Elliptic boundary value problems in convex polygons with low regularity boundary data via the unified method

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by  A. C. L. Ashton

ELLIPTIC BOUNDARY VALUE PROBLEMS IN CONVEX
POLYGONS WITH LOW REGULARITY BOUNDARY
DATA VIA THE UNIFIED METHOD
A.C.L. ASHTON AND A.S. FOKAS
Abstract. Weusenovelintegralrepresentationsdevelopedbythesec-
ondauthortoprovecertainrigorousresultsconcerningellipticboundary
3 value problems in convex polygons. Central to this approach is the so-
1 calledglobalrelation,whichisanon-localequationintheFourierspace
0 thatrelatestheknownboundarydatatotheunknownboundaryvalues.
2
Assuming that the global relation is satisfied in the weakest possible

n sense, i.e. in a distributional sense, we prove there exist solutions to
a
Dirichlet, Neumann and Robin boundary value problems with distri-
J
butional boundary data. We also show that the analysis of the global
8
relationcharacterisesinastraightforwardmannerthepossibleexistence

] of both integrable and non-integrable corner-singularities.
P
A
.
h
at 1. Introduction
m
In this paper we present a new means to study rigorous aspects of bound-
[
ary value problems for elliptic PDEs in convex polygons. We concentrate

1
on the basic elliptic equation
v
0
9 (1) −∆q+β2q = 0,
4
1 where β2 > 0 (modified Helmholtz). The case β = 0 (Laplace) was studied
.
1 in [9] for continuous boundary data and can be extended to distributional
0
boundary data using the methods presented here. All our results hold if
3
1 β2 ≤ 0 (Helmholtz or Laplace), but for economy of presentation we state
: our results for β2 > 0 only.
v
i We study (a) existence of solutions with distributional boundary values,
X
(b) a priori estimates for boundary data regularity and (c) corner singu-
r
a larities. The approach we use is based on the formal results obtained in
[6] where novel integral representations for solutions to the basic elliptic
equations in a convex polygon were derived, under the assumption that the
solution exists. The rigorous methodology is based on the following pro-
cedure: we define a function using the novel integral representations of [6]
and show that this function satisfies the relevant PDE and converges to the
desired data on the boundary provided that the global relation is satisfied.
Date: December 11, 2013.
The first author is grateful for the support of Emmanuel College, University of
Cambridge.
1
2 A.C.L.ASHTONANDA.S.FOKAS
Regarding (a), we first generalise the integral representations in [6] so
they make sense for distributional boundary values. We then show that
the function defined by the associated integral representation satisfies the
boundary value problem with distributional boundary values provided that
theunknownboundaryvaluessatisfytheglobalrelationinthedistributional
sense.
Regarding (b), we provide classical a priori estimates for the unknown
boundary data by analysing the global relation.
Regarding (c), we use the global relation to prove the existence of corner
singularities when certain mixed boundary data are prescribed at adjacent
edges of the polygon. The arguments are based on simple asymptotic eval-
uation of terms in the global relation.
In this paper we do not address the fundamental question of existence
of solution to the global relation. It is known that if Dirichlet data is in
Hs+1/2(∂Ω) with s ∈ [−1, 1], then the global relation uniquely defines the
2 2
unknown boundary values, which belongs to Hs−1/2(∂Ω) [2]. A new, more
direct method for solving the Dirichlet-Neumann map via the global rela-
tion was presented in [3] for Dirichlet Data in H1(∂Ω). This method can
be extended to boundary data with lower regularity. We expect that if one
works on a suitable quotient space, then regularity results for the Dirichlet-
Neumann map will hold for general distributional boundary data. The rele-
vant space quotients out finite linear combinations of Dirac measures at the
vertices (the relevance of this space is discussed in Theorem 1).
For a more classical approach to boundary value problems with distribu-
tional boundary data, we refer the reader to [4, 5] and references therein.
Severalnumericaltechniquesfortheimplementationoftheunifiedmethod
of [6, 7] applied to linear elliptic PDEs are discussed in [8, 9, 15, 16].
2. The Spaces of distributions
Let Ω ⊂ R2 (cid:39) C denote the interior of a polygon with vertices {z }n
i i=1
and sides Γ = (z ,z ). Let α = arg(z −z ) denote the angle the side
i i i+1 i i+1 i
Γ makes with the positive real axis and let |Γ | be the length of the side
i i
Γ . It was shown in [6] that if one assumes that there exists a solution
i
to (1) in Ω with sufficient smoothness, then the solution has the following
representation:
n (cid:90)
(2) q(z,z¯) = 1 (cid:88) eiλz−iβ2z¯/λρ (λ) dλ,
i
4πi λ
i=1 (cid:96)i
where the functions {ρ (λ)}n , called spectral functions, are defined by
i i=1
(cid:90) (cid:20)(cid:18)∂q (cid:19) (cid:18)iβ2 ∂q (cid:19) (cid:21)
(3) ρ (λ) = e−iλz(cid:48)+iβ2z¯(cid:48)/λ +iλq dz(cid:48)+ q− dz¯(cid:48)
i ∂z(cid:48) λ ∂z¯(cid:48)
Γi
and {(cid:96) }n denote the rays in the complex plane with arg(λ| ) = −α ,
i i=1 (cid:96)i i
orientated towards infinity. Furthermore, the spectral functions satisfy the
DISTRIBUTIONAL BOUNDARY DATA 3
global relation
n
(cid:88)
(4) ρ (λ) = 0.
i
i=1
In what follows we will first generalise these results so that they make sense
fordistributions(generalisedfunctions). Inthisrespectwewillusestandard
results from the theory of distributions, a comprehensive account of which
can be found in [10]. We will be interested in distributions supported on the
closure of the edges Γ (the edges Γ do not contain the vertices). We use
i i
E[0,1] to denote the space of smooth functions from [0,1] to C. This space
is endowed with a topology generated from the semi-norms
(cid:12) (cid:12)
δ (ϕ) = max (cid:12)ϕ(n)(τ)(cid:12), n = 0,1,2,...,
n (cid:12) (cid:12)
τ∈[0,1]
a sequence of functions ϕ → 0 in E[0,1] if and only if δ (ϕ ) → 0 for all
k n k
n ∈ N∪{0}. This is a Frech´et space and we denote its topological dual by
E(cid:48)[0,1], which is the space of continuous linear forms from E[0,1] to C. The
pairing between E(cid:48)[0,1] and E[0,1] is denoted by (cid:104)·,·(cid:105). It is well known that
a linear form u : E[0,1] → C is an element of E(cid:48)[0,1] if and only if there are
constants N and C such that
(cid:88)
(5) |(cid:104)u,ϕ(cid:105)| ≤ C δ (ϕ),
n
n≤N
for all ϕ ∈ E[0,1].
We will also make use of the less familiar space of Schwartz distributions
on an open interval. To define this space of distributions, we first setup the
appropriate space of test functions. Let S(0,1) denote the space of smooth
functions from (0,1) to C such that
σ (ϕ) = sup |τ−m(1−τ)−mϕ(n)(τ)| < ∞, n,m = 0,1,2,...,
mn
τ∈(0,1)
so that these functions vanish to all orders at the boundary of [0,1]. The
topology generated by these semi-norms makes S(0,1) a Frech´et space. The
topological dual, consisting of continuous linear forms from S(0,1) to C, is
denoted by S(cid:48)(0,1). A linear form v : S(0,1) → C is an element of S(cid:48)(0,1)
if and only if there exist constants D and M such that
(cid:88)
|(cid:104)v,ϕ(cid:105)| ≤ D σ (ϕ),
mn
m,n≤M
for all ϕ ∈ S(0,1). For ϕ ∈ S(0,1) we define the Fourier transform by
(cid:90) 1
ϕˆ(λ) = e−iλτϕ(τ)dτ.
0
If ϕˇ(x) = ϕ(−x), the Fourier inversion theorem can be written as 2πϕ =
((ϕˆ)ˆ)ˇ. Using the natural extension of an element of E(cid:48)[0,1] to E(cid:48)(R), the
4 A.C.L.ASHTONANDA.S.FOKAS
space of distributions on R with compact support, we can define the Fourier
transform on E(cid:48)[0,1] via
(cid:68) (cid:69)
uˆ(λ) = u ,e−iλτ .
τ
See[10,Ch. 7]forthestandardtreatmentoftheFouriertransformonE(cid:48)(R).
We now generalise the integral representation (2) so it makes sense for
distributional boundary values. To facilitate this generalisation we need a
local description of the edges Γ so that we can define distributions on them.
i
We use the local parametrisations ψ : [0,1] → Γ , with
i i
ψ : τ (cid:55)→ τz +(1−τ)z .
i i+1 i
For a function f : Γ → C we write its pullback by ψ via
i i
ψ∗(f)(τ) = f(ψ(τ)).
i
1-forms are pulled back by ψ via
i
ψ∗(dz) = eiαi|Γ |dτ, ψ∗(dz¯) = e−iαi|Γ |dτ.
i i i i
We now define the spectral functions ρ (λ) by
i
ρ (λ) (cid:68) (cid:16) (cid:17)(cid:69)
(6) i = ∂ q ,ψ∗ e−iλz(cid:48)+iβ2z¯(cid:48)/λ
i|Γ | n i i
i
(cid:18) β2 (cid:19)(cid:68) (cid:16) (cid:17)(cid:69)
+ λeiαi + q ,ψ∗ e−iλz(cid:48)+iβ2z¯(cid:48)/λ ,
λeiαi i i
where ∂ q ,q ∈ E(cid:48)[0,1] for i = 1,...,n. The first term is motivated by the
n i i
identification
(cid:90) (cid:18)∂q ∂q (cid:19) (cid:90) 1 (cid:20) ∂q(cid:21)
e−iλz(cid:48)+iβ2z¯(cid:48)/λ dz(cid:48)− dz¯(cid:48) = |Γ | ψ∗ e−iλz(cid:48)+iβ2z¯(cid:48)/λ (τ)dτ
∂z(cid:48) ∂z¯(cid:48) i i ∂n
Γi 0
(cid:68) (cid:16) (cid:17)(cid:69)
(cid:32) |Γ | ∂ q ,ψ∗ e−iλz(cid:48)+iβ2z¯(cid:48)/λ .
i n i i
In this calculation we have used the identities
∂q (cid:12)(cid:12) e−iαi (cid:18)∂q ∂q(cid:19)(cid:12)(cid:12) ∂q (cid:12)(cid:12) eiαi (cid:18)∂q ∂q(cid:19)(cid:12)(cid:12)
(cid:12) = +i (cid:12) , (cid:12) = −i (cid:12) ,
∂z(cid:48)(cid:12) 2 ∂t ∂n (cid:12) ∂z¯(cid:48)(cid:12) 2 ∂t ∂n (cid:12)
Γi Γi Γi Γi
where∂/∂tand∂/∂narederivativesinthetangentialand(outward)normal
directions along Γ . The second term in (6) is similarly motivated, e.g.
i
iβ2 (cid:90) e−iλz(cid:48)+iβ2z¯(cid:48)/λqdz¯(cid:48) = iβ2|Γi|e−iαi (cid:90) 1ψ∗(cid:104)e−iλz(cid:48)+iβ2z¯(cid:48)/λq(cid:105)(τ)dτ
λ λ i
Γi 0
(cid:32) iβ2|Γi|e−iαi (cid:68)q ,ψ∗(cid:16)e−iλz(cid:48)+iβ2z¯(cid:48)/λ(cid:17)(cid:69).
λ i i
So the formal identifications are
(cid:18) (cid:19)
∂q
∂ q (cid:32) ψ∗ , q (cid:32) ψ∗(q).
n i i ∂n i i
DISTRIBUTIONAL BOUNDARY DATA 5
Γ
i
Γ(cid:15) (cid:15)
i
∂Ω
∂Ω
(cid:15)
Figure 1.
The boundaries ∂Ω and ∂Ω .
(cid:15)
3. Distributional Boundary Data
In this section we will outline how to setup a given boundary value prob-
lem so that the boundary data can be distributional. Our aim is to show
that the function defined by (2), where the {ρ }n are given in (6), satisfies
i i=1
the following boundary value problem:
(7a) −∆q+β2q = 0 in Ω
(7b) lim q = q in S(cid:48)(Γ ) for i = 1,...,n,
i i
z→Γi
where q ∈ E(cid:48)[0,1]. We need to specify a notion of convergence, via which
i
the solution (2) converges to a distribution on Γ .
i
For a given polygon Ω we inscribe within it a one parameter family of
polygons
Ω = {z ∈ Ω : dist(z,∂Ω) ≥ (cid:15)}.
(cid:15)
For (cid:15) > 0 sufficiently small we have ∂Ω (cid:39) ∂Ω and we can define the edges
(cid:15)
of ∂Ω by Γ(cid:15) for i = 1,...,n as shown in Figure 3. Now suppose q ∈ C∞(Ω)
(cid:15) i
is a given function. Using a local parametrisation ψ(cid:15) : [0,1] → Γ(cid:15), we can
i i
define a distribution q(cid:15) ∈ S(cid:48)(0,1) by
i
(cid:90) 1
(8) (cid:104)q(cid:15),ϕ(cid:105) = (ψ(cid:15))∗(q)(τ)ϕ(τ)dτ,
i i
0
for all ϕ ∈ S(0,1). As (cid:15) ↓ 0 we have Γ(cid:15) → Γ and ψ(cid:15) → ψ . This limit
i i i i
and its equivalence to others modes of convergence for boundary values
are discussed in [4] (cf. [14]). Our definition of q ∈ C∞(Ω) tending to a
distribution on Γ is
i
(cid:110) (cid:111) (cid:110) (cid:111)
lim q = q in S(cid:48)(Γ ) ⇔ limq(cid:15) = q in S(cid:48)(0,1) .
i i i i
z→Γi (cid:15)→0
These limits are understood in the topology of S(cid:48)(0,1). Recall that our
boundary datum is an element of E(cid:48)[0,1]; on the other hand we are taking
limits in the topology of S(cid:48)(0,1) and not in the topology of E(cid:48)[0,1] (the test
6 A.C.L.ASHTONANDA.S.FOKAS
function appearing in (8) is an element of S(0,1) and not of E[0,1]). Thus,
we must make sure that we do not lose any information in this limit process.
In this respect we note S(0,1) ⊂ E[0,1] and that the inclusion is con-
tinuous1. Hence E(cid:48)[0,1] (cid:44)→ S(cid:48)(0,1). Therefore there is a natural projection
π : E(cid:48)[0,1] → S(cid:48)(0,1), which is obtained by restricting u ∈ E(cid:48)[0,1] to test
functions in S(0,1). Given a linear form v ∈ S(cid:48)(0,1), the Hahn-Banach the-
oremimpliesthatwecanextendthislinearformtoalinearformu ∈ E(cid:48)[0,1]
0
with π(u ) = v. The general solution to the equation π(u) = v is [13, p.
0
178]
N N
(cid:88) (k) (cid:88) (k)
u = u + a δ + a δ .
0 k 0 k 1
k=0 k=0
Thisobservationmeansthatthedifferencebetweenu ∈ E(cid:48)[0,1]andπ(u)can
only be a collection of Dirac measures and derivatives thereof supported at
the boundary of [0,1]. Intuitively speaking this is because the test functions
in S(0,1) vanish to all orders at the boundary of [0,1], so they are unable
to “detect” a distribution in E(cid:48)[0,1] with support contained in the boundary
of [0,1]. These observations imply that knowledge of
lim q = q in S(cid:48)(Γ )
i i
z→Γi
yields
N
(cid:88)(cid:16) (cid:17)
(9) lim q = q + a δ(j)+b δ(j) in E(cid:48)(Γ ),
z→Γi i j zi j zi+1 i
j=0
for some constants {a ,b }N and some integer N. The possibility of the
j j j=0
existence of the above additional contributions to the right hand side of (9)
does not affect the answer to the questions of existence of a solution to (7),
since we can always consider the new function (distribution)
n N
(cid:88)(cid:88)
q˜= q− c δ(j),
ij zi
i=1 j=0
for appropriate {c } which satisfies the relevant PDE in Ω and converges to
ij
the desired distribution in E(cid:48)[0,1].
The following result shows that S(cid:48)(0,1) is, in some sense, the right space
of distributions to take limits in.
Theorem 1. Let ∆[0,1] denote the space of all finite linear combinations
of Dirac measures and derivatives thereof supported at 0 and 1. Given q ∈
i
E(cid:48)[0,1] (i = 1,...,n), there is at most one of each ∂ q ∈ E(cid:48)[0,1]/∆[0,1]
n i
(i = 1,...,n) such that the global relation is satisfied.
This result implies that a solution of the global relation is unique modulo
Dirac measures at the vertices of the polygon. By taking limits in S(cid:48)(0,1)
1Meaningthatifasequence{ϕ } tendstozeroinS(0,1)(i.e. σ (ϕ )→0forall
k k≥1 mn k
m,n), it also tends to zero in E[0,1] (i.e. δ (ϕ )→0 for all n).
n k
DISTRIBUTIONAL BOUNDARY DATA 7
we effectively quotient out this degeneracy. In this sense, the topology on
S(cid:48)(0,1) is the right one to take limits in for distributional boundary data. A
moredetailedtreatmentofdistributionalsolutionstotheglobalrelationwill
be presented elsewhere, but the interested reader may look at [3, Lemma 4]
for an idea of the proof.
To prove our main result we will need to interchange the order of some
integrals. To this end, the following simple lemma is essential.
Lemma 1. For z ∈ Ω and for each i, the function
1
eiλz−iβ2z¯/λρ (λ)
i
λ
decays exponentially as |λ| → 0 and |λ| → ∞ along the ray (cid:96) .
i
Proof. Fromthesemi-normestimates(5)weknowthatthereexistconstants
C,N such that
|ρ (λ)| ≤ C(cid:88)N (cid:18)|λ|+ β2(cid:19)nmax(cid:12)(cid:12)e−iλz(cid:48)+iβ2z¯(cid:48)/λ(cid:12)(cid:12).
i n=0 |λ| cl(Γi)(cid:12) (cid:12)
The proof now follows from calculations analogous to those in [9, Lemma
1]. (cid:4)
From this observation it is clear that the double integrals
(cid:34) (cid:35)
(cid:90) 1 (cid:90) dλ
ϕ(τ)(ψ(cid:15))∗ eiλz−iβ2z¯/λρ (λ) (τ)dτ
i j λ
0 (cid:96)j
convergeabsolutelyforeach1 ≤ i,j ≤ nand(cid:15) > 0. HenceapplyingFubini’s
theorem we can interchange the order of integration
(cid:90) (cid:20)(cid:90) 1 (cid:16) (cid:17) (cid:21) dλ
ρ (λ) (ψ(cid:15))∗ eiλz−iβ2z¯/λ ϕ(τ)dτ .
j i λ
(cid:96)j 0
Lemma 2. Let (cid:96)ˆ denote the ray for which arg(λ| ) = π−α , so that (cid:96)ˆ is
i (cid:96)ˆi i i
the continuation of the ray (cid:96) . Then, for j (cid:54)= i:
i
(cid:90) (cid:20)(cid:90) 1 (cid:16) (cid:17) (cid:21) dλ
ρ (λ) ψ∗ eiλz−iβ2z¯/λ ϕ(τ)dτ
j i λ
(cid:96)j 0
(cid:90) (cid:20)(cid:90) 1 (cid:16) (cid:17) (cid:21) dλ
= ρ (λ) ψ∗ eiλz−iβ2z¯/λ ϕ(τ)dτ .
(cid:96)ˆi j 0 i λ
Proof. In order to be able to deform contours in the complex plane, we need
to establish some estimates for the integrand when |λ| is either small or
large (Cauchy’s theorem will then provide the result since the integrand is
analytic in any punctured disc centred at the origin). In this respect, for
8 A.C.L.ASHTONANDA.S.FOKAS
each m ≥ 0 and some constant C we have the following estimates:
m
(cid:12)(cid:12)(cid:12)(cid:90) 1ψ∗(cid:16)eiλz−iβ2z¯/λ(cid:17)ϕ(τ)dτ(cid:12)(cid:12)(cid:12)
(cid:12) i (cid:12)
0
≤ Cm(cid:18)|λ|+ |βλ2|(cid:19)−m(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) 1ψi∗(cid:16)eiλz−iβ2z¯/λ(cid:17)ϕ(m)(τ)dτ(cid:12)(cid:12)(cid:12)(cid:12)
0
(cid:18) β2(cid:19)−m (cid:12) (cid:12)
≤ C |λ|+ max(cid:12)eiλz−iβ2z¯/λ(cid:12).
m |λ| cl(Γi)(cid:12) (cid:12)
This estimate follows from repeated integration by parts, using the fact that
ϕ and its derivatives vanish to all orders at the boundary ∂Γ . The semi-
i
norm estimates used in Lemma 1 imply that for m and C as above, we
m
have
(cid:12)(cid:12)(cid:12)(cid:12)ρj(λ)(cid:90) 1ψi∗(cid:16)eiλz−iβ2z¯/λ(cid:17)ϕ(τ)dτ(cid:12)(cid:12)(cid:12)(cid:12)
0
(cid:18) β2(cid:19)−m (cid:12) (cid:12)
≤ C |λ|+ max max (cid:12)eiλ(z−z(cid:48))−iβ2(z¯−z¯(cid:48))/λ(cid:12).
m |λ| z∈cl(Γi)z(cid:48)∈cl(Γj)(cid:12) (cid:12)
Observe that
(10)
(cid:12) (cid:12) (cid:18) (cid:18) β2(cid:19) (cid:19)
(cid:12)eiλ(z−z(cid:48))−iβ2(z¯−z¯(cid:48))/λ(cid:12) = exp −|z−z(cid:48)| |λ|+ sin(arg(λ(z−z(cid:48)))) .
(cid:12) (cid:12) |λ|
If |α −α | = π, then (cid:96) and (cid:96)ˆ coincide, thus there is nothing to prove. Let
i j j i
us suppose 0 < α −α < π. This condition together with convexity implies
i j
α −π < α < arg(z−z(cid:48)) < α +π.
i j j
Hence, if λ satisfies π−α ≤ argλ ≤ −α , we find
i j
0 ≤ arg(λ(z−z(cid:48))) ≤ π.
It follows now from (10) that the ray (cid:96) can be deformed onto (cid:96)ˆ. The case
j i
0 < α −α < π is similar. (cid:4)
j i
Theorem 2. Define the function Q = Q(z,z¯) by
n (cid:90)
Q(z,z¯) = 1 (cid:88) eiλz−iβ2z¯/λρ (λ) dλ
i
4πi λ
i=1 (cid:96)i
with {ρ (λ)}n defined in (6), where q ,∂ q ∈ E(cid:48)[0,1] for i = 1,...,n. If
i i=1 i n i
the global relation (4) is satisfied, then Q solves the following boundary value
problem:
(11a) −∆Q+β2Q = 0 in Ω
(11b) lim Q = q in S(cid:48)(Γ ) for i = 1,...,n.
i i
z→Γi
Proof. It is straightforward to show that Q satisfies the relevant equation
at any interior point of Ω, since exponential convergence justifies differenti-
ating under the integral sign. Thus, we focus on the behaviour of Q at the
boundary. Without loss of generality we position the vertex z at the origin
i
DISTRIBUTIONAL BOUNDARY DATA 9
and align the side Γ with the positive real z-axis so that α = 0. Our aim
i i
is to compute the limit
(cid:90) 1
lim(cid:104)Q(cid:15),ϕ(cid:105) = lim (ψ(cid:15))∗(Q)(τ)ϕ(τ)dτ.
i i
(cid:15)→0 (cid:15)→0 0
We have already established that the double integral converges absolutely,
so interchanging the order of integration is justified. Thus, our task is to
compute the limit
lim 1 (cid:88)n (cid:90) ρ (λ)(cid:20)(cid:90) 1(ψ(cid:15))∗(cid:16)eiλz−iβ2z¯/λ(cid:17)ϕ(τ)dτ(cid:21) dλ.
(cid:15)→0 4πi j=1 (cid:96)j j 0 i λ
It is simple to bound each of the integrands by an absolutely integrable
function (independent of (cid:15)), employing the same integration by parts type
estimates used in Lemma 2. So the dominated convergence theorem applies
and we can pass the limit through the integral signs. Hence,
lim(cid:104)Q(cid:15),ϕ(cid:105) = 1 (cid:88)n (cid:90) ρ (λ)(cid:20)(cid:90) 1ψ∗(cid:16)eiλz−iβ2z¯/λ(cid:17)ϕ(τ)dτ(cid:21) dλ
(cid:15)→0 i 4πi j=1 (cid:96)j j 0 i λ
1 (cid:90) (cid:20)(cid:90) 1 (cid:16) (cid:17) (cid:21) dλ
= ρ (λ) ψ∗ eiλz−iβ2z¯/λ ϕ(τ)dτ
4πi i i λ
(cid:96)i 0
+ 1 (cid:88)(cid:90) ρ (λ)(cid:20)(cid:90) 1ψ∗(cid:16)eiλz−iβ2z¯/λ(cid:17)ϕ(τ)dτ(cid:21) dλ.
4πi j i λ
j(cid:54)=i (cid:96)j 0
Using Lemma 2 and the global relation (4) we find
1 (cid:88)(cid:90) ρ (λ)(cid:20)(cid:90) 1ψ∗(cid:16)eiλz−iβ2z¯/λ(cid:17)ϕ(τ)dτ(cid:21) dλ
4πi j i λ
j(cid:54)=i (cid:96)j 0
(cid:32) (cid:33)
= 1 (cid:90) (cid:88)ρ (λ) (cid:20)(cid:90) 1ψ∗(cid:16)eiλz−iβ2z¯/λ(cid:17)ϕ(τ)dτ(cid:21) dλ
4πi (cid:96)ˆi j(cid:54)=i j 0 i λ
1 (cid:90) (cid:20)(cid:90) 1 (cid:16) (cid:17) (cid:21) dλ
= − ρ (λ) ψ∗ eiλz−iβ2z¯/λ ϕ(τ)dτ .
4πi (cid:96)ˆi i 0 i λ
Hence
(12)
1 (cid:18)(cid:90) (cid:90) (cid:19) (cid:20)(cid:90) 1 (cid:16) (cid:17) (cid:21) dλ
lim(cid:104)Q(cid:15),ϕ(cid:105) = − ρ (λ) ψ∗ eiλz−iβ2z¯/λ ϕ(τ)dτ .
(cid:15)→0 i 4πi (cid:96)i (cid:96)ˆi i 0 i λ
Since α = 0, we can parametrise the λ-integrals in (12) so that (cid:96) is the
i i
positive real axis and (cid:96)ˆ is the negative real axis:
i
4πi lim(cid:104)Q(cid:15),ϕ(cid:105)
i
(cid:15)→0
= (cid:90) ∞ϕˆ(cid:16)|Γ |(cid:16)β2 −λ(cid:17)(cid:17)ρ (λ) dλ −(cid:90) −∞ϕˆ(cid:16)|Γ |(cid:16)β2 −λ(cid:17)(cid:17)ρ (λ) dλ
i λ i λ i λ i λ
0 0
= (cid:90) ∞ϕˆ(cid:16)|Γ |(cid:16)β2 −λ(cid:17)(cid:17)ρ (λ) dλ −(cid:90) ∞ϕˆ(cid:16)|Γ |(cid:16)λ− β2(cid:17)(cid:17)ρ (−λ) dλ.
i λ i λ i λ i λ
0 0
10 A.C.L.ASHTONANDA.S.FOKAS
In the first integral we make the change of variables k = |Γ |(λ−β2/λ), i.e.
i
1 (cid:16) (cid:112) (cid:17) dλ dk
2λ = k+ k2+4|Γ |2β2 , = ,
i (cid:112)
|Γi| λ k2+4|Γi|2β2
where we have chosen the positive square root so that λ ≥ 0. Similarly, in
the second integral we make the change of variables k = −|Γ |(λ−β2/λ),
i
i.e.
1 (cid:16) (cid:112) (cid:17) dλ dk
2λ = −k+ k2+4|Γ |2β2 , = − .
i (cid:112)
|Γi| λ k2+4|Γi|2β2
Hence
(cid:32) (cid:33)
(cid:90) ∞ i|Γ |(∂ q )ˆ(k)
4πi lim(cid:104)Q(cid:15),ϕ(cid:105) = ϕˆ(−k) i n i +iqˆ(k) dk
(cid:15)→0 i −∞ (cid:112)k2+4|Γi|2β2 i
(cid:32) (cid:33)
(cid:90) ∞ i|Γ |(∂ q )ˆ(k)
(13) − ϕˆ(−k) i n i −iqˆ(k) dk.
(cid:112) i
−∞ k2+4|Γi|2β2
We note that the above integrals are well defined for q ,∂ q ∈ E(cid:48)[0,1],
i n i
since by the semi-norm estimates (5), the corresponding Fourier transforms
are entire functions of polynomial growth on the real line, . We find the
contributionsfromthederivativescancel,andweareleftwiththeexpression
1 (cid:90) ∞
(14) lim(cid:104)Q(cid:15),ϕ(cid:105) = ϕˆ(−k)qˆ(k)dk.
(cid:15)→0 i 2π −∞ i
Now ϕˆ ∈ S(R) and since q ∈ E(cid:48)[0,1] (cid:44)→ S(cid:48)(R), we can apply the Fourier
i
inversion theorem on S(cid:48)(R) to find
1
lim(cid:104)Q(cid:15),ϕ(cid:105) = (cid:104)qˆ,(ϕˆ)ˇ(cid:105) = (cid:104)q ,ϕ(cid:105).
(cid:15)→0 i 2π i i
Hence lim Q = q in S(cid:48)(Γ ). (cid:4)
z→Γi i i
Remark 1. By performing a similar calculation it is possible to determine
the limit of ∂Q/∂n, where n is the normal to Γ(cid:15) (cid:39) Γ . In this case the
i i
corresponding terms for qˆ in (13) cancel and we find
i
lim(cid:104)(∂Q/∂n)(cid:15),ϕ(cid:105) = (cid:104)∂ q ,ϕ(cid:105).
i n i
(cid:15)→0
Thus, undertheassumptionthattheglobalrelationissatisfied, itispossible
by using (2), to prove existence for the Dirichlet, the Neumann or the Robin
boundary value problems.
Remark 2. Central to this argument was the statement that our domain
can be placed in the z-plane so that the vertex at z = z lies at the origin.
i
We certainly do not lose any generality by enforcing this situation. Indeed,
suppose a priori that one is interested in the behaviour of the boundary
values at a particular edge of the domain boundary. Then, after this edge
has been singled out, the domain can be embedded into C (cid:39) R2 in any way
we find convenient. Alternatively, one can check directly that the integral
representation (2) is invariant under rotations and translations in the z-
plane.