Martin Costabel IRMAR, Université de Rennes 1 CMR9 Gala¸ti ... · Nous appellerons ce théorème...

Spectral Properties of the Double Layer Potential

Martin Costabel

IRMAR, Université de Rennes 1

CMR9

Galati, 28 June – 03 July 2019

Martin Costabel (Rennes) Spectral Properties of the Double Layer Potential Galati, 30/06/2019 1 / 20

The classical boundary integral operators, definition

Ω bounded Lipschitz domain in Rd , Γ = ∂ ΩSingle layer potential for Laplace operator in R3 (Names: Helmholtz 1835)

S φ(x) =∫

Γ

φ(y)ds(y)

4π|x−y |

Double layer potential for Laplace operator in R3

Dv(x) =∫

Γv(y)∂n(y)

14π|x−y |

ds(y)

Boundary integral operators

Vφ = S φ |Γ , K ′φ = ∂nS φ |Γ ,

Kv = Dv |Γ , Wv =−∂nDv |Γ .

Standard generalizations:Laplace operator −→ strongly elliptic system in Rd , d ≥ 2

14π|x | −→ fundamental solution∂n −→ conormal derivativeMartin Costabel (Rennes) Spectral Properties of the Double Layer Potential Galati, 30/06/2019 2 / 20

Time frame (My personal version, simplified)

• Gauss 1813 (jump relations), 1838–1840 (first kind integral equation, variational method, analysis and numerics)• Green 1828 (representation formula)• C. Neumann 1870–1877 (analysis of double layer potential, Neumann series for convex domains)• Poincaré 1896 (energy method for double layer potential, Neumann series for smooth domains)→Korn, Stekloff, Plemelj 1899–1904

• Fredholm 1900–1903 (general theory of integral equations of second kind, Fredholm alternative)→Hilbert 1908 and many others

• Calderón – Zygmund 1956–1957 (singular integral equations)→Coifman-McIntosh-Meyer 1981, Verchota, Jerison-Kenig, ...

• Nedelec – Planchard 1973 (energy method for first kind integral equation, analysis and numerics)• W. Wendland’s group 1980 – (non-smooth domains, variational methods and Fourier analysis, numerics)→Costabel, Stephan, Steinbach

• Khavinson–Putinar-Shapiro 2007 (Poincaré revisited, spectral theory on smooth domains)→many others 2007–present (“Neumann-Poincaré operator”)


3 cases of “Memory Lost and Retrieved”: 1st case, Gauss

Gauss 1839: First kind integral equation for the gravity potential (d = 3)

Vφ(x)≡∫

Γ

φ(y)ds(y)

4π|x−y |= f (x), x ∈ Γ

Variational approach: Minimize 12 〈φ ,Vφ〉−〈f ,φ〉.

Needed: Bilinear form 〈φ ,Vψ〉 is positiv definite.• 2 principal methods: With or without looking at the integral operator.

[Gauss 1839] Looking at the kernel, obvious estimate∫Γ

∫Γ

φ(x)φ(y)

|x−y |ds(y)ds(x)≥

‖φ‖2L1(Γ)

diam(Γ)if φ ≥ 0

2 problems:1. Existence of minimum impossible to prove with 19th century maths.2. Minimum over positive functions→ Integral inequality (deplored by Gauss himself)

Completion: O. Frostman 1937 (using compactness properties of spaces of positive measures)


1st case, Gauss. Proof with energy argument (without looking at the kernel)

[Nedelec-Planchard 1973]

‖φ‖2V = 〈φ ,Vφ〉 defines a norm on H−

12 (Γ), equivalent to the Sobolev norm.

V : H−12 (Γ)→ H

12 (Γ) is an isomorphism.

Proof using the jump relation for the single layer potential (Ω− = Ω, Ω+ = R3 \Ω):

u = S φ =⇒ φ =−[∂nu]Γ = ∂−n u−∂

+n u

Green’s formula

∆u = 0 in Ω−∪Ω+ =⇒∫

Ω−+|∇u|2 dx = +

−∫

Γu∂

−+

n u ds

Adding up (Vφ = u|Γ, φ =−[∂nu]Γ) :

〈φ ,Vφ〉=∫R3|∇S φ |2 dx This is > 0 if φ 6= 0.


2nd case, Neumann – Poincaré

Dv(x) =1

4π

∫Γ

v(y)∂n(y)|x−y |−1ds(y) , Kv = Dv |Γ

Jump relations for the double layer potential u = Dv

[∂nu]Γ = 0 ; [γu]Γ = v ; γ−+

u = (−+12 + K )v

2nd kind integral equation for the Dirichlet problem ∆u = 0 in Ω, u = g on Γ

( 12 −K )v =−g or (1−N)v =−2g with N = 2K

If one can show that N is a contraction in some Banach space, one gets aunique solution by successive approximation (“Neumann series”)

v =−2∞

∑`=0

N`g .


2nd case, C. Neumann

First approach (looking at the kernel)

dθx (y) =−n(y) · (y−x)

2π|x−y |3ds(y)

is for x ∈ Γ a measure (solid angle) of total mass 1 on Γ,positive if Ω is convex.

[C. Neumann 1877] Using hard analysis

If Ω is convex, but not the intersection of 2 convex cones, then N = 2K is acontraction on L∞(Γ)/R in a norm equivalent to the L∞ norm.


Energy and the double layer potential

We have seen for u = S φ :∫

Ω−+ |∇u|2 dx = +

−∫Γ u∂

−+n u ds = 〈Vφ ,( 1

2 +−K ′)φ〉

Hence with the Dirichlet trace w = Vφ∫R3|∇u|2 dx = 〈Vφ ,φ〉= 〈w ,V−1 w〉

∫Ω−|∇u|2 dx−

∫Ω+|∇u|2 dx = 2〈Vφ ,K ′φ〉= 〈N w ,V−1 w〉

Energy in Ω−+: ∫

Ω−+|∇u|2 dx = 〈( 1

2 +−K )w ,V−1w〉

Rayleigh quotient for 12 +−K in H

12 (Γ) with norm ‖ · ‖V−1 :∫

Ω−+|∇u|2 dx∫

R3|∇u|2 dx

=〈( 1

2 +−K )w ,V−1w〉〈w ,V−1 w〉


2nd case, PoincaréSecond approach to Neumann series (without integral operators)

[Poincaré 1896] An energy inequality

There exists a constant µ > 0 depending on Ω such that1 If u is a double layer potential, then

1µ

∫Ω+|∇u|2 ≤

∫Ω|∇u|2 ≤ µ

∫Ω+|∇u|2

2 If u is a single layer potential, then∫Ω|∇u|2 ≤ µ

∫Ω+|∇u|2 and if

∫Γ

u = 0 then∫

Ω+|∇u|2 ≤ µ

∫Ω|∇u|2

Poincaré: Proved for simply connected smooth domains.Korn, Stekloff. . . : For Lyapunov domains.Nowadays easy exercise for Lipschitz domains (Ω+ connected), using trace lemma,Lax-Milgram, extension theorem for H1.

[Steklov 1900]

Nous appellerons ce théorème théorème fondamental. . . .Nous verrons dans ce qui va suivre, que la solution de tous les problèmesfondamentaux de la Physique mathématique se ramène à la démonstrationcomplète du théorème fondamental.


2nd case, Poincaré et. al.

[Co 2007] Corollary of the “Théorème fondamental”

The operators A = 12 −K ′ and B = 1

2 + K ′ are bounded selfadjoint

operators on the space H−12 (Γ) with norm ‖ · ‖V satisfying A + B = 1.

1 A is positive definite, hence B is a contraction, with norm

‖B‖ ≤ µ

1+µ.

2 On the subspace H− 1

20 = φ | 〈φ ,1〉= 0, B is positive definite, hence

both A and N ′ = A−B are contractions, and the Neumann series (for K ′ orN ′) converges in the norm ‖ · ‖V .

3 Same results for 12−+K in the space H

12 (Γ) with norm ‖ · ‖V−1 .

Proof of 1 : Poincaré⇒ 〈Vφ ,Bφ〉 ≤ µ〈Vφ ,Aφ〉 ⇒‖φ‖2

V = 〈Vφ ,φ〉= 〈Vφ ,(A + B)φ〉≤ (1 + µ)〈Vφ ,Aφ〉 ⇒ A pos. def. and〈Vφ ,Bφ〉= 〈Vφ ,φ〉−〈Vφ ,Aφ〉≤(1− 1

1+µ)‖φ‖2

V = µ

1+µ‖φ‖2

V


Steinbach – Wendland 2001, idea of proof

The first proof of the convergence of the Neumann series for 12−+K and

12−+K ′ in the energy spaces H−+

12 (Γ) was given by Steinbach and Wendland

2001.

[O. Steinbach – W. Wendland 2001]

Let the classical boundary integral operators for a positive elliptic 2nd orderpartial differential operator be defined on the boundary Γ of a boundedLipschitz domain in Rd , d ≥ 2.Then the operators 1

2 + K and 12 + K ′ are contractions in H

12 (Γ) and

H−12 (Γ) with norms ‖ · ‖V−1 and ‖ · ‖V , respectively.

For the operators 12 −K and 1

2 −K ′ the same holds in certain

finite-codimensional invariant subspaces H−+ 1

20 (Γ)

Remark: 2 versions of Neumann series, for example for the equation

( 12 −K )v =−g ⇐⇒ (1−2K )v =−2g ⇐⇒ (1− ( 1

2 + K ))v =−g

v =−2∞

∑`=0

(2K )`g =−∞

∑`=0

( 12 + K )`g


Convergence of Neumann series in H−+12 , Steinbach – Wendland 2001

Starting point: Calderón projector. The 2×2 matrix of integral operators

C− =

( 12 −K V

W 12 + K ′

)projects in H

12 (Γ)×H−

12 (Γ) onto the Cauchy data of harmonic functions in the

interior domain Ω. It satisfies (C−)2 = C−, implying in particular12−+K = ( 1

2−+K )2 + VW .

The operator VW is positive definite on H12

0 (Γ) with norm ‖ · ‖V−1 , because of

(VWu,u)V−1 = (Wu,u) =∫

Ω−∪Ω+|∇Du|2 dx ≥ α‖u‖2V−1 .

One concludes with the following observation:

Lemma

Let A and B be bounded selfadjoint operators on a Hilbert space. If

B = B2 + A and A is positive definite, A≥ α I,

then B is a contraction with norm ‖B‖ ≤ 12 +

√14 −α


3rd case: Mellin analysis in the 1980’s

Problem:On a non-smooth boundary, K is not a compact operator in the usualfunction spaces, there is a non-trivial essential spectrum.On piecewise smooth curves in R2 (curved polygons), the essentialspectrum is determined by the corner angles. By localization, it is sufficientto consider a curve with one corner, where Γ coincides with

Γω = R+∪eiωR+ , 0 < ω < 2π .

On Γω , K and K ′ have kernels homogeneous of degree −1, they arediagonalized by Mellin transformation, which we define here by

u(ξ ) =∫

∞

0x iξ−1u(x)dx .

Thus there is a Mellin symbol K (ξ ), which is a (2×2)-matrix valuedfunction that provides a criterion for Fredholmness:



Theorem [Costabel-Stephan 1981–1985]

For − 12 < s < 3

2 , the operator λ −K : Hs(Γ)→ Hs(Γ) is a Fredholmoperator if and only if the determinant of the Mellin symbol ξ 7→ λ − K (ξ )does not vanish on the line

ξ ∈ C | Imξ = s− 12 .

On Γω , K is defined by the operator Kω on the half-axis

Kω u(x) = 12π

∫∞

0Im 1

xeiω−x u(y)dy

hence K (ξ ) =

(0 Kω (ξ )

Kω (ξ ) 0

)with

Kω (ξ ) =− sinh(π−ω)ξ

2sinhπξ



Corollary

For − 12 < s < 3

2 , the essential spectrum of the operator K on Γ is given bythe values of the function Kω on the corresponding line

σess(K |Hs ) =−+

sinh(π−ω)ξ

2sinhπξ| Imξ = s− 1

2

.

In particular, for s = 12 , the essential spectrum is real (not surprising...), and

σess(K |H

12

) = [−|π−ω

2π], |π−ω

2π|]⊂ (− 1

2 ,12 ) .

Remark: For s = 1 or s = 0, one finds that the Fredholm radius is12 sin |π−ω|

2 < 12 , but the more general question

whether 12 + K or 2K on a Lipschitz boundary are contractions in L2(Γ),

seems to be still open (?).For s < 0 or s > 1, there are counterexamples, see the following graph.


Essential spectrum on polygon in Hs, angle ω = 0.2π

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-0.4

-0.2

0

0.2

0.4s=0.5s=0.75s=1s=1.1


Spectral symmetry in 2D, Costabel-Darrigrand-Sakly 2015

Motivation from 2D Maxwell equations:Analysis of strongly singular domain integral equation with principal part

B E = F with B E(x) = 12π

curl∫

Ωlog |x−y |curlE(y)dy

One can show that B on the space H(curl(Ω)∩H(div0,Ω) is spectrallyequivalent to the system of boundary integral operators

D =

( 12 + K ′ −∂tV∂tV 1

2 + K ′

)on the space H−

12 (Γ)2 .

Namely, there are operators S and T such that B = ST and D = TS.On a polygon, one can determine σess(D) by Mellin transformation. Theresult was

σess(D) = 0,1 .

One also sees curlB E = curlE in Ω, which implies B(BE) = BE , that is,B is a projection.

Corollary: D is a projection.


Spectral symmetry in 2D, Costabel-Darrigrand-Sakly 2015

D =

( 12 + K ′ −∂tV∂tV 1

2 + K ′

)is a projection

This means in particular:

∂tV ( 12 + K ′) + ( 1

2 + K ′)∂tV = ∂tV

or∂tV K ′+ K ′ ∂tV = 0.

On H− 1

20 (Γ) = φ ∈ H−

12 (Γ) | 〈φ ,1〉= 0, ∂tV (Hilbert transform) is an

isomorphism, hence one can write

−K ′ = (∂tV )−1 K ′ ∂tV .

Corollary: On H− 1

20 (Γ), K ′ and −K ′ have the same spectrum.

Likewise, on H12 (Γ)/C, K and −K have the same spectrum.

Remark: There exist other proofs for the symmetry of the point spectrum orfor the symmetry of the essential spectrum [Co-Dar-Sak 2015].


Some References

M. COSTABEL, E. STEPHAN: Boundary integral equations for mixed boundaryvalue problems in polygonal domains and Galerkin approximation. inMathematical models and methods in mechanics, vol. 15 of Banach CenterPubl., pp. 175–251, PWN, Warsaw, 1985.

M. COSTABEL, E. STEPHAN: A direct boundary integral equation method fortransmission problems. J. Math. Anal. Appl. 106 (1985) 367–413.

O. STEINBACH, W. L. WENDLAND:On C. Neumann’s method for second-order elliptic systems in domains withnon-smooth boundaries. J. Math. Anal. Appl. 262 (2001) 733–748.

M. COSTABEL:Some historical remarks on the positivity of boundary integral operators.In Boundary element analysis, volume 29 of Lect. Notes Appl. Comput. Mech.,pages 1–27. Springer, Berlin, 2007.

D. KHAVINSON, M. PUTINAR, H. SHAPIRO:On Poincaré’s variational problem in potential theory.Arch. Ration. Mech. Anal. 185 (2007), 143-184.

M. COSTABEL, E. DARRIGRAND, H. SAKLY:Volume integral equations for electromagnetic scattering in two dimensions.Computers and Mathematics with Applications 70(8) (2015), 2087-2101.


Thank you for your attention!


Martin Costabel IRMAR, Université de Rennes 1 CMR9 Gala¸ti ... · Nous appellerons ce théorème...

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