(t−κ0 ). Also, Theorem 2.1 was once proved in [2] by assuming C ∞ -smoothness on the boundary∂Ω.
The second one is a characterization of the essential spectrum when Ω is a planar domain with acorner. This result was obtained by a joint work with Eric Bonnetier (Université Grenoble–Alpes),Charles Dapogny (Université Grenoble–Alpes) and Hyeonbae Kang.
Let α be the opening angle of the corner, and assume that 0 < α < 2π. We define a set Σ(k0 , α)by˜ ξ) = 0 for some ξ > 0},
Σ(k0 , α) := {p ∈ (0, 1 − k0 )|d(p,where˜ ξ) =16d+ (p, ξ)d− (p, ξ),d(p,d± (p, ξ) =f1,± (p, ξ)f2,± (p, ξ) + g(p, ξ),f1,± (p, ξ) = sinh(αξ)(p − 1) ± ξ sin α,f2,± (p, ξ) = sinh((2π − α)ξ)(p + k0 ) ± ξ sin α,g(p, ξ) =p(p − 1 + k0 ) sinh2 ((π − α)ξ).
Then, we have the following relation.
Theorem 2.2. Let 0 < α < 2π. Then, we haveσess (K∗ ) =11−Σ(k0 , α).
2 1 − k0

A proof of Theorem 2.2 is based on [3], where we discuss the essential spectrum of the NP
operator for the Laplace equation.

References[1] K. Ando, Y.-G. Ji, H. Kang, K. Kim and S. Yu, Spectral properties of the Neumann-Poincaré
operator and cloaking by anomalous localized resonance for the elasto-static system, Eur. J.
App. Math., 29 (2018), pp. 189-225.
[2] K. Ando, H. Kang and Y. Miyanishi, Elastic Neumann-Poincaré Operators on three dimensional smooth domains: polynomial compactness and spectral structure, Int. Math. Res. Not.,
2019 (2017), no. 12, pp. 3883-3900.
[3] E. Bonnetier and H. Zhang, Characterization of the essential spectrum of the NeumannPoincaré operator in 2D domains with corner via Weyl sequences, Rev. Mat. Iberoam., 35(2019), no. 3, pp. 925-948.

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