A Explicit kernel formulas

# The spectra of harmonic layer potential operators on domains with rotationally symmetric conical points\tnotereffund

## Abstract

We study the adjoint of the double layer potential associated with the Laplacian (the adjoint of the Neumann–Poincaré operator), as a map on the boundary surface of a domain in with conical points. The spectrum of this operator directly reflects the well-posedness of related transmission problems across . In particular, if the domain is understood as an inclusion with complex permittivity , embedded in a background medium with unit permittivity, then the polarizability tensor of the domain is well-defined when belongs to the resolvent set in energy norm. We study surfaces that have a finite number of conical points featuring rotational symmetry. On the energy space, we show that the essential spectrum consists of an interval. On , i.e. for square-integrable boundary data, we show that the essential spectrum consists of a countable union of curves, outside of which the Fredholm index can be computed as a winding number with respect to the essential spectrum. We provide explicit formulas, depending on the opening angles of the conical points. We reinforce our study with very precise numerical experiments, computing the energy space spectrum and the spectral measures of the polarizability tensor in two different examples. Our results indicate that the densities of the spectral measures may approach zero extremely rapidly in the continuous part of the energy space spectrum.

Résumé

Nous étudions l’adjoint du potentiel de double couche associé à l’opérateur de Laplace (l’adjoint de l’opérateur de Neumann-Poincaré) défini sur la frontière d’un domaine de contenant des points coniques. Le spectre de cet opérateur est intimement lié à la résolution de problèmes de transmission à travers . En particulier, dans le contexte de la propagation des ondes électromagnétiques, si le domaine délimité par représente une inclusion contenant un matériau de permittivité complexe , immergé dans un milieu infini de permittivité égale à 1, on peut définir le tenseur de polarisabilité dès que le rapport appartient à l’ensemble résolvent de l’opérateur au sens de la norme d’énergie. Nous étudions des surfaces qui possèdent un nombre fini de points coniques à symétrie de rotation. Lorsque l’opérateur est défini sur l’espace d’énergie, nous montrons que son spectre essentiel est un intervalle. Lorsqu’il est défini dans l’espace , i.e. pour des fonctions de carré intégrable sur , nous montrons que son spectre est constitué d’une union de courbes, en dehors desquelles on peut calculer l’indice de Fredholm de l’opérateur, comme l’indice par rapport à ces courbes. Nous donnons des formules explicites, en fonction de l’angle d’ouverture des points coniques. Nous complétons notre étude par des expériences numériques très précises, où, pour deux exemples, nous calculons le spectre de l’opérateur au sens de l’espace d’énergie et les mesures spectrales du tenseur de polarisabilité. Nos résultats suggèrent que les densités des mesures spectrales approchent zéro extrêmement rapidement dans la partie continue du spectre au sens de l’espace d’énergie.

###### keywords:
layer potential, Neumann–Poincaré operator, spectrum, polarizability
###### Msc:
[2010] 31B10, 45B05, 45E05
1\tnotetext

[fund]This work was supported by the Swedish Research Council under contract 621-2014-5159.

\cortext

[cor1]Corresponding author

## 1 Introduction

Let be a connected Lipschitz surface, enclosing a bounded open domain and with surface measure . We are interested in the spectrum of the layer potential operator

 KΓf(r)=12π∫ΓKΓ(r,r′)f(r′)dσ(r′),r∈Γ, (1)

based on the normal derivative of the Newtonian kernel

 KΓ(r,r′)=⟨r−r′,νr⟩|r′−r|3, (2)

where denotes the outward unit normal of . may also be considered for planar Lipschitz curves , in which case the kernel is given by .

Knowledge about the spectrum of leads to existence and uniqueness results for boundary value problems involving the Laplacian on the interior and exterior domains and of . For example, layer potential techniques may be used to solve the classical Dirichlet and Neumann problems for by understanding the Fredholm theory of and , respectively (51).

When is non-smooth, for example if has corners in 2D, or edges or conical points in 3D, the spectrum of is highly dependent on the space . For example, suppose that is a curvilinear polygon in the plane. is always invertible (51) when is Lipschitz, but in the polygonal case there always exist , depending on the opening angles of the corners of , such that is not Fredholm (34); (46). The underlying explanation for this is that when is an infinite wedge, the model domain to analyze corners; then, by homogeneity of its kernel, may be realized as a block matrix of Mellin convolution operators. These convolution kernels depend on , accounting for the dependence on of the spectrum (18). In 3D, similar results were shown in (19) in the idealized cases of being an infinite straight cone or an infinite three-dimensional wedge. We refer also to (40) for an extensive account of the -theory in 2D, although with results only stated for .

In this paper we will, for surfaces , consider the action of on two different spaces: and the energy space . The energy space consists of the distributions on whose single layer potentials have finite energy in . It is identifiable with the Sobolev space of index on the boundary. The energy space stands out as the most natural space on which to consider for many reasons, one of them being that is self-adjoint and therefore, in contrast with the -theory, has a real spectrum.

Our interest in the entire spectrum of arises from the transmission problem

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩∫R3|∇U|2dV<∞,ΔU(r)=0,r∈R3∖Γ,TrintU(r)=TrextU(r),r∈Γ,∂extνU(r)=ϵ∂intνU(r)−g(r),r∈Γ. (3)

Here , and and denote the boundary trace and normal derivative of interior approach, and the corresponding operators of exterior approach. If , it turns out that there exists a solution of (3) satisfying if and only if there is such that

 (KΓ−z)f=g1−ϵ,z=−1+ϵ1−ϵ.

In the special case that for a vector , then solving the transmission problem is involved in computing the polarizability tensor (10); (20); (28); (47) of . In this setting, the domain is an inclusion with complex permittivity in an infinite space of permittivity 1. The polarizability tensor is associated with a set of spectral measures that arise from the spectral measure of , see Section 2.2. Atoms in these spectral measures correspond to values of for which surface plasmon resonances can be excited (2); (3). However, not every eigenvalue of necessarily produces a singularity in the polarizability tensor; we call such eigenvalues dark plasmons. In Section 7.3 we will observe an abundance of dark spectra for the type of surface that we will consider. More precisely, the described relationship between the transmission problem (3) and plasmonic resonances holds in the quasi-static approximation of the Maxwell equations. In the setting of smooth surfaces , detailed analysis of plasmonic resonances using the full Maxwell equations and justification of the quasi-static approximation can be found in (1); (4). Note that the spectrum of is pure point when is smooth.

For a plane polygon and , , the spectrum of (a more general version of) the transmission problem (3) was studied in (14). In (31), the spectral resolution of was determined in a model case where is constructed from two intersecting circles (equivalent to the infinite wedge). For a general curvilinear polygon in 2D, the essential spectrum of was determined in (44).

###### Theorem 1.1 ((44)).

Suppose that is a curvilinear polygon with corners of angles . Then the spectrum of consists of an interval and a sequence of eigenvalues with no limit point outside the interval,

 σ(KΓ,E)={x∈R:|x|≤max1≤j≤N∣∣∣1−αjπ∣∣∣}∪{λk}.

See also (24) for a numerical experiments in agreement with this theorem. In the special case that coincides with two line segments in a neighborhood of each corner, a different approach to Theorem 1.1, yielding more information, very recently appeared in (6). In three dimensions, only a few results concerning the entire spectrum seem to be available. As mentioned, the -theory for infinite straight cones and wedges was considered in (19). Also for the infinite straight cone, the generalized eigensolutions to the transmission problem (3) were explicitly computed in (32); (41); (48) – these will be important in our determination of the spectrum of . For the more general type of infinite cone , where is a smooth curve on the sphere, the invertibility of on certain weighted Sobolev spaces has via Mellin convolutions been reduced to the invertibility of a parametric system of operators on (9); (45).

In the current contribution, we will characterize the spectrum of and in the case that is a rotationally symmetric surface with a conical point, see Figure 1. Our main results straightforwardly generalize to surfaces with a finite number of conical points, each of which is locally rotationally symmetric around some axis. However, since the level of complexity is already quite high, we will never do so explicitly.

We now state our main theorems, beginning with our result on the -spectrum. In the statement, denotes an associated Legendre function of the first kind (see the Appendix), and denotes its derivative in .

###### Theorem 5.28.

Let be a closed surface of revolution with a conical point of opening angle , obtained by revolving a -curve . For , denote by the closed curve

 Πn=⎧⎨⎩Pniξ(cosα)˙Pniξ(−cosα)−Pniξ(−cosα)˙Pniξ(cosα)Pniξ(−cosα)˙Pniξ(cosα)+Pniξ(cosα)˙Pniξ(−cosα):−∞≤ξ≤∞⎫⎬⎭,

with orientation given by the -variable. Then the operator has essential spectrum

 σess(KΓ,L2)=∞⋃n=−∞Πn.

If , then has Fredholm index

 index(KΓ−z)=∞∑n=−∞W(z,Πn)=W(z,Π0)+2∞∑n=1W(z,Πn)

where denotes the winding number of with respect to and the right-hand side is always a finite sum. In particular, every point lying inside one of the curves belongs to the spectrum .

Whenever is not a real number, it holds that , so that

 index(KΓ−z)=−codimranKΓ,z∈C∖R.

In particular, if (so that lies outside every curve ), then either is invertible or is real and an eigenvalue of .

Theorem 5.28 is illustrated in Figure 2. After reversing the signs of the winding numbers, the first paragraph of the theorem applies equally well to the adjoint operator , known as the Neumann–Poincaré operator. As a consequence, the number of eigenfunctions of to the eigenvalue is equal to the winding number of with respect to the essential spectrum, except at certain exceptional real values .

Next, we state our characterization of the -spectrum.

###### Theorem 6.33.

Let be a closed surface of revolution with a conical point of opening angle , obtained by revolving a -curve . For , denote by the closed interval

 Σn=⎧⎪⎨⎪⎩Pniξ−1/2(cosα)˙Pniξ−1/2(−cosα)−Pniξ−1/2(−cosα)˙Pniξ−1/2(cosα)Pniξ−1/2(−cosα)˙Pniξ−1/2(cosα)+Pniξ−1/2(cosα)˙Pniξ−1/2(−cosα):−∞≤ξ≤∞⎫⎪⎬⎪⎭.

Then the self-adjoint operator , where is the energy space of , has essential spectrum

 σess(KΓ,E)=∞⋃n=−∞Σn.

Hence, the spectrum of consists of this interval and a sequence of real eigenvalues with no limit point outside of it,

 σ(KΓ,E)={λk}∪σess(KΓ,E).

In Section 7 we will develop a method to numerically determine the polarizability tensor and spectrum of for rotationally symmetric surfaces . We offer one of our numerical results already here, which at the same time illustrates Theorem 6.33. In the proof of Theorem 6.33 we decompose according to its Fourier modes, . Figure 3 demonstrates the indicator function for mode , which detects the spectrum of , for a surface of opening angle . The set where the indicator function is equal to coincides with the interval of Theorem 6.33, i.e. the essential spectrum of . The points where the indicator function is correspond to eigenvalues. It turns out (only numerically demonstrated) that in this case there is an infinite sequence of eigenvalues outside the essential spectrum, and every eigenvalue but one yields a plasmon resonance.

We now explain the layout of the paper, with some remarks on the content of each section. Section 2 contains preliminary material on layer potentials, the energy space, the transmission problem, limit polarizability, Fredholm operators, Mellin transforms, Sobolev spaces and singular integral operators.

In Section 3 we study the model case in which is an infinite straight cone. We provide the spectral resolution of both operators and . The first case is quite straightforward, and the relevant analysis appears implicitly in (19). Each modal operator is in this case unitarily equivalent to a Mellin convolution operator, and this leads to the spectral resolution. On the energy space we make use of a special norm related to the single layer potential which has several advantages. For one, this norm decomposes orthogonally with respect to the Fourier modes. Secondly, it allows us to exploit that we can calculate the action of the single layer potential on the generalized eigenfunctions of on the infinite cone. We remark that for the case of intersecting disks, the same norm was used in (31) to realize the spectral theorem of .

In Section 4 we show, in a certain sense, that is a compact perturbation of , where is a straight cone of the same opening angle. The proof proceeds by writing the difference of kernels as a sum of products of Riesz kernels with smooth, small kernels. The Riesz transforms are however not bounded on ; since this would contradict ((39), Eq. (6.50)). Hence, the indicated argument provides compactness on , but for we have to work harder. We combine certain algebraic identities (the Plemelj formula) with further estimates and real interpolation in this case.

In Section 5 we prove Theorem 5.28. The index formula is proven by showing that the modal operators are pseudodifferential operators of Mellin type, for which there is a well-developed symbolic calculus (17); (35); (36). Theorem 6.33 is proven in Section 6. The method of proof is to first show that on the infinite cone, the singularities of at the origin and at infinity contribute equally to the essential spectrum. The theorem is then pieced together from the results in Sections 3 and 4.

Section 7 contains our numerical results. We first define the indicator function, and establish its properties. Then we give an overview of the numerical method, after which we present numerical results on the polarizability tensor and the spectrum of for the two surfaces illustrated in Figure 1.

Finally, the Appendix contains explicit expressions for the various modal kernels we will consider, in terms of special functions. Our theory and numerical method both depend on these explicit formulas. In particular, we will defer the proof of the technical Lemma 3.10 to the Appendix.

#### Notation

If and are two non-negative quantities depending on some variables, we write to signify that there is a constant such that . If and , we write .

## 2 Background, definitions and notation

### 2.1 Single and double layer potentials

For a function on , its single layer potential is given by

 SΓf(r)=∫ΓSΓ(r,r′)f(r′)dσ(r′),r∈R3, (4)

where

 SΓ(r,r′)=12π1|r−r′|.

Note that is a harmonic function for . If is any reasonable function or distribution, will have traces from both the interior domain and the exterior domain . Due to the weak singularity of the kernel, these traces coincide with evaluating directly on ,

 TrintSΓf(r)=TrextSΓf(r)=SΓf(r),r∈Γ.

In the most general case, these traces may be understood in the sense of convergence in nontangential cones for almost every point of (51), or in a distributional sense (43). Most of the time we will consider as map directly on , since the well-posedness of the interior and exterior Dirichlet problems ensure that can be uniquely identified with its values on , see (43).

The layer potential , evaluated on the boundary, is given by the principal value integral (1) with kernel defined in (2). The adjoint operator (with respect to ) is usually referred to as the boundary double layer potential, or the Neumann–Poincaré operator. Note also that the choice of normalizing constant in front of (1) may be different in other works.

We will consider two different function/distribution spaces for the action of . First, we will consider as an operator on . is always bounded on (51), but note that is not a self-adjoint operator in this space. When has singularities, so that is not a compact operator, the spectrum of on is typically not real. This is illustrated by our main theorem on the -spectrum, Theorem 5.28. See also (40) for the 2D-case.

The second space we will consider is the Hilbert space , obtained by completing in the positive definite scalar product

 ⟨f,g⟩E=⟨SΓf,g⟩L2(Γ).

By applying the classical jump formulas for the interior and exterior normal derivatives of and Green’s formula, we have that

 ⟨f,f⟩E=12∫R3|∇SΓf|2dV. (5)

A proof, which carries over verbatim to the Lipschitz case (see (43)) may be found in ((33), Lemma 1). Here denotes the usual volume element on . Hence, we refer to as the energy space, as it consists of charges generating single layer potentials with finite energy in . In light of this physical interpretation, it is not a complete surprise that is self-adjoint as an operator on . Indeed, from the Plemelj formula

 SΓKΓf(r)=(KΓ)∗SΓf(r),r∈Γ (6)

it follows that

 ⟨KΓf,g⟩E=⟨f,KΓg⟩E,

see (31); (33); (43). By considering trace theorems and well-posedness of Dirichlet problems, it can be deduced that the -norm is equivalent to the Sobolev norm of index on the boundary (see Section 2.5),

 ∥f∥E≃∥f∥H−1/2(Γ). (7)

Again we refer the reader to (33), or to (43) for a treatment explicitly for the Lipschitz case. By interpolating between and , where is bounded by the classical theory (51), it now follows that is bounded.

It is known ((11), Theorem 2.5) that as an operator on the spectrum of is contained in ,

 σ(KΓ,E)⊂(−1,1]. (8)

However, without additional hypotheses on such as convexity or smoothness, it is not even known if the essential norm of is less than ,

 ∥KΓ∥L2(Γ)→L2(Γ),ess≤1?

We refer to (52) for a discussion.

In addition to bounded domains, we will consider one instance of an unbounded surface. Namely, we will consider an infinite straight cone of opening angle , , . In general the layer potential theory for domains with non-compact boundary is rather delicate, but in our particular case is a Lipschitz graph. In any case, since will be our model for studying domains with axially symmetric conical points, we will make precise calculations from which the boundedness and other basic properties of and will be apparent. All of the properties of , , and mentioned in this subsection continue to hold, except that we (the authors) are not entirely sure about the available results on the Dirichlet problem. In particular, we are not sure if (7) holds. However, in view of (5) and the boundedness of the trace (38), we at least have that

 ∥f∥E=sup∥g∥E=1⟨SΓαg,f⟩L2(Γα)≤sup∥g∥E=1∥TrSΓαg∥H1/2(Γα)∥f∥H−1/2(Γα)≲∥f∥H−1/2(Γα). (9)

Furthermore, if is a smooth compactly supported function, then

 ∥ρf∥E≃∥ρf∥H−1/2(Γα), (10)

with implicit constants possibly depending on .

### 2.2 The transmission problem and limit polarizability

In the transmission problem (3), with , the normal derivatives and of exterior and interior approach need to be understood in a distributional sense. Making the ansatz , the jump formulas

 ∂intνSΓf(r)=f(r)−KΓf(r),∂extνSΓf(r)=−f(r)−KΓf(r),r∈Γ (11)

imply that solves the transmission problem if and only if and

 (KΓ−z)f=g1−ϵ,z=−1+ϵ1−ϵ.

In fact, in the case that is a bounded surface, any solution to the transmission problem which satisfies must be of this form, as mentioned in the introduction. See ((28), Proposition 5.1).

To define the polarizability tensor of we understand as an inclusion with permittivity , embedded in infinite space of permittivity . For a unit field , we seek a potential such that

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩∫R3|∇U−e|2dV<∞,ΔU(r)=0,r∈R3∖Γ,TrintU(r)=TrextU(r),r∈Γ,∂extνU(r)=ϵ∂intνU(r),r∈Γ.

The single layer potential ansatz

 U(r)=e⋅r+SΓρ(r)

yields ((28), Section 2) the equation

 (KΓ−z)ρ=ge,ge(r)=(e⋅νr),z=−1+ϵ1−ϵ.

If the solution exists uniquely for all , then the polarizability tensor , a linear map on , scaled by the volume of , is defined by

 ω(z)e=(ϵ−1)|int(Γ)|∫int(Γ)∇U(r)dV(r).

To evaluate the polarizability, we make use of Green’s formula

 ∫int(Γ)⟨∇U,∇V⟩dV=∫ΓU¯¯¯¯¯¯¯¯¯¯¯¯¯∂intνVdσ, (12)

valid for and harmonic in and of sufficient smoothness.

We suppose now that is rotationally symmetric around the -axis. Then is diagonal, and its first two diagonal entries are equal, . We refer to as polarizability in the -direction, . Applying (12) and the jump formulas (11) yields that

 ωjj(z)=ej⋅ω(z)ej=∫Γρ(r)hej(r)dσ(r)=⟨(KΓ−z)−1gej(r),hej(r)⟩L2(Γ), (13)

where denotes the th unit vector in the standard basis of , and

 he(r)=−2e⋅r|int(Γ)|.

In Section 7.1 we will see that (13) is associated with a spectral measure ,

 ωjj(z)=∫1−1dμj(s)s−z. (14)

This statement is a little more subtle than it appears, since is not a self-adjoint operator in the -pairing. An appropriate formalism was developed in ((28), Section 5), and we shall carry out the corresponding details for our situation in Section 7.1. Alternative approaches may be found in (10); (20); (21); (42). Some of these references concern the effective permittivity tensor rather than polarizability. However, the polarizability tensor may be viewed as a limiting case of the effective permittivity tensor.

By the representation of the polarizability as a Cauchy integral (14), the limit

 ω−jj(x)=limy→0−ωjj(x+iy)

exists almost everywhere , even when lies in the support of . We refer to as the limit polarizability. When lies outside the support of the limit polarizability and polarizability coincide. For axially symmetric domains with a conical point, we will find that the spectral measure typically has an absolutely continuous part, in addition to a possible singular part. The absolutely continuous part is recognized by the fact that almost everywhere it holds that

 μ′j(x)=−Imω−jj(x)π.

By ((28), Remark 5.1 and Theorem 5.2), is a positive measure, and and satisfy that

 ∫Γρ(r)dσ(r)=0, (15)
 ∫1−1dμj(s)=−2. (16)

Let be the pure point part of . By ((28), Theorem 5.6) there are eigenvectors and of and , respectively, normalized so that

 ⟨ϕi,ψk⟩L2(Γ)=δik,

such that

 ∫Γdμppj(s)=∑iuivi,ui=⟨ϕi,hej⟩L2(Γ),vi=⟨ψi,gej⟩L2(Γ).

In particular, if has no singular continuous part, then (16) takes the form

 ∑iuivi+∫1−1μ′j(x)dx=−2. (17)

We strongly believe, but will not prove, that never has a singular continuous part for the surfaces we consider. We shall use the rules (15) and (17) to verify the accuracy of our numerical results.

### 2.3 Fredholm operators

Recall that a bounded operator on a Hilbert space is Fredholm if it has closed range and both its kernel and cokernel are finite-dimensional. Equivalently, is Fredholm if and only if it is invertible modulo compact operators. If is Fredholm, its index is given by

 indexS=dimkerS−codimranS.
###### Definition 2.2.

If two operators and on Hilbert spaces and are unitarily equivalent, we write that .

###### Definition 2.3.

We write if there exist Hilbert spaces and and a compact operator such that is similar to .

The point of the above definition is that if and , it holds that is Fredholm if and only if is Fredholm and then the Fredholm indices satisfy

 index(S−z)=index(T−z).

For a (not necessarily self-adjoint) operator we will denote its essential spectrum in the sense of invertibility modulo compacts by .

###### Definition 2.4.

The essential spectrum of is the set

 σess(S,H1)={z∈C:K−z%isnotFredholm}.

We will also make use of another concept of essential spectrum, also invariant under compact perturbations. We say that a bounded sequence is a singular sequence for the operator and spectral point if has no convergent subsequences and in as .

###### Definition 2.5.

The point belongs to if and only if there is a singular sequence for and .

Note that if is a self-adjoint operator, then the two type of essential spectra agree by Weyl’s criterion, . Furthermore, in this case whenever .

### 2.4 Mellin transforms

For , let be its Mellin transform,

 Mg(ζ)=^g(ζ)=∫∞0tζg(t)dtt.

The -hypothesis on implies that is well-defined and bounded at least for . We will denote Mellin convolution by ,

 u⋆v(t)=∫∞0u(t/t′)v(t′)dt′t′,t>0.

The Mellin transform is the Fourier transform of the multiplicative group of ; for sufficiently nice functions and and appropriate it holds that

 M(u⋆v)(ζ)=Mu(ζ)Mv(ζ).

Young’s inequality for the Mellin transform says that

 ∥u⋆v∥L2(dt/t)≤∥u∥L1(dt/t)∥v∥L2(dt/t). (18)

Another way to see this is by noting that is a unitary operator,

 Wv(ξ)=1√2πMv(iξ), (19)

with inverse

 W−1ψ(t)=1√2π∫∞−∞t−iξψ(ξ)dξ.

In particular, Plancherel’s formula takes the form

 12π∫∞−∞∣∣∣∫∞0tiξv(t)dtt∣∣∣2dξ=∫∞0|v(t)|2dtt. (20)

### 2.5 Singular integral estimates on Sobolev spaces

Suppose first that is a Lipschitz graph

 Γ={(x,y,φ(x,y)):(x,y)∈R2}.

The parametrization then induces tangential derivatives on on . The (inhomogeneous) Sobolev space consists of those functions such that

 ∥f∥2H1(Γ)=∥f∥L2(Γ)+∥∂xf∥L2(Γ)+∥∂yf∥L2(Γ)<∞.

This also allows us to define in the case that is a bounded Lipschitz surface, via its Lipschitz manifold structure. In this setting, we will make use of the fact that is characterized by single layer potentials.

###### Lemma 2.6 ((51), Theorem 3.3).

Let be a bounded and simply connected Lipschitz surface. Then

 SΓ:L2(Γ)→H1(Γ)

is a continuous isomorphism.

For we define the Sobolev-Besov space via the Gagliardo-Slobodeckij norm

 ∥u∥2Hs(Γ)=∥u∥2L2(Γα)+∫Γ∫Γ|u(r)−u(r′)|2|r−r′|2+2sdσ(r)dσ(r′). (21)

The spaces , , coincide with the real interpolation scale between and , see for instance (50). For we define the space of distributions as the dual space of with respect to the scalar product of . Recall that coincides with the energy space , with equivalent norms.

Our goal in Section 4 is to view the operator , where has a single axially symmetric conical point, as a perturbation of , where is a straight cone. In doing so we will encounter many integral operators with weakly singular kernels. It is well known that such kernels generate compact operators, see for example (8) and (49). However, we have been unable to locate a precise statement which covers all of our cases. We therefore sketch a proof of a statement which is far from sharp, but sufficient for our purposes.

###### Lemma 2.7.

Let be a bounded and simply connected Lipschitz surface, and let be a kernel on satisfying

 |H(r,r′)|≲1|r−r′|, (22)
 |H(r,r′)−H(r∗,r′)|≲|r−r∗||r−r′|2,|r−r∗|<12|r−r′|, (23)

and

 |H(r′,r)−H(r′,r∗)|≲|r−r∗||r−r′|2,|r−r∗|<12|r−r′|. (24)

Then the integral operator

 Hf(r)=∫ΓH(r,r′)f(r′)dσ(r′)

defines compact operators , , and .

###### Proof.

For it is easy to show that the operator

 Gβf(r)=∫Γ1|r−r′|βf(r′)dσ(r′)

is bounded on , for instance by interpolating between the spaces and , on which the boundedness property is evident. Next, inequalities (22) and (23) imply that

 |H(r,r′)−H(r∗,r′)|≲|r−r∗|3/4|r−r′|7/4+|r−r∗|3/4|r∗−r′|7/4,r,r∗,r′∈Γ.

Hence, for

 |Hf(r)−Hf(r∗)|≲|r−r∗|3/4(G7/4|f|(r)+G7/4|f|(r∗)).

From this estimate we obtain that

 ∥Hf∥H1/2(Γ)≲2∫Γ[G7/4|f|(r)]2∫Γ1|r−r∗|3/2dσ(r∗)dσ(r)≲∫Γ[G7/4|f|(r)]2dσ(r)≲∥f∥L2(Γ).

Hence is bounded. In particular is compact, since is compactly contained in . By (24) the same argument yields that the -adjoint also maps into . Equivalently, by duality, maps into boundedly. Since is compactly contained in it follows that is compact. By duality, this is equivalent to saying that is compact. Since the statement of the lemma is symmetric with respect to and , it follows that also is compact. ∎

###### Remark 2.8.

If is a -surface, then satisfies the hypotheses of the lemma. Hence is a compact operator in this case (as is well known). Another example we have in mind is given by the kernel

 Hj(r,r′)=rj−r′