Volume 2012, Article ID 340310,18pages doi:10.1155/2012/340310
Research Article
The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges
P. A. Krutitskii
KIAM, Miusskaya Square 4, Moscow 125047, Russia
Correspondence should be addressed to P. A. Krutitskii,[email protected] Received 30 March 2012; Accepted 3 May 2012
Academic Editor: Vladimir Mityushev
Copyrightq2012 P. A. Krutitskii. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The Dirichlet problem for the 2D Helmholtz equation in an exterior domain with cracks is studied.
The compatibility conditions at the tips of the cracks are assumed. The existence of a unique classical solution is proved by potential theory. The integral representation for a solution in the form of potentials is obtained. The problem is reduced to the Fredholm equation of the second kind and of index zero, which is uniquely solvable. The asymptotic formulae describing singularities of a solution gradient at the edgesendpointsof the cracks are presented. The weak solution to the problem may not exist, since the problem is studied under such conditions that do not ensure existence of a weak solution.
1. Introduction
The 2D Dirichlet boundary value problem for the Helmholtz equation in an exterior multiply connected domain bounded by closed curves is considered in monographs on mathematical physics, for instance, in1–3. The review on studies of the Dirichlet problem for this equation in the exterior of cracks is given in4. The present paper is an attempt to combine these problems and to consider exterior domains containing cracks. From a practical stand point such domains have great significance, because cracks model both cracks in solids and wings or double-sided screens in fluids.
So, we study Dirichlet problem in an exterior domain bounded by closed curves and cracks. The theorems on existence and uniqueness of a classical solution are proved. The integral representation for a solution to a problem in the form of potentials is obtained. The problem is reduced to the uniquely solvable Fredholm integral equation of the second kind and index zero. To derive uniquely solvable integral equation on the whole boundary we use
the modified integral equation approach5,6on the closed curves. Since the derived integral equation of the 2nd kind is uniquely solvable, we may obtain its numerical solution in a very simple way, just by discretization and inversion of the matrix. Substituting numerical solution of the integral equation into potentials, we obtain numerical solution to the exterior Dirichlet problem in a very simple way as well. The integral representation for a solution presented in the present paper enables us to derive asymptotic formulae for singularities of a gradient of the solution at the tips of the cracks.
The Dirichlet problem for the Helmholtz equation in the exterior of several closed curves in a plane and the Dirichlet problem for the Helmholtz equation in the exterior of several curvilinear cracks in a plane are particular cases of our problem.
It is important to stress that the boundary data on the closed curves in the present paper is assumed to be only continuous. This means that weak solution may not exist in our problem, though classical solution exists. In other words, the problem in this paper is studied under conditions, which are not sufficient for existence of a weak solution inHloc1 and weak solution may not exist, but these conditions are sufficient for existence of a classical solution.
This curious fact follows from the Hadamard example of a nonexistence of a weak solution to the Dirichlet problem for Laplacian in the unit disc with continuous boundary dataclassical solution exists in this example. Roughly speaking, continuity of a Dirichlet data on smooth closed curves does not ensure existence of a weak solution in the Dirichlet problem. The Hadamard example of existence of a classical solution and nonexistence of a weak solution is presented and is discussed in the book7, section 12.5by Sobolev himself, who invented Sobolev’s spaces for analysis of weak solvability of boundary value problems.
Numerical methods for the Dirichlet and Neumann problems for the Laplace and Helmholtz equations in the exterior of cracks in a plane have been developed in 8,9on the basis of boundary integral equations. Numerical simulation for engineering problems with cracks is presented in10–12using boundary element method. Problems with a crack in electromagnetoelasticity have been reduced to integral equations in13, and numerical solutions for some model problems have been obtained. The Dirichlet problem for elasticity equations in an exterior of several arbitrary curvilinear cracks in a plane has been reduced to the uniquely solvable integral equations in14.
2. Formulation of the Problem
By an open curve we mean a simple smooth nonclosed arc of finite length without self- intersections15. In the planex x1, x2∈R2we consider an exterior multiply connected domain bounded by simple open curves Γ11, . . . ,Γ1N1 ∈ C2,λ and simple closed curves Γ21, . . . ,Γ2N
2 ∈ C2,λ, λ ∈ 0,1, so that the curves do not have common points; in particular, they do not have common endpoints. Suppose that N1 N2 > 0. We set Γ1 N1
n1Γ1n, Γ2N2
n1Γ2n, andΓ Γ1∪Γ2. The exterior connected domain bounded byΓ2will be denoted byD. We assume that each curveΓknis parametrized by the arc lengths:
Γkn
x:xxs x1s, x2s, s∈
akn, bkn
, n1, . . . , Nk, k1,2, 2.1
so that a11 < b11 < · · · < a1N1 < b1N1 < a21 < b12 < · · · < a2N2 < b2N2 and the domainD is to the right when the parameter sincreases on Γ2n. Therefore points x ∈ Γ and values of the parameters are in one-to-one correspondence except fora2n andb2n, which correspond
Γ21 Γ22
Γ11
Γ12
Figure 1: An example of an exterior domain.
to the same point x for n 1, . . . , N2. Below the sets of the intervals on the Os axis N1
n1a1n, b1n, N2
n1a2n, b2n, 2
k1Nk
n1akn, bknwill be denoted byΓ1,Γ2, andΓalso.
We setC0Γ2n {Fs:Fs∈C0a2n, b2n, Fa2n Fb2n}, and
C0 Γ2
N2
n1
C0 Γ2n
. 2.2
The tangent vector toΓat the pointxsis denoted byτx cosαs,sinαs, where cosαs x1s, and sinαs x2s. Let nx sinαs,−cosαsbe the normal vector toΓ atxs. The direction of nxis chosen such that it will coincide with the direction ofτxif nxis rotated anticlockwise through an angle ofπ/2.
We considerΓ1as a set of cracks. The side ofΓ1which is on the left, when the parameter sincreases, will be denoted byΓ1 and the opposite side will be denoted byΓ1−.
We say that the functionwxbelongs to the smoothness class K if 1w∈C0D \Γ1∩C2D \Γ1,
2∇w ∈ C0D \Γ1 \ Γ2 \X, where X is a point-set, consisting of the endpoints ofΓ1 :X N1
n1xa1n∪xb1n,
3in the neighbourhood of any endpointxd∈X for some constantsC> 0, >−1 the inequality
|∇w| ≤ C|x−xd| 2.3
holds, wherex → xdandda1nordbn1forn1, . . . , N1.
Remark 2.1. ByC0D \Γ1 \Γ2\ Xwe denote the class of continuous in D \Γ1 functions, which are continuously extensible to the sides of the cracksΓ1\Xfrom the left and from the right, but their limiting values onΓ1\Xcan be different from the left and from the right, so that these functions may have a jump onΓ1\X. The functions of class C0D \Γ1\Γ2\X belong to the classC0D \Γ1if they are continuously extensible toΓ2fromDand if they are continuously extensible to the tips of the cracksΓ1.
Let us formulate the Dirichlet problem for the Helmholtz equation in the exterior domainD \Γ1seeFigure 1.
ProblemU. Find a functionuxof the class K which satisfies the Helmholtz equation:
ux1x1x ux2x2x β2ux 0, x∈ D \Γ1, β const >0, 2.4a
the boundary conditions:
uxs|Γ1 F s, uxs|Γ1−F−s, uxs|Γ2Fs, 2.4b
and the radiating conditions at infinity:
uO
|x|−1/2
, ∂u
∂|x|−iβuo
|x|−1/2
, |x|
x12 x22−→ ∞. 2.4c
All conditions of the ProblemUmust be satisfied in the classical sense. Problem U includes two particular cases. In the first particular case, there are no cracksΓ1i.e.,Γ1 ∅, then we get the Dirichlet problem for the Helmholtz equation in the exterior of several closed curvesΓ2in a planesee1–3. In another particular case, there are no closed curvesΓ2i.e., Γ2 ∅, and we obtain the Dirichlet problem for the Helmholtz equation in the exterior of several curvilinear cracksΓ1in a plane4.
By
Γk· · ·dσ we meanNk
n1
bk
n
akn· · ·dσ. On the basis of the Rellich lemma 1, energy equalities2, and the regularity of the solution to the homogeneous Dirichlet problem near the boundaryΓ2see16, lemma 6.18, we can easily prove the following assertion.
Theorem 2.2. IfΓ∈C2,λ, λ∈0,1, then the ProblemUhas at most one solution.
Proof. It is sufficiently to prove that the homogeneous ProblemUadmits the trivial solution only. Letu0xbe a solution to the homogeneous ProblemUwith F s ≡ F−s ≡ 0 and Fs≡0. LetSr be an open disc with a center in the origin and with sufficiently large radius r. Assume thatΓ⊂Sr0for somer0and assume thatr > r0.
SinceΓ2 ∈ C2,λ,u0x ∈ C0D \Γ1 remind that u0x ∈ K, and sinceu0|Γ2 0 ∈ C2,λΓ2, and owing to the lemma on regularity of solutions of elliptic equations near the boundary16, lemma 6.18, we obtainu0x∈C1D \Γ1. Sinceu0x ∈K, we observe that u0x ∈C1D \Γ1\X. We envelope each crackΓ1nn 1, . . . , N1by a closed contour and write first Green’s formula for u0x in a domain, bounded by these contours, by Γ2 and by∂Sr. Then we allow to shrink closed contours onto cracksΓ1 and use smoothness of the functionu0x. In this way we arrive at the identity foru0xin the domainD ∩Sr\Γ1
∇u02L
2D∩Sr\Γ1−β2u02L
2D∩Sr\Γ1
Γ1u0 ∂u0
∂nx
ds−
Γ1u0− ∂u0
∂nx
− ds
−
Γ2u0∂u0
∂nxds
∂Sr
u0∂u0
∂|x|dl,
∗
which is true for anyr > r0, wherer0 is some constant. The curviliniear integral of the 1st kind is taken over∂Sr. Byu0xthe complex conjugate function tou0xis denoted. Clearly, u0xbelongs to class K and satisfies homogeneous boundary conditions.
By the superscripts and−we denote the limiting values of functions onΓ1 and onΓ1−, respectively. Sinceu0xsatisfies the homogeneous boundary condition2.4bonΓ, we rewrite identity∗in the following form:
∇u02L
2D∩Sr\Γ1−β2u02L
2D∩Sr\Γ1
∂Sr
u0x∂u0x
∂|x| dl. 2.5
Taking the imaginary part, we obtain the identity
Im
∂Sr
u0x∂u0x
∂|x| dl0, 2.6
which is true for anyr > r0. Tendingr → ∞in this identity and taking into account condi- tions2.4cat infinity, we obtain
rlim→ ∞Im
∂Sr
u0x∂u0x
∂|x| dlβlim
r→ ∞
∂Sr
|u0|2dl0. 2.7
Sinceβ >0, we have
rlim→ ∞
∂Sr
|u0|2dl0, 2.8
whenceu0x≡0 inD \Γ1on the basis of the Rellich lemma1, section 229. Thus,u0xis a trivial solution to the homogeneous ProblemU. Consequently, the homogeneous ProblemU has only the trivial solution, and the theorem is proved owing to the linearity of the Problem U.
3. Integral Equations at the Boundary
To prove existence of a solution to the ProblemU, we assume that
F s∈C1,λ Γ1
, F−s∈C1,λ Γ1
, Fs∈C0 Γ2
, λ∈0,1, 3.1a F
a1n F−
a1n
, F
b1n F−
bn1
, n1, . . . , N1. 3.1b
The conditions3.1bare compatibility conditions for functionsF sandF−sat the tips of the cracks. To solve ProblemUwe discuss some preliminary matter.
If B1Γ1 and B2Γ2 are Banach spaces of functions given on Γ1 and Γ2, then for functions given on Γ we introduce the Banach space B1Γ1 ∩ B2Γ2 with the norm · B1Γ1∩B2Γ2 · B1Γ1 · B2Γ2.
We consider the angular potential from4for2.4aonΓ1:
v1νx i 4
Γ1νσVx, σdσ. 3.2 The kernelVx, σis defined on each curveΓ1n, n1, . . . , N1, by
Vx, σ σ
a1n
∂
∂nyH10
βx−yξdξ, σ∈ a1n, b1n
, 3.3
whereH10 zis the Hankel function of the first kind3:
H10 z
√2 expiz−iπ/4 π√
z
∞
0
exp−tt−1/2
1 it 2z
−1/2 dt, yyξ
y1ξ, y2ξ , x−yξ
x1−y1ξ2
x2−y2ξ2 .
3.4
Here in after we suppose that νσ belongs toC0,λΓ1 and satisfies the following additional conditions:
b1n
a1n
νσdσ0, n1, . . . , N1. 3.5
As shown in4, for suchνσthe angular potentialv1νxbelongs to the class K. In particular, the condition2.3is satisfied for any∈0,1. Moreover, integratingv1νxby parts and using3.5we express the angular potential in terms of a double-layer potential:
v1νx −i 4
Γ1ρσ ∂
∂nyH10
βx−yσdσ 3.6
with the density
ρσ σ
a1n
νξdξ, σ∈ a1n, b1n
, n1, . . . , N1. 3.7
Consequently,v1νxsatisfies both equation2.4aoutsideΓ1and the conditions at infinity 2.4c.
Let us construct a solution to the ProblemU. This solution can be obtained with the help of potential theory for the Helmholtz equation 2.4a. We look for a solution to the problem in the following form:
u ν, μ
x v1νx w μ
x, 3.8
wherev1νxis given by3.2,3.6, and
w μ
x w1
μ
x w2
μ x, w1
μ x i
4
Γ1μσH10
βx−yσdσ, w2
μ x i
4
Γ2μσ ∂
∂ny −i
H10
βx−yσdσ.
3.9
The densityνσmust belong toC0,λΓ1and must satisfy conditions3.5.
We will look forμsin the Banach spaceCqωΓ1∩C0Γ2, ω∈0,1, q∈0,1with the norm · CωqΓ1∩C0Γ2 · CωqΓ1 · C0Γ2. We say thatμsbelongs to the Banach space CωqΓ1with someω∈0,1andq∈0,1if
μsN1
n1
s−a1nqs−bn1q∈C0,ω Γ1
, 3.10
whereC0,ωΓ1is a H ¨older space with the exponentω. The norm in the Banach spaceCqωΓ1 is defined by
μ·
CωqΓ1 μ·N1
n1
· −a1nq· −b1nq
C0,ωΓ1
. 3.11
It can be checked directly with the help of4that for suchμsthe functionw1μx satisfies2.4aand belongs to the class K. In particular, the inequality2.3holds with−q ifq∈0,1. The potentialw2μxsatisfies2.4aand belongs toC0D∩C2D. It is clear that the function3.8satisfies conditions at infinity2.4c. So, the function3.8with densities μsandνssubject to requirements described before satisfies all conditions of the Problem Uexcept for the boundary conditions2.4b.
To satisfy the boundary conditions we substitute3.8in2.4band arrive at the system of integral equations for the densitiesμsandνs:
±1
2ρs i 4
Γ1νσVxs, σdσ i 4
Γ1μσH10
βxs−yσdσ i
4
Γ2μσ ∂
∂ny −i
H10
βxs−yσdσF±s, s∈Γ1, 3.12a
i 4
Γ1νσVxs, σdσ i 4
Γ1μσH10
βxs−yσdσ 1 2μs i
4
Γ2μσ ∂
∂ny −i
H10
βxs−yσdσFs, s∈Γ2, 3.12b whereρsis defined in terms ofνsin3.7.
To derive limit formulas for the angular potential, we used its expression in the form of a double-layer potential3.6.
Equation 3.12a is obtained as x → xs ∈ Γ1± and comprises two integral equations. The upper sign denotes the integral equation onΓ1 , and the lower sign denotes the integral equation onΓ1−.
In addition to the integral equations written before we have the conditions3.5.
Subtracting the integral equations3.12aand using3.7, we find ρs
F s−F−s
∈C1,λ Γ1
,
νs
F s−F−s
∈C0,λ Γ1
, F±s d dsF±s.
3.13
We note that νs is found completely and satisfies all required conditions, in particular, conditions3.5. Hence, the angular potential of3.2and3.6is found completely as well.
We introduce the functionfsonΓby fs Fs− i
4
Γ1
F σ−F−σ
Vxs, σdσ, s∈Γ, 3.14
whereFsis specified onΓ2in2.4bandFs 1/2F s F−sifs∈Γ1. As shown in 4, ifs∈Γ1, thenfs∈C1,p0Γ1wherep0λif 0< λ <1 andp01−0for any0∈0,1 ifλ1. Consequently,fs∈C1,p0Γ1∩C0Γ2.
We set
δs
0 ifs∈Γ1,
1 ifs∈Γ2. 3.15
Adding the integral equations3.12aand taking into account3.12bwe obtain the integral equation forμsonΓ:
i 4
Γ1μσH10
βxs−yσdσ 1
2δsμs i
4
Γ2μσ ∂
∂ny −i
H10
βxs−yσdσfs, s∈Γ,
3.16
wherefsis given in3.14.
Thus, ifμsis a solution of3.16in the spaceCωqΓ1∩C0Γ2, ω∈0,1, q ∈0,1, then the potential 3.8with νs from3.13satisfies all conditions of the ProblemU. We arrive at the following statement.
Theorem 3.1. IfΓ∈C2,λ, if conditions3.1aand3.1bhold, and if equation3.16has a solution μsfrom the Banach spaceCωqΓ1∩C0Γ2, ω∈0,1, q ∈0,1, then a solution to the Problem Uexists and is given by formula3.8, whereνsis defined in3.13.
Ifs ∈ Γ2, then3.16 is an equation of the second kind. Ifs ∈ Γ1, then3.16is an equation of the first kind and its kernel has a logarithmic singularity, because
H10 z 2i π lnz
β hz, 3.17
wherehzis a smooth function. Indeed, asz → 0 0,
hz const O z2lnz
, hz Ozlnz, hz Olnz. 3.18
Our further treatment will be aimed to the proof of the solvability of equation3.16 in the Banach space CqωΓ1∩C0Γ2. Moreover, we reduce equation 3.16to a Fredholm equation of the second kind and of index zero, which can be easily computed by classical methods.
By differentiating equation3.16onΓ1we reduce it to the following singular integral equation onΓ1:
∂
∂sw μ
xs 1 2π
Γ1μσsinϕ0
xs, yσ xs−yσ dσ i
4
Γ1μσ∂
∂sh
βxs−yσdσ i
4
Γ2μσ ∂
∂ny −i ∂
∂sH10
βxs−yσdσfs, s∈Γ1, 3.19a
where the functionhzis defined by3.17, andϕ0x, yis the angle between the vectorxy and the direction of the normal nx. The angleϕ0x, yis taken to be positive if it is measured anticlockwise fromnxand negative if it is measured clockwise from nx. Besides,ϕ0x, yis continuous inx, y∈Γifx /y.
Equation3.16onΓ2we rewrite in the following form:
μs
ΓμσA2s, σdσ2fs, s∈Γ2, 3.19b
where
A2s, σ i
21−δσH10
βxs−yσ i 2δσ
∂
∂ny −i
H10
βxs−yσ i
πδσlnxs−yσ I1
xs, yσ
i
πδσ I2
xs, yσ xs−yσ1/3 I1
xs, yσ .
3.20
HereI1xs, yσ∈C0Γ2×Γ see3, page 339,
I2
xs, yσ
xs−yσ1/3lnxs−yσ∈C0
Γ2×Γ2
. 3.21
We note that3.19ais equivalent to3.16onΓ1if and only if3.19ais accompanied by the following additional conditions:
w μ
x a1n
f a1n
, n1, . . . , N1. 3.22
The system of3.19a,3.19b, and3.22is equivalent to3.16.
It can be easily provedsee4for detailsthat sinϕ0
xs, yσ xs−yσ − 1
σ−s
∈C0,λ
Γ1×Γ1
. 3.23
Therefore we can rewrite3.19ain the following form:
2 ∂
∂sw μ
xs 1 π
Γ1μσ dσ σ−s
ΓμσYs, σdσ2fs, s∈Γ1, 3.24 wheresee4
Ys, σ
1−δσ 1
π
sinϕ0
xs, yσ xs−yσ − 1
σ−s i
2
∂
∂sh
βxs−yσ i
2δσ ∂
∂ny −i ∂
∂sH10
βxs−yσ
∈C0,p0 Γ1×Γ
, p0 λif 0< λ <1, andp01−0 for any0∈0,1if λ1.
3.25
4. The Fredholm Integral Equation and the Solution to the Problem
Inverting the singular integral operator in3.24, we arrive at the following integral equation of the second kind4,15:
μs 1 Q1s
ΓμσA1s, σdσ 1 Q1s
N1−1 n0
Gnsn 1
Q1sΦ1s, s∈Γ1, 4.1 where
A1s, σ −1 π
Γ1
Yξ, σ
ξ−s Q1ξdξ, Q1s N1
n1
s−a1n
b1n−s
sign s−a1n
,
Φ1s −1 π
Γ1
2Q1σfσ σ−s dσ,
4.2
andG0, . . . , GN1−1are arbitrary constants. We set signs−a1n 1 assa1n; then the function signs−a1nbelongs toC∞Γ1insvariable forn1, . . . , N1.
It can be shown using the properties of singular integrals15thatΦ1sandA1s, σ are H ¨older continuous functions if s ∈ Γ1 and σ ∈ Γ. Consequently, any solution of 4.1 belongs toCω1/2Γ1with someω ∈ 0,1, and here in after we look for μsonΓ1 in this space.
We set
Qs 1−δsQ1s δs, s∈Γ. 4.3
Instead of μs ∈ Cω1/2Γ1 ∩ C0Γ2 we introduce the new unknown function μ∗s μsQs∈C0,ωΓ1∩C0Γ2 and rewrite 4.1 and 3.19b in the form of one equation:
μ∗s
Γμ∗σQ−1σAs, σdσ 1−δsN1−1
n0
Gnsn Φs, s∈Γ, 4.4
where
As, σ 1−δsA1s, σ δsA2s, σ, Φs 1−δsΦ1s 2δsfs. 4.5 To derive equations forG0, . . . , GN1−1we substituteμsfrom4.1and3.19bin the conditions3.22; then in terms ofμ∗swe obtain
ΓQ−1ξμ∗ξlnξdξ N1−1
m0
BnmGmHn, n1, . . . , N1, 4.6
where
lnξ −w
Q−1·A·, ξ a1n
, Hn−w
Q−1·Φ·
a1n f
a1n , Bnm−w
Q−1·1−δ··m a1n
.
4.7
By·we denote the variable of integration in the potentialwμxin3.9.
Thus, the system of3.19a,3.19b, and3.22forμshas been reduced to the system of 4.4 and 4.6 for the function μ∗s and constants G0, . . . , GN1−1. It is clear from our consideration that any solution of system of4.4and4.6generates a solution to the system of3.19a,3.19b, and3.22.
As noted before,Φ1sandA1s, σ are H ¨older continuous functions ifs ∈ Γ1, and σ ∈ Γ. More precisely see4, Φ1s ∈ C0,pΓ1, p min{1/2, λ}, andA1s, σbelongs toC0,pΓ1in suniformly with respect to σ ∈ Γ. Using these properties we can prove the following.
Lemma 4.1. IfΓ∈C2,λ, λ∈0,1, Φs∈C0,pΓ1∩C0Γ2, pmin{λ,1/2}, and ifμ∗sfrom C0Γsatisfies equation4.4, thenμ∗s∈C0,pΓ1∩C0Γ2.
The conditionΦs∈C0,pΓ1∩C0Γ2holds if conditions3.1aand3.1bhold.
Hence here in after we will look forμ∗sfromC0Γ.
SinceA1s, σ ∈ C0Γ1×Γ, and due to the special representation for A2s, σ from 3.19b, the integral operator from4.4
Aμ∗
Γμ∗σQ−1σAs, σdσ
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
Γμ∗σQ−1σA1s, σdσ if s∈Γ1,
Γμ∗σQ−1σA2s, σdσ if s∈Γ2,
4.8
is a compact operator mappingC0Γinto itself. Indeed, one can check using Arzela theorem 17 that the integral operator with the kernel A1s, σ is a compact operator mapping C0ΓintoC0Γ1, while the integral operator with the kernelA2s, σis a compact operator mappingC0Γ1intoC0Γ2. Moreover, it can be verified directly with the help of the Arzela theorem that the integral operator
Γμ∗σQ−1σA2s, σdσis a compact operator mapping C0Γ2intoC0Γ2. To verify equicontinuity in the Arzela theorem, we may use the property of uniform continuity inxonΓ2for the functions|x−y|1/3and|x−y|1/3ln|x−y|. In doing so, we may use the Cauchy-Bunyakovski inequality for estimates.
We rewrite4.4in the following operator form:
I Aμ∗ P G Φ, 4.9
where P is the operator of multiplication of the row P 1−δs s0, . . . , sN1−1 by the columnG G0, . . . , GN1−1T. The operatorP is finite-dimensional fromEN1 intoC0Γand therefore compact18.
Now we rewrite4.6in the following form:
IN1G Lμ∗ B−IN1GH, 4.10 whereH H1, . . . , HN1T is a column ofN1 elements,IN1 is the identity operator inEN1, Bis aN1×N1 matrix consisting of the elementsBnm from4.7. The operatorLacts from C0ΓintoEN1, so thatLμ∗ L1μ∗, . . . , LN1μ∗T, where
Lnμ∗
ΓQ−1ξμ∗ξlnξdξ. 4.11 The operatorsB−IN1andLare finite-dimensional and therefore compact19.
We consider the columns μ
μ∗
G
, Φ
Φ H
4.12 in the Banach spaceC0Γ×EN1with the normμC0Γ×EN1 μ∗C0Γ GEN
1. We write the system of4.9and4.10in the form of one equation:
I Rμ Φ, R
A P L B−IN1
, 4.13
where I is the identity operator in the spaceC0Γ×EN1. It is clear that R is a compact operator mappingC0Γ×EN1 into itself, since it consists of compact operators. Therefore,4.13is a Fredholm equation of the second kind and of index zero in the space C0Γ ×EN1 see 17,18,20.
Let us show that homogeneous equation 4.13 has only a trivial solution. Then, according to Fredholm’s alternative 17, 18,20, the inhomogeneous equation4.13has a unique solution for any right-hand side. Let
μ0 μ0∗
G0
∈C0Γ×EN1 4.14
be an arbitrary solution of the homogeneous equation4.13. According toLemma 4.1,
μ0 μ0∗
G0
∈C0,p Γ1
∩C0 Γ2
×EN1, pmin
"
λ,1 2
#
. 4.15
Therefore the functionμ0s μ0∗sQ−1s ∈ C1/2p Γ1∩C0Γ2and the columnG0 convert the homogeneous equations4.1,3.19band4.6into identities. For instance,3.19btakes the following form:
x→xs∈Γlim 2w μ0
x 0, x∈ D. 4.16a
Using the homogeneous identities 4.1 and 3.19b, we check that the homogeneous identities4.6are equivalent to
w μ0
a1n
0, n1, . . . , N1. 4.16b
Besides, acting on the homogeneous identity4.1with a singular operator with the kernel s−t−1we observe thatμ0ssatisfies the homogeneous equation3.24:
∂
∂sw μ0
xs
Γ1 0. 4.16c It follows from 4.16a,4.16band 4.16c that μ0s satisfies the homogeneous equation 3.16. On the basis ofTheorem 3.1,u0, μ0x≡wμ0xis a solution to the homogeneous ProblemU. According toTheorem 2.2,wμ0x≡0 forx∈ D \Γ1. Using the limit formulas for normal derivatives of a single-layer potential onΓ1, we have
lim
x→xs∈Γ1
∂
∂nxw μ0
x− lim
x→xs∈Γ1−
∂
∂nxw μ0
x μ0s≡0, s∈Γ1. 4.17
Hence,wμ0x w2μ0x≡ 0 forx∈ D, andμ0ssatisfies4.16a, which can be written as
1
2μ0s i 4
Γ2μ0σ ∂
∂ny −i
H10
βxs−yσdσ0, s∈Γ2. 4.18 It is shown in5, page 502–504and6, page 187–189that μ0s ≡ 0 is the unique solution of4.18inC0Γ2.
Consequently, ifs ∈ Γ, thenμ0s ≡ 0 andμ0∗s μ0sQ−1s ≡ 0. It follows from the homogeneous identity4.1forμ0sandG00, . . . , G0N1−1 thatG0 G00, . . . , G0N1−1T ≡ 0.
Hence,μ0 ≡0. Thus, the homogeneous Fredholm equation4.13has only a trivial solution inC0Γ×EN1.
We have proved the following assertion.
Theorem 4.2. IfΓ∈C2,λ, λ∈0,1, then4.13is a Fredholm equation of the second kind and of index zero in the spaceC0Γ×EN1. Moreover,4.13has a unique solution
μ μ∗
G
∈C0Γ×EN1 4.19
for any
Φ Φ
H
∈C0Γ×EN1. 4.20
As a consequence ofTheorem 4.2and theLemma 4.1we obtain the following corollary.
Corollary 4.3. IfΓ∈C2,λ, λ∈0,1, then equation4.13has a unique solution:
μ μ∗
G
∈C0,p Γ1
∩C0 Γ2
×EN1 4.21
for any
Φ Φ
H
∈C0,p Γ1
∩C0 Γ2
×EN1, 4.22
wherepmin{λ,1/2}.
We recall that Φ belongs to the class of smoothness required in Corollary 4.3 if conditions 3.1a and 3.1b hold. Besides, 4.13 is equivalent to the system of 4.4and 4.6. As mentioned before, if{μ∗s, G0, . . . , GN1−1} is a solution of the system of4.4and 4.6, andμ∗s∈C0,pΓ1∩C0Γ2, then the functionμs μ∗sQ−1s∈Cp1/2Γ1∩C0Γ2 is a solution of the system of3.19a,3.19band3.22, and thereforeμssatisfies3.16. We obtain the following statement.
Theorem 4.4. IfΓ ∈ C2,λ, and if conditions 3.1a and 3.1b hold, then equation3.16 has a solutionμs ∈ Cp1/2Γ1∩C0Γ2, p min{1/2, λ}. This solution is expressed by the formula μs μ∗sQ−1s, where the functionμ∗s∈C0,pΓ1∩C0Γ2is found by solving the Fredholm equation4.13, which is uniquely solvable according toCorollary 4.3.
On the basis ofTheorem 3.1we arrive at the solvability theorem for the ProblemU.
Theorem 4.5. IfΓ∈C2,λ, and if conditions3.1aand3.1bhold, then the solution to the Problem Uexists and is given by3.8, whereνsis defined in3.13andμsis a solution of equation3.16 inCp1/2Γ1∩C0Γ2, pmin{1/2, λ}ensured byTheorem 4.4.
It can be checked directly that the solution to the ProblemUsatisfies condition2.3 with−1/2. Explicit expressions for singularities of the solution gradient at the endpoints of the cracks will be presented in the next section.
Theorem 4.5ensures existence of a classical solution to the ProblemUwhenΓ∈C2,λ, λ ∈ 0,1, and conditions3.1a and3.1b hold. The uniqueness of the classical solution follows fromTheorem 2.2. On the basis of our consideration we suggest the following scheme for solving the Problem U. First, we find the unique solution of the Fredholm equation 4.13 from C0Γ × EN1. This solution automatically belongs to C0,pΓ1∩ C0Γ2 ×EN1, pmin{λ,1/2}. Second, we construct the solution of 3.16fromCp1/2Γ1∩C0Γ2by the formulaμs μ∗sQ−1s. Finnaly, substitutingνsfrom3.13andμsin3.8we obtain the solution to the ProblemU.
Remark 4.6. It is important to stress that the solution ux to the Problem U, ensured by Theorem 4.5, is a classical solution, but it may be not a weak solution to the ProblemU.
In other words, classical solution to the ProblemUexists and is ensured byTheorem 4.5, but weak solution to the ProblemUmay not exist inHloc1 D \Γ1space. This follows from the fact that Dirichlet data on the closed curvesΓ2is assumed to be continuous only. Continuity of a Dirichlet boundary data on closed curves is not sufficient for existence of a weak solution
in Hloc1 D \Γ1 space. The Hadamard example of a nonexistence of a weak solution to a harmonic Dirichlet problem in a disc with continuous boundary data is given in the book7, section 12.5by Sobolev himselfthe classical solution exists in this example.
5. Singularities of the Gradient of the Solution at the Endpoints of the Cracks
As noted at the end ofSection 4, the gradient of the solution to ProblemUcan be unbounded at the endpoints of the cracksΓ1, so that the gradient of the solution to the ProblemUsatisfies estimate 2.3 with the exponent −1/2. We will now make a detailed analysis of the behaviour of∇uxat the endpoints ofΓ1.
Letuxbe a solution to the ProblemUensured byTheorem 4.5and given by3.8.
Letxd ∈Xbe one of the endpoints ofΓ1. In the neighbourhood ofxd, we introduce the system of polar coordinates:
x1x1d |x−xd|cosϕ, x2x2d |x−xd|sinϕ. 5.1
We will assume thatϕ∈αd, αd 2πifda1nandϕ∈αd−π, αd πifdb1n. We recall thatαsis the angle between the direction of theOx1axis and the tangent vectorτxto Γ1at the pointxs.
Hence,αd αa1n 0ifda1nandαd αbn1−0ifdb1n.
Thus, the angleϕvaries continuously in the neighbourhood of the endpointxd, cut alongΓ1.
We will use the notation μ1s μs|s−d|1/2 Q−1sμ∗s|s−d|1/2 and put μ1d μ1a1n μ1a1n 0ifda1n, andμ1d μ1b1n μ1b1n−0ifdb1n.
Using the representation of the derivatives of harmonic potentials in terms of Cauchy type integralssee4and using the properties of these integrals near the endpoints of the integration line, presented in15, we can prove the following assertion.
Theorem 5.1. Letux be a solution to the Problem Uensured by Theorem 4.5. Letxd be an arbitrary endpoint of the cracksΓ1, that is,xd∈Xandda1nordb1nfor somen1, . . . , N1. Then the derivatives of the solution to the ProblemUin the neighbourhood ofxdhave the following asymptotic behaviour.
Ifda1n, then
∂
∂x1ux μ1
a1n 2x−xa1n1/2 sin
ϕ α a1n 2
−ν a1n 2π
−sinα a1n
lnx−x
a1n ϕcosα a1n
O1,
∂
∂x2ux − μ1
a1n
2x−xa1n1/2cos
ϕ α a1n 2
−ν a1n 2π
cosα
a1n
lnx−x
a1n ϕsinα a1n
O1.
5.2
Ifdb1n, then
∂
∂x1ux − μ1
b1n 2x−x
b1n1/2cos
ϕ α b1n 2
ν b1n 2π
−sinα b1n
lnx−x
b1n ϕcosα bn1
O1,
5.3
∂
∂x2ux − μ1
bn1 2x−xb1n1/2sin
ϕ α bn1 2
ν b1n 2π
cosα
b1n
lnx−x
b1n ϕsinα b1n
O1.
5.4
ByO1one denotes functions which are continuous at the endpointxd. Moreover, the functions denoted byO1 are continuous in the neighbourhood of the endpoint xdcut along Γ1 and are continuously extensible toΓ1 and toΓ1−from this neighbourhood.
The formulas of the theorem demonstrate the following curious fact. In the general case, the derivatives of the solution to the ProblemUnear the endpointxdof cracksΓ1 behave as
O
|x−xd|−1/2 O
ln
1
|x−xd|
. 5.5
However, if μ1d νd 0, then ∇ux will be bounded and even continuous at the endpointxd∈X.
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