Proof of Theorems 1.1 and 1.2 in the 2-d case

4.2.1 The norm and numerical range of the double-layer potential operator on a periodic Lipschitz graph

As the main step in the proofs of Theorems1.1and1.2, we calculate in this section lower bounds on the norm and numerical range (and their essential variants) for the double-layer potential operator D on a particular “sawtooth” periodic Lipschitz graph.

Definition 4.4 (The “sawtooth” Lipschitz graph Γ^M) Given M > 0 (the Lipschitz constant) define fM : R → R by

fM(s) :=

 M s − 2m, 2m ≤ M s ≤ 2m + 1,

2m + 2 − M s, 2m + 1 ≤ M s ≤ 2m + 2, (4.17) for m ∈ Z, so that f^M is periodic with period 2/M and

|f_M⁰ (s)| = M,

for almost all s ∈ R. Let Γ^M := {(s, fM(s)) : s ∈ R} be the graph of f^M, so that Γ^M = [

m∈Z

Γ_m,

where Γ_mis the open line segment connecting ((1−m)/M, 0) and (−m/M, 1), for m odd, connecting ((1 − m)/M, 1) and (−m/M, 0), for m even. (See Figure4.2for Γ_M when M = 1.)

Let D^M : L²(Γ^M) → L²(Γ^M) denote the double-layer potential operator on Γ^M, defined with the unit normal n on Γ^M such that n2(x) > 0 for almost all x ∈ Γ^M. Define V₊^M ⊂ V^M ⊂ L²(Γ^M)

by

The next two lemmas do most of the quantitative work for us in proving Theorems 1.1 and 1.2. In particular, Lemma 4.6 provides precise characterisations for kD_VMk_VM ≤ kD^Mk_L2(Γ^M)

and W (D_VM) ⊂ W (D^M) in terms of symbols of associated infinite Toeplitz matrices.

Lemma 4.5 For M > 0,

In the following lemma we use the notations of§2.3. In particular, wr(CN) denotes the numer-ical abscissa of a square matrix CN, and BN is the matrix defined by (2.25). We denote the norm on L^∞(−π, π) by k · k_∞.

Lemma 4.6 (The double-layer operator on Γ^M) Given N ∈ N, define the orthonormal set {ψ1, ..., ψ_N} ⊂ V₊^M ⊂ L²(Γ^M) by (4.7), but with Γ_m as in Definition4.4, and define the Galerkin matrix D_N by (4.1), where D = D^M is the double-layer potential operator on Γ^M. Then:

(a) where E : `<sup>2</sup>(N) → `²(N) is (multiplication by) the infinite Toeplitz matrix defined by

(E)_jm:= sign(m − j)d⁰_m−j, j, m ∈ N,

where er∈ C(R) is 2π-periodic with er(0) = 0, so that

kD_VMk_VM = kEk_`2(N)= kek_∞≥ M

2 . (4.26)

(c) For N ≥ 3,

W (DN) ∩ R ⊃ [−a, a], where a :=√

2 d⁰₁, (4.27)

with equality when N = 3. Further, where D^θ_N denotes the real part of e^iθD_N, w_r(e^iθD_N) = w_r(e^−iθD_N) = w_r(−e^iθD_N) = kD_N^θk₂, θ ∈ R, and

wr(e^iθDN) = kD_N^θk2→ kH^θk2= kh^θk∞, as N → ∞, (4.28) where H^θ: `<sup>2</sup>(N) → `²(N) is (multiplication by) the infinite Toeplitz matrix defined by

H^θ

jm:=1 − e^−2iθ(−1)^j−m

2 sign(m − j)d⁰_m−j, j, m ∈ N, and h^θ∈ L^∞(−π, π) is its symbol, given by

h^θ(t) = −i

∞

X

m=1

(1 − e^−2iθ(−1)^m)d⁰_msin(mt) = −iMsign(t)(π − |t|) + e^−2iθt

4π + h^θ_r(t), (4.29) for −π < t < π, where h^θ_r∈ C(R) is 2π-periodic with h^θr(0) = 0, so that

lim

N →∞wr(e^iθDN) = kh^θk_∞≥M

4 , θ ∈ R. (4.30)

Moreover,

W (D_VM) =

∞

[

N =1

W (DN) = \

0≤θ≤2π

λ ∈ C : <(e^iθλ) ≤ kh^θk∞ ⊃ {λ ∈ C : |λ| ≤ M/4}.

(4.31) Proof. Part (a). Arguing as in the proof of Lemma4.2, (D_N)_jm is given by the right hand side of (4.8), but with djm replaced by d⁰_jm, defined by (4.9) and (4.10) with Γm as in Definition4.4.

By symmetry of Γ^M and since fM is periodic with period 2/M , d⁰_jm = d⁰_|j−m|,0 = d⁰_|j−m|, for 1 ≤ j, m ≤ N . That d⁰_{−`</sub> = d<sup>0</sup><sub>`} = d⁰<sub>`,0</sub> → 1/2 as M → ∞, for each ` ∈ N, so that (4.22) holds, follows exactly as in the proof of Lemma4.2, and the asymptotics (4.23) follow easily from the definitions (4.20) and (4.21).

Part (b). kD_VMk_VM = kD_VM

+ k_VM by Lemma 4.5, and kDNk2 → kD_VM

+ k_VM as N → ∞ by (4.4), since H1⊂ H2⊂ ..., where HN is the space spanned by {ψ1, ..., ψN}, and V₊^M = ∪^∞_{N =1}HN. Moreover, it is easy to see that, for each N ∈ N, kD^Nk2= kENk2, where EN is the order N finite section of E. That kENk2 → kEk2 = kek_∞, with e given by the first of the equalities in (4.25), are standard properties of infinite Toeplitz matrices with bounded symbols [10]. The last equation in (4.25), with

er(t) := −2i

∞

X

m=1

(d⁰_m− M/(2πm)) sin(mt), t ∈ R, (4.32) follows since

sign(t)(π − |t|) = 2

∞

X

m=1

sin(mt)

m , −π ≤ t ≤ π, (4.33)

and that e_r is continuous and e_r(0) = 0 follows since the series (4.32) is absolutely and uniformly convergent by (4.23). The bound (4.26) follows since kek_∞≥ limt→0|e(t)| = M/2.

Part (c). Where C_N is the matrix in Lemma 2.8, D_N = C_N if we set e_jm = d⁰_|j−m|, for 1 ≤ j, m ≤ N . Thus (4.27) follows from Lemma2.8(iii), and that wr(e^iθDN) = wr(e^−iθDN) = wr(−e^iθDN) = kD^θ_Nk2, for θ ∈ R, from Lemma 2.8(ii) and (iv). Using (2.27), we see also that

Figure 4.3: Graphs of 2|e(t)|/M and 4 max_0≤θ≤2π|h^θ(t)|/M against t/π, where e and h^θ are the symbols of the infinite Toeplitz matrices E and H^θ, given by (4.25) and (4.29), respectively.

H^θ

jm= e^−iθ(−1)^m+1(D_N^θ)jm, 1 ≤ j, m ≤ N , so that kD_N^θk2 = kH_N^θk2, where H_N^θ is the order N finite section of H^θ. The remaining results up to and including (4.30) follow in the same way as we proved (4.24)-(4.26), using (4.33) and that

−t = 2

∞

X

m=1

(−1)^msin(mt)

m , −π < t < π.

The final statement (4.31) follows from (4.4), (2.26), and (4.30).

Combining Lemma4.5and4.6we see that, for M > 0, kD^Mk_L2(Γ^M) ≥ lim

N →∞kD_Nk₂= kD_VMk_VM ≥M

2 and (4.34)

W (D^M) ⊃

∞

[

N =1

W (DN) = W (D_VM) ⊃ {λ ∈ C : |λ| ≤ M/4} , (4.35)

so that w(D^M) ≥ M/4. On the other hand, by (2.4) and Theorem3.3, for some C, µ > 0, 1

2kD^Mk_L2(Γ^M)≤ w(D^M) ≤ kD^Mk_L2(Γ^M)≤ CM (1 + M )^µ, M > 0.

For sufficiently large M > 0 it appears that the last of the bounds in (4.34) and the last inclusion in (4.35) are in fact equalities. Indeed, it follows from (4.26) and (4.30) that (4.34) holds with the last “≥” replaced by “=” if kek_∞= M/2, and (4.35) holds with the last “⊃” replaced by “=” if kh_∞k_∞= M/4, for θ ∈ R, and the plots in Figure4.3suggest the conjecture that

kek∞= M/2, at least for M ≥ 1, and kh^θk∞= M/4, for θ ∈ R, at least for M ≥ 2.

4.2.2 The proof of Theorems 1.1and 1.2

We now use the above results to prove Theorems1.1and1.2in the 2-d case, as Theorem4.9below.

Let

Γ^M,m:=

2m

[

j=1

Γ_j, m ∈ N, (4.36)

where Γj is as defined in Definition4.4(and see Figure4.2).

Definition 4.7 (Γ^M_d and Ω^M_d for d = 2) Choose β ∈ (0, 1) and, given M > 0, define (γj)^∞_j=1 ⊂ (0, ∞) by

γ_j :=M (β^j−1− β^j)

2j , j ∈ N.

0 ... β⁴ β³ β² β 1

Figure 4.4: The curve Γ^M₂ , as specified in Definition4.7, in the case M = 1 and β = 0.6. The labels are the x1-coordinates of the point 0 = (0, 0) and of the first 5 of the points (β^j, 0), j = 0, 1, ....

All these points lie on Γ^M₂ .

Define f_β^M : R → [0, ∞) by

f_β^M(s) :=

 0, if s ≤ 0 or s ≥ 1,

γ_jf_M((s − β^j)/γ_j), if β^j ≤ s < β^j−1, (4.37) for j ∈ N, where fM is defined by (4.17), and note that the definition we make for γ_j ensures that f_β^M ∈ C^0,1(R), with |(fβ^M)⁰(s)| = M for almost all s ∈ (0, 1), and that

{f_β^M(s) : β^j≤ s ≤ β^j−1} = (β^j−1, 0) + γ_jΓ^M,j, j ∈ N, (4.38) where Γ^M,j is defined by (4.36). Let ε := 0.1 and let

Γ^M₂ := 

s, f_β^M(s)
: −ε ≤ s ≤ 1 + ε

= {(s, 0) : −ε ≤ s ≤ 0 or 1 ≤ s ≤ 1 + ε} ∪ [

j∈N

(β^j−1, 0) + γ_jΓ^M,j
; (4.39)

see Figure4.4for Γ¹₂ when β = 0.6. Set x⁰:= (−ε, −ε) and x⁰⁰:= (1 + ε, −ε), and let Ω^M₂ :=(x1, x₂) : −ε < x₁< 1 + ε and − 2ε < x₂< f_β^M(x₁) ∪ Bε(x⁰) ∪ B_ε(x⁰⁰);

see Figure 1.1for Ω¹₂ when β = 0.6. Note that Ω^M₂ ⊂ R² is a simply-connected Lipschitz domain with Lipschitz constant M , and the boundary Γ of Ω^M₂ contains Γ^M₂ and is C¹except at a countable set of points on

s, f_β^M(s)
: 0 ≤ s ≤ 1 ⊂ Γ^M₂ .

The proofs of Theorems1.1and1.2, in both the 2-d and 3-d cases, depend on (4.34) and (4.35), and on the localisation result Theorem3.2. They also depend on the simple observation that the norm and numerical range of the double-layer potential on a curve or surface Γ⁰ are the same as those of the double-layer potential operator on its translate, Γ⁰+ x, for x ∈ R^d. These quantities are also invariant under scaling in the sense of the following lemma.

Lemma 4.8 Suppose that Γ⁰ := {(y⁰, f (y⁰)) : y⁰ = (y₁, ..., y_d−1) ∈ N }, for some open N ⊂ R^d−1, d = 2 or 3, and some f ∈ C^0,1(R^d−1), and let D_κ denote the double-layer potential operator on κΓ⁰ for κ > 0. Then

kDκk_L2(κΓ⁰)= kD1k_L2(Γ⁰) and W (Dκ) = W (D1), κ > 0.

Proof. Define V : L²(κΓ⁰) → L²(Γ⁰) by V φ(y) = κ^(d−1)/2φ(κy), for φ ∈ L²(Γ⁰), y ∈ Γ⁰. Then, arguing exactly as in the proofs of Lemmas3.12and3.13, we see that V is an isometric isomorphism and D1V = V Dκ, so that D1 and Dκ are unitarily equivalent, and the result follows.

Theorem 4.9 (Theorems 1.1 and 1.2 in the 2-d case) Suppose that Γ is the boundary of Ω^M₂ , defined as in Definition4.7, for some M > 0 and β ∈ (0, 1). Then

kDk_L2(Γ),ess≥ M

2 and W_ess(D) ⊃ {λ ∈ C : |λ| ≤ M/4}.

Proof. Let x^∗:= 0 ∈ Γ^M₂ ⊂ Γ. By Theorem3.2, kDk_L2(Γ),ess≥ lim

δ→0kDx^∗,δk_L2(Γ) and Wess(D) ⊃ \

δ>0

W (Dx^∗,δ).

Thus the result follows if we show that kDx^∗,δk_L2(Γ) ≥ M/2 and W (Dx^∗,δ) ⊃ {λ ∈ C : |λ| ≤ M/4}. By (4.34) and (4.35), this in turn follows if we can show that kDx^∗,δk_L2(Γ)≥ kDNk2 and W (Dx^∗,δ) ⊃ W (DN), for every N ∈ N and δ > 0, where D^N is as defined in Lemma4.6. Given m ∈ N, set eΓ := Γ^M,m, defined by (4.36). Then, by (4.3) and (4.2), kDNk2 ≤ k eDk_L2(eΓ) and W (DN) ⊂ W ( eD), for N = 1, ..., 2m, where eD denotes the double-layer potential operator on eΓ.

But, by construction of Γ^M₂ in Definition4.7 (see (4.38)), for every δ > 0 there exists κ > 0 and s ∈ R such that bΓ := (s, 0) + κeΓ ⊂ Bδ(x^∗) ∩ Γ, so that

k bDk_L2(bΓ)≤ kDx^∗,δkL²(Γ) and W ( bD) ⊂ W (D_x^∗_,δ),

by (2.5), where bD denotes the double-layer potential operator on bΓ. Furthermore, by Lemma4.8, k bDk_L2(bΓ)= k eDk_L2(eΓ) and W ( bD) = W ( eD),

so that

kD_x^∗_,δk_L2(Γ)≥ kD_Nk₂ and W (D_x^∗_,δ) ⊃ W (D_N),

for N = 1, ..., 2m. Since this holds for every m ∈ N and δ > 0, the proof is complete.