Volume 2012, Article ID 269607,20pages doi:10.1155/2012/269607
Research Article
The Dirichlet Problem for the 2D Laplace Equation in a Domain with Cracks without Compatibility Conditions at Tips of the Cracks
P. A. Krutitskii
KIAM, Miusskaya Square 4, Moscow 125047, Russia
Correspondence should be addressed to P. A. Krutitskii,[email protected] Received 21 March 2012; Accepted 11 June 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.
We study the Dirichlet problem for the 2D Laplace equation in a domain bounded by smooth closed curves and smooth cracks. In the formulation of the problem, we do not require compatibility conditions for Dirichlet’s boundary data at the tips of the cracks. However, if boundary data satisfies the compatibility conditions at the tips of the cracks, then this is a particular case of our problem. The cases of both interior and exterior domains are considered. The well- posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integral representation for a solution is obtained. It is shown that weak solution of the problem does not typically exist, though the classical solution exists. The asymptotic formulae for singularities of a solution gradient at the tips of the cracks are presented.
1. Introduction
Boundary value problems in 2D domains with cracks double-sided open arcs are very important for applications1.
It is known that if the Dirichlet problem for the Laplace equation is considered in a 2D domain bounded by sufficiently smooth closed curves, and if the function specified in the boundary condition is smooth enough, then existence of a classical solution follows from existence of a weak solution. In the present paper, we consider the Dirichlet problem for the Laplace equation in both interior and exterior 2D domain bounded by closed curves and double-sided open arcscracksof an arbitrary shape. The Dirichlet condition is specified on the whole boundary, that is, on both closed curves and on the cracks, so that different functions may be specified on opposite sides of the cracks. The case of this problem, when the Dirichlet boundary data satisfies compatibility conditions at the tips of the cracks,
has been previously studied in 2–6, where theorems on existence and uniqueness of a classical solution have been proved, and the integral representation for a classical solution has been obtained. In the present paper, this problem is considered in case when the Dirichlet boundary data may not satisfy the compatibility conditions at the tips of the cracks. However if boundary data satisfies the compatibility conditions at the tips of the cracks, then this is a particular case of our problem. We prove that there exists a unique classical solution to our problem and obtain an integral representation for the classical solution. In addition, we prove that a weak solution to our problem may not exist even if both cracks and functions in the boundary conditions are sufficiently smooth. This result follows from the fact that the square of the gradient of a classical solution, basically, is not integrable near the ends of the cracks, since singularities of the gradient are rather strong there. This result is very important for numerical analysis, it shows that finite elements and finite difference methods cannot be applied to numerical treatment of the Dirichlet problem in question directly, since all these methods imply existence of a weak solution. To use difference methods for numerical analysis, one has to localize all strong singularities first and next to use difference method in a domain excluding the neighbourhoods of the singularities. The asymptotic formulae for singularities of a solution gradient at the tips of the cracks are presented.
2. Formulation of the Problem
By an open curve we mean a simple smooth nonclosed arc of finite length without self- intersections7.
LetΓbe a set of curves, which may be both closed and open. We say thatΓ ∈C2,λ or Γ∈C1,λif curvesΓbelong to the classC2,λorC1,λwith the H ¨older exponentλ∈0,1.
In a plane in Cartesian coordinates x x1, x2 ∈ R2, we consider a multiply connected domain bounded by simple open curves Γ11, . . . ,Γ1N1 ∈ C2,λ and simple closed curvesΓ21, . . . ,Γ2N
2∈C2,λ,λ∈0,1, in such a way that all curves do not have common points, in particular, endpoints. We will consider both the case of an exterior domain and the case of an interior domain when the curveΓ21encloses all others. Set
Γ1 N1
n 1
Γ1n, Γ2 N2
n 1
Γ2n, Γ Γ1∪Γ2. 2.1
The connected domain bounded by Γ2 and containing curves Γ1 will be calledD, so that
∂D Γ2, Γ1⊂ D. We assume that each curveΓjnis parametrized by the arc lengths:
Γjn
x: x xs x1s, x2s, s∈
ajn, bjn
, n 1, . . . , Nj, j 1,2, 2.2
so that,
a11< b11<· · ·< a1N
1 < b1N
1 < a21 < b12<· · ·< a2N
2< b2N
2, 2.3
and the domainDis placed to the right when the parameters increases onΓ2n. The points x∈ Γand values of the parametersare in one-to-one correspondence except the pointsa2n, b2n, which correspond to the same pointxforn 1, . . . , N2. Further on, the sets of the intervals
N1
n 1
a1n, b1n
,
N2
n 1
a2n, b2n
, 2 j 1
Nj
n 1
ajn, bjn
2.4
on theOs-axis will be denoted byΓ1,Γ2, andΓalso.
SetC0Γ2n {Fs: Fs∈C0a2n, b2n,Fa2n Fb2n}, andC0Γ2 N2
n 1C0Γ2n. The tangent vector toΓin the pointxs, in the direction of the increment ofs, will be denoted by τx cosαs,sinαs, while the normal vector coinciding withτxafter rotation through an angle ofπ/2 in the counterclockwise direction will be denoted by nx sinαs,−cosαs.
According to the chosen parametrization cosαs x1s, sinαs x2s. Thus, nx is an inward normal toDonΓ2. ByXwe denote the point set consisting of the endpoints ofΓ1 : X N1
n 1xa1n∪xbn1.
We considerΓ1as a set of cracksor double-sided open arcs. The side of the crackΓ1, which is situated on the left when the parametersincreases, will be denoted byΓ1, while the opposite side will be denoted byΓ1−.
We say that the functionuxbelongs to the smoothness class K1, if 1u∈C0D \Γ1\X∩C2D \Γ1, ∇u∈C0D \Γ1\Γ2\X, 2in the neighbourhood of any pointxd∈X, both the inequality
|ux|<Const 2.5
and the equality
rlim→0 ∂Sd,r
∂ux
∂nx
dl 0 2.6
hold, where the curvilinear integral of the first kind is taken over a circumference
∂Sd, rof a radiusr with the center in the pointxd, in addition, nxis a normal in the pointx∈∂Sd, r, directed to the center of the circumference andd a1n or d b1n, n 1, . . . , N1.
Remark 2.1. ByC0D \Γ1\Xwe denote the class of continuous inD \Γ1functions, 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. To obtain the definition of the classC0D \Γ1\Γ2\X we have to replaceC0D \Γ1\XbyC0D \Γ1\Γ2\XandD \Γ1byD \Γ1in the previous sentence.
Γ21 Γ22 Γ23
Γ11
Γ12
Figure 1: An example of an interior domain.
Γ11
Γ12
Γ21 Γ22
Figure 2: An example of an exterior domain.
Let us formulate the Dirichlet problem for the Laplace equation in a domainD \Γ1 interior or exterior, see Figures1and2.
Problem D1
Find a functionuxfrom the class K1, so thatuxobeys the Laplace equation
ux1x1x ux2x2x 0 2.7
inD \Γ1and satisfies the boundary conditions
ux|xs∈Γ1 Fs, ux|xs∈Γ1− F−s, ux|xs∈Γ2 Fs. 2.8
IfDis an exterior domain, then we add the following condition at infinity:
|ux| ≤const, |x|
x21x22−→ ∞. 2.9
All conditions of the problem D1must be satisfied in a classical sense. The boundary conditions2.8onΓ1must be satisfied in the interior points ofΓ1, their validity at the ends ofΓ1is not required.
Theorem 2.2. IfΓ∈C2,λ, λ∈0,1, then there is no more than one solution to the problem D1. Proof. It is sufficient to prove that the homogeneous problem D1 admits the triv- ial solution only. Let u0x be a solution to the homogeneous problem D1 with Fs≡F−s≡0,Fs≡0. LetSd, be a disc of a small enough radius with the center in the pointxd d a1nord b1n, n 1, ..., N1. LetΓ1n,be a set consisting of such points of the curveΓ1nwhich do not belong to discsSa1n, andSb1n, . We choose a number0so small that the following conditions are satisfied:
1for any 0 < ≤ 0 the set of points Γ1n, is a unique non-closed arc for each n 1, ..., N1,
2the points belonging toΓ\Γ1nare placed outside the discsSa1n, 0,Sb1n, 0for any n 1, ..., N1,
3discs of radius0with centers in different ends ofΓ1do not intersect.
SetΓ1, ∪Nn 11Γ1n,, S ∪Nn 11Sa1n, ∪Sbn1, , D D \Γ1,\S. IfDis an exterior domain, then we setD,R D∩SR, whereSR is a disc with a center in the origin and with sufficiently large radiusR.
SinceΓ2 ∈C2,λ, u0x ∈ C0D \Γ1 remind thatu0x ∈K1, and sinceu0|Γ2 0 ∈ C2,λΓ2, and owing to the lemma on regularity of solutions of elliptic equations near the boundary8, lemma 6.18, we obtainu0x∈C1D \Γ1. Sinceu0x∈K1, we observe that u0x ∈ C1Dfor any ∈ 0, 0. ByC1Dwe meanC1D∪Γ2∪Γ1,∪Γ1,−∪∂S. Analogously, in case of exterior domainD: u0x ∈ C1D,Rfor ∈ 0, 0. Let D be an interior domain. Since the boundary of a domain D is piecewise smooth, we write out Green’s formula9, page 328for the functionu0x:
∇u02
L2D Γ1,
u0
∂u0
∂nx
ds−
Γ1,
u0−
∂u0
∂nx −
ds
− Γ2u0∂u0
∂nxds
∂S
u0∂u0
∂nxdl.
2.10
By nx the exterior with respect to D normal on ∂S at the point x ∈ ∂S is denoted.
By the superscripts and − we denote the limiting values of functions on Γ1 and on
Γ1−, respectively. Sinceu0xsatisfies the homogeneous boundary condition2.8onΓ, we observe thatu0|Γ2 0 andu0±|Γ1, 0 for any∈0, 0. Therefore, identity2.10takes the form
∇u02
L2D ∂S
u0∂u0
∂nxdl, ∈0, 0. 2.11
Setting → 0 in2.11, taking into account thatu0x ∈ K1 and using the relationships 2.6,2.5we obtain
∇u02
L2D\Γ1 lim
→0
∇u02
L2D 0. 2.12
From the homogeneous boundary conditions2.8we conclude thatu0x≡0 inD\Γ1, where Dis an interior domain.
LetDbe an exterior domain. Since the boundary of a domainD,Ris piecewise smooth and sinceu0x∈C1D,Rfor any∈0, 0, we may write Green’s formula in a domainD,R
for a harmonic functionu0x 9, c.328:
∇u02
L2D,R Γ1,
u0
∂u0
∂nx
ds−
Γ1,
u0−
∂u0
∂nx −
ds
− Γ2u0∂u0
∂nxds
∂S
u0∂u0
∂nxdl
∂SR
u0∂u0
∂|x|dl.
2.13
By nx on∂S we denote an outwardwith respect toD,Rnormal in the pointx ∈ ∂S. It follows from condition2.9and from the theorem on behaviour of a gradient of a harmonic function at infinity9, page 373that
∂u0x
∂|x| O 1
|x|2
, as|x| −→ ∞. 2.14
Consequently,
Rlim→ ∞ ∂SR
u0x∂u0x
∂|x| dl 0, 2.15
and formula2.13transforms to the formula2.10 asR → ∞. Repeating all arguments, presented above for the case of an interior domainD, we arrive to formula 2.12. Taking into account homogeneous boundary conditions2.8, we obtain from2.12thatu0x ≡0 inD \Γ1, whereDis an exterior domain. Thus, in all casesu0x≡ 0 inD \Γ1. Theorem is proved.
Remark 2.3. The maximum principle cannot be used for the proof of the theorem even in the case of an interior domainD, since the solution to the problem may not satisfy the boundary
condition 2.8 at the ends of the cracks, and it may not be continuous at the ends of the cracks.
3. Properties of the Double Layer Potential on the Open Curve
Let γ be an open curve of class C1,λ, λ ∈ 0,1. Assume that γ is parametrized by the arc length s: γ {x: xs x1s, x2s, s ∈ a, b}. The points x ∈ Γ and values of the parameter s are in one-to-one correspondence, so the segment a, b will be also denoted byγ. The tangent vector to γ in the point xs, in the direction of the increment ofs, will be denoted byτx cosαs,sinαs, while the normal vector toγ in the point xswill be denoted by nx sinαs,−cosαs. According to the chosen parametrization cosαs x1s, sinαs x2s. The side of the crackγ, which is situated on the left when the parametersincreases, will be denoted byγ, while the opposite side will be denoted by γ−. LetXγ xa∪xbbe a set of the ends ofγ.
Setμs∈C0,λa, b, and consider the double layer potential for Laplace equation in a plane
W μ
x − 1 2π
b a
μσ ∂
∂nylnx−yσdσ. 3.1
Setz x1ix2, t tσ y1σ iy2σ∈γ, μt μtσ μσ. Ifμs ∈C0,λa, b, thenμt ∈C0,λγ, since
μt2−μt 1 μtσ2−μtσ 1 μσ2−μσ1≤c|σ2−σ1|λ c
|σ2−σ1|
|tσ2−tσ1| λ
|tσ2−tσ1|λ c·cλ0|t2−t1|λ,
3.2
wherecandc0are constants,t2 tσ2∈γ, t1 tσ1∈γ. We took into account in deriving the latter inequality that
|σ2−σ1|
|tσ2−tσ1| ∈C0a, b×a, b, 3.3 see lemma 1 in10, whence
|σ2−σ1|
|tσ2−tσ1| ≤c0. 3.4
Consider the integral of the Cauchy type with the real densityμt:
Φz 1
2πi γμt dt
t−z, 3.5
then Wμx −ReΦz. It follows from properties of the Cauchy type integral that if μσ ∈ C0,λa, b, thenWμx ∈ C0R2\γ \Xγ. This means that the potential Wμx
is continuously extensible toγ from the left and from the right in interior pointsthough its values onγfrom the left and from the right may be different. If, in addition,μd 0, then the potentialWμxis continuously extensible to the endxd, whered aord bsee 7, Section 15.2. Set
cosψ
x, y x1−y1
x−y −x−y
y1, sinψ
x, y x2−y2
x−y −x−y
y2, 3.6 thenψx, yis a polar angle of the coordinate system with the origin in the pointy. Formulae for cosψx, y, sinψx, ydefine the angle ψx, ywith indeterminacy up to 2πmmis an integer number. LetSd, be a disc of a sufficiently small radiuswith the center inxd d aord b. From asymptotic formulae describing behavior ofΦzat the ends ofγ 7, Section 22, we may derive the asymptotic formulae forWμx −ReΦzat the ends of γ. Namely, for anyx∈Sd, andx /∈γ, the formula holds:
W μ
x ±μd
2π ψx, xd Ωx. 3.7
Here by ψx, xdwe mean some fixed branch of this function, so that the branch varies continuously inxin a neighbourhood of the pointxd, cut alongγ. The upper sign is taken ifd a, while the lower sign is taken ifd b. The functionΩxis continuous asx → xd.
Moreover,Ωxis continuous inSd, outsideγand is continuously extensible from the left and from the right to the part ofγ lying inSd, . It follows from formula3.7that for any x∈Sd, andx /∈γthe inequality holds
W μ
x≤const. 3.8
Now we will study properties of derivatives of the double layer potential. It follows from Cauchy-Riemann relations that
dΦz
dz ReΦx1−iReΦx2 −Wx1iWx2. 3.9 On the other hand, ifμσ∈C1,λa, b, then forz /∈γ:
dΦz dz
1
2πi γμt d dz
1
t−zdt − 1
2πi γμt d
dt 1 t−z
dt
− 1 2πi
μb
tb−z− μa ta−z−
γ
μt t−zdt
,
3.10
where
dμt dt
dμσ dσ
dσ dt
μσ
tσ e−iασμσ. 3.11
Since γ ∈ C1,λ, then e−iασ ∈ C0,λa, b, so one can show that μt ∈ C0,λγ the proof repeats the given above proof of the fact that μt ∈ C0,λγ, if μσ ∈ C0,λa, b. From 3.9and3.10and from properties of the Cauchy type integral7, Section 15, it follows that ifμσ∈C1,λa, b, then∇Wμx∈C0R2\γ\Xγ, that is,∇Wμxis continuously extensible toγ from the left and from the right in interior points, though the limiting values of∇Wμxonγfrom the left and from the right can be different. We can write3.10in the form
dΦz dz
1 2πi
μbe−iψx,xb
|x−xb| − μae−iψx,xa
|x−xa|
Ω0z, 3.12
where
Ω0z 1 2πi γ
μt
t−zdt. 3.13
It follows from7,§22that for allz∈Sd, d aord b, such thatz /∈γ, the inequality holds
|Ω0z| ≤c0μdln 1
|x−xd|1
≤cln 1
|x−xd|, 3.14
wherec0andcare constants.
Comparing formulae3.9and 3.12, we obtain that forx ∈ Sd, andx /∈ γ, the formulae hold
∂W μ
x
∂x1
1 2π
∓μd
|x−xd|sinψx, xd Ω1x,
∂W μ
x
∂x2
1 2π
±μd
|x−xd|cosψx, xd Ω2x,
3.15
where
Ωjx≤c1
μdln 1
|x−xd|1
≤c2ln 1
|x−xd|, j 1,2, 3.16
c1, c2are constants. The upper sign in formulae is taken ifd a, while the lower sign is taken ifd b. It follows from7, Section 15.2that ifμd 0, then the functionsΩ1xandΩ2x
are continuously extensible to the endxd. Moreover, ifx∈Sd, andx /∈ γ, then for the functionsΩ1xandΩ2x, the formulae hold:
Ω1x −ReΩ0z
∓μd 2π
sinαdln|x−xd| −cosαdψx, xd
Ω10x, Ω2x ImΩ0z
±μd 2π
cosαdln|x−xd|sinαdψx, xd
Ω20x,
3.17
which can be derived, using the asymptotics for Ω0z from 7,§ 22. The upper sign in formulae is taken ifd a, while the lower sign is taken ifd b. FunctionsΩ10xandΩ20x are continuously extensible to the endxd. Byψx, xdwe mean some fixed branch of this function, which varies continuously inxin a neighbourhood of a pointxd, cut alongγ.
Letμσ∈C1,λa, b, and let nxbe a normal in the pointx∈∂Sd, , directed to the center of the circumference∂Sd, , that is, nx −cosψx, xd,−sinψx, xd, then we obtain from3.15forx /∈γ
∂Wμx
∂nx
∂Sd, −Ω1xcosψx, xd−Ω2xsinψx, xd. 3.18
Therefore, according to3.16:
∂Wμx
∂nx
∂Sd,≤const ln 1
|x−xd|
∂Sd, const ln1
, 3.19
since|x−xd| on∂Sd, . From here we obtain that
∂Sd,
∂W μ
x
∂nx dl
2π 0
∂W μ
x
∂nx
dψ≤2πconstln1
−→0, 3.20
if → 0, so
lim→0 ∂Sd,
∂W μ
x
∂nx
dl 0. 3.21
Now letbe a fixed positive numbersufficiently small. Using formulae3.15and setting r |x−xd|, ψ ψx, xd, we consider the integral over the discSd, :
Sd,
∇W μ
x2dx
2π 0
0
μd 2πr
2
μd πr
∓Ω1xsinψ±Ω2xcosψ
Ω21x Ω22x
rdrdψ I1I2, 3.22
I1 1 2π
0
1
rμ2ddr, I2
2π 0
0
μd π
∓Ω1xsinψ±Ω2xcosψ r
Ω21x Ω22x drdψ.
3.23
The integralI2converges according to estimates3.16:
|I2| ≤4c2 0
ln1 r
μdc2πrln1 r
dr≤const. 3.24
Hence, if integral3.22converges, then the integralI1converges as wellas a difference of two convergent integrals, but the integralI1converges if and only ifμd 0, while in other casesI1 diverges. Thus, the integral3.22converges if and only ifμd 0. Consequently
|∇Wμx|belongs toL2Sd, with small > 0 if and only ifμd 0. Let us formulate obtained results in the form of the theorem.
Theorem 3.1. Letγ be an open curve of classC1,λ,λ∈0,1. LetSd, be a disc of a sufficiently small radiuswith the center in the pointxd∈Xγ(d aord b).
iIfμs∈C0,λa, b, thenWμx∈C0R2\γ\Xγand for anyx∈Sd, , such that x /∈γ, the relationships3.7and3.8hold.
iiIfμs∈C1,λa, b, then
1∇Wμx∈C0R2\γ\Xγ;
2for anyx∈Sd, , such thatx /∈γ, the formulae3.15hold, in which the functions Ω1xandΩ2xsatisfy relationships3.16and3.17;
3forWμxthe property3.21holds;
4|∇Wμx|belongs toL2Sd, for sufficiently small >0 if and only ifμd 0.
Remark 3.2. Each function of class C0R2\γ \Xγis continuous inR2 \γ, is continuously extensible toγ\Xγ from the left and from the right, but limiting values of such a function on γ\Xγfrom the left and from the right can be different, that is, the function may have a jump onγ\Xγ.
Let us study smoothness of the direct value of the double layer potential on the open curve.
Lemma 3.3. Letγbe an open curve of classC2,λ,λ∈0,1, and letμs∈C0a, b. Let
I1s − 1
2π γμσ∂lnxs−yσ
∂ny dσ 3.25
be the direct value of the double layer potentialWμxonγ. Then
I1s∈C1,λ/4a, b. 3.26
Proof. Let us prove thatI1s∈C1,λ/4a, b. Taking into account that ny y2σ,−y1σ, we find
∂lnxs−yσ
∂ny
Ts, σ
gs, σ, gs, σ xs−yσ2 s−σ2 , Ts, σ
x2s−y2σ
y1σ−
x1s−y1σ y2σ s−σ2 .
3.27
Note thatyσ is a point onΓ corresponding tos σ. So, we may putxσ yσ. For j 1,2, we have10,§3
xjs−xjσ s−σZj1s, σ −xjσσ−s σ−s2Zj2σ, s, 3.28
where
Z1js, σ 1
0
xjσξs−σdξ∈C1,λa, b×a, b,
Zj2σ, s 1
0
ξxjsξσ−sdξ∈C0,λa, b×a, b.
3.29
Note that the function
gs, σ |xs−xσ|2 s−σ2
Z11s, σ2
Z12s, σ2
∈C1,λa, b×a, b 3.30
does not equal zero anywhere onΓandgs, s 1; therefore, 1
gs, σ ∈C1a, b×a, b. 3.31
Further,
∂
∂s 1 gs, σ
∂
∂s
s−σ2
|xs−xσ|2 −gss, σ g2s, σ
−2Z11s, σ
Z11s, σ
sZ21s, σ
Z12s, σ
s
g2s, σ ∈C0,λa, b×a, b.
3.32
Consequently, 1/gs, σ∈C1,λa, b×a, b. Similarly,
Ts, σ x2s−x2σx1σ−x1s−x1σx2σ s−σ2
Z22σ, sx1σ−Z21σ, sx2σ
∈C0,λa, b×a, b.
3.33
Consider∂Ts, σ/∂s J1s, σ−2J2s, σ, where
J1s, σ x2sx1σ−x1sx2σ s−σ2
x2s−x2σ
x1σ−
x1s−x1σ x2σ s−σ2
1 s−σ
x1σ 1
0
x2sξσ−sdξ−x2σ 1
0
x1sξσ−sdξ
;
J2s, σ x2s−x2σx1σ−x1s−x1σx2σ s−σ3
1 s−σ
x1σ 1
0
ξx2sξσ−sdξ−x2σ 1
0
ξx1sξσ−sdξ
.
3.34
Then
∂Ts, σ
∂s
1 s−σ
x1σ 1
0
1−2ξx2sξσ−sdξ−x2σ 1
0
1−2ξx1sξσ−sdξ
Ks, σ s−σ ,
3.35
where Ks, σ ∈ C0,λa, b ×a, b and Ks, s 0. According to 7, Section 5.7, the following representation holds
∂Ts, σ
∂s
K∗s, σ
|s−σ|1−λ/4, 3.36
K∗s, σ ∈ C0,3λ/4a, b×a, b. Using properties of H ¨older functions 7, we obtain the representation
∂
∂s
∂lnxs−yσ
∂ny
1 gs, σ
∂Ts, σ
∂s Ts, σ∂
∂s 1 gs, σ K1s, σ
|s−σ|1−λ/4 K2s, σ,
3.37
whereK1s, σ∈C0,3λ/4a, b×a, b, K2s, σ∈C0,λa, b×a, b. By formal differentia- tion under the integral, we find
dI1s ds − 1
2π γμσ∂
∂s
∂lnxs−yσ
∂ny dσ
− 1
2π γμσ K1s, σ
|s−σ|1−λ/4dσ− 1
2π γμσK2s, σdσ.
3.38
The validity of differentiation under the integral can be proved in the same way as at the end of9, Section 1.6 Fubini theorem on change of integration order is used. Taking into account the obtained representation fordI1s/dsand applying results of7, Section 51.1, we obtain thatdI1s/ds∈C0,λ/4a, b. Lemma is proved.
4. Existence of A Classical Solution
We will construct the solution to the problem D1in assumption thatFs, F−s∈C1,λΓ1, λ ∈0,1,Fs∈ C0Γ2. Note that we do not require compatibility conditions at the tips of the cracks, that is, we we do not require thatFd F−dfor anyxd ∈X. We will look for a solution to the problem D1in the form
ux −w
F−F−
x vx, 4.1
where
w
F−F−
x − 1 2π Γ1
Fσ−F−σ ∂
∂nylnx−yσdσ 4.2
is the double layer potential. The potentialwF−F−xsatisfies the Laplace equation2.7 inD\Γ1and belongs to the class K1according toTheorem 3.1. Limiting values of the potential wF−F−xonΓ1±are given by the formula
w
F−F− x
xs∈Γ1± ∓Fs−F−s
2 w
F−F−
xs, 4.3
wherewF−F−xsis the direct value of the potential onΓ1.
The functionvxin4.1must be a solution to the following problem.
Problem D
Find a functionvx ∈ C0D∩C2D \Γ1, which obeys the Laplace equation2.7in the domainD \Γ1and satisfies the boundary conditions
vx|xs∈Γ1
Fs F−s
2 w
F−F−
xs fs, vx|xs∈Γ2 Fs w
F−F−
xs fs.
4.4
Ifxs∈Γ1, thenwF−F−xsis the direct value of the potential onΓ1. IfDis an exterior domain, then we add the following condition at infinity:
|vx| ≤const, |x|
x12x22−→ ∞. 4.5 All conditions of the problem D have to be satisfied in a classical sense. Obviously, wF−F−xs∈C0Γ2. It follows fromLemma 3.3thatwF−F−xs∈C1,λ/4Γ1 here bywF−F−xswe mean the direct value of the potential onΓ1. So,fs ∈ C1,λ/4Γ1 andfs∈C0Γ2.
We will look for the functionvxin the smoothness class K.
We say that the functionvxbelongs to the smoothness class K if
1vx∈C0D∩C2D\Γ1, ∇v∈C0D \Γ1\Γ2\X, whereXis a pointset consisting of the endpoints ofΓ1.
2in a neighbourhood of any point xd ∈ X for some constantsC > 0,δ > −1 the inequality |∇v| ≤ C|x−xd|δ holds, where x → xd and d a1n or d b1n, n 1, . . . , N1.
The definition of the functional classC0D \Γ1\Γ2\Xis given in the remark to the definition of the smoothness class K1. Clearly, K⊂K1, that is, ifvx∈K, thenvx∈K1.
It can be verified directly that ifvxis a solution to the problem D in the class K, then the function4.1is a solution to the problem D1.
Theorem 4.1. LetΓ∈C2,λ/4,fs∈C1,λ/4Γ1,λ∈0,1,fs∈C0Γ2. Then the solution to the problem D in the smoothness class K exists and is unique.
Theorem 4.1has been proved in the following papers:
1in2,3, ifDis an interior domain;
2in4, ifDis an exterior domain andΓ2/∅;
3in5,6, ifΓ2 ∅and soD R2is an exterior domain.
In all these papers, the integral representations for the solution to the problem D in the class K are obtained in the form of potentials, densities in which are defined from the uniquely solvable Fredholm integroalgebraic equations of the second kind and index zero.
Uniqueness of a solution to the problem D is proved either by the maximum principle or by the method of energyintegralidentities. In the latter case, we take into account that a solution to the problem belongs to the class K. Note that the problem D is a particular case of more general boundary value problems studied in3–6.
Note that conditions of Theorem 4.1 hold if Γ ∈ C2,λ, Fs ∈ C1,λΓ1, F−s ∈ C1,λΓ1,λ ∈ 0,1, Fs ∈ C0Γ2. From Theorems 3.1and 4.1we obtain the solvability of the problem D1.
Theorem 4.2. LetΓ∈C2,λ, Fs∈C1,λΓ1, F−s∈C1,λΓ1, λ∈0,1, Fs∈C0Γ2. Then a solution to the problem D1exists and is given by the formula4.1, wherevxis a unique solution to the problem D in the class K ensured byTheorem 4.1.
Uniqueness of a solution to the problem D1 follows from Theorem 2.2. In fact, the solution to the problem D1found inTheorem 4.2is a classical solution. Let us discuss, under which conditions this solution to the problem D1is not a weak solution.
5. Nonexistence of a Weak Solution
Let ux be a solution to the problem D1 defined in Theorem 4.2 by the formula 4.1.
Consider a disc Sd, with the center in the pointxd ∈ X and of radius > 0 d a1n ord b1n, n 1, . . . , N1. In doing so,is a fixed positive number, which can be taken small enough. Sincevx ∈K, we havevx∈ L2Sd, and|∇vx| ∈ L2Sd, this follows from the definition of the smoothness class K. Letx ∈ Sd, andx /∈ Γ1. It follows from 4.1that|∇wμx| ≤ |∇ux||∇vx|, whence
∇w μ
x2≤ |∇ux|2|∇vx|22|∇ux| · |∇vx|
≤2
|∇ux|2|∇vx|2 ,
5.1
since 2|∇ux| · |∇vx| ≤ |∇ux|2|∇vx|2. Assume that|∇ux|belongs toL2Sd, , then, integrating this inequality overSd, , we obtain∇w2|L2Sd, ≤ 2∇u2|L2Sd,
∇v2|L2Sd,. Consequently, if|∇ux| ∈ L2Sd, , then|∇w| ∈ L2Sd, . However, according to Theorem 3.1, if Fd− F−d/0, then |∇w| does not belong to L2Sd, . Therefore, ifFd/F−d, then our assumption that|∇u| ∈L2Sd, does not hold, that is,
|∇u|∈/L2Sd, . Thus, if among numbersa11, . . . , a1N
1, b11, . . . , b1N
1there exists such a number d thatFd/F−d, then for some > 0, we have|∇u| ∈/ L2Sd, L2Sd, \Γ1, so u /∈ H1Sd, \ Γ1, where H1 is a Sobolev space of functions from L2, which have generalized derivatives fromL2. We have proved the following theorem.
Theorem 5.1. Let conditions ofTheorem 4.2hold and among numbersa11, . . . , a1N
1, b11, . . . , b1N
1there exists such a numberd, that Fd/F−d (i.e., compatibility condition does not hold at the tip xd ∈ X). Then the solution to the problem D1, ensured by Theorem 4.2, does not belong to H1Sd, \Γ1for some > 0, whence it follows that it does not belong to Hloc1 D \Γ1. Here Sd, is a disc of a radiuswith the center in the pointxd∈X.
ByHloc1 D \Γ1we denote a class of functions, which belong toH1 on any bounded subdomain of D \ Γ1. If conditions of Theorem 5.1 hold, then the unique solution to the problem D1, constructed inTheorem 4.2, does not belong to Hloc1 D \Γ1, and so it is not a weak solution. We arrive to the following corollary.
Corollary 5.2. Let conditions ofTheorem 5.1hold, then a weak solution to the problem D1in the class of functionsHloc1 D \Γ1does not exist.
Remark 5.3. It should be stressed that even if closed curves and cracks are very smooth and if boundary data is very smooth as well, but if there exists a tip of the crack, where the compatibility condition does not hold, then a weak solution of the problem D1in the class of functionsHloc1 D \Γ1does not exist.
Remark 5.4. Even if the numberd, mentioned inTheorem 5.1, does not exist, then the solution uxto the problem D1, ensured byTheorem 4.2, may not be a weak solution to the problem D1. The Hadamard example of a nonexistence of a weak solution to a harmonic Dirichlet problem in a disc with continuous boundary data is given in11, Section 12.5 the classical solution exists in this example.
Clearly,L2D \Γ1 L2D, sinceΓ1is a set of zero measure.
6. Singularities of the Gradient of the Solution at the Endpoints of the Cracks
It follows from Theorems4.2and4.1that the gradient of the solution of problem D1given by formula4.1can be unbounded at the endpoints of the cracksΓ1.
Letvxbe a solution of the Problem D ensured byTheorem 4.1. Letuxbe a solution of the Problem D1 ensured by Theorem 4.2 and given by formula 4.1. Let xd ∈ X be one of the endpoints ofΓ1. In the neighbourhood ofxd, we introduce the system of polar coordinates
x1 x1d |x−xd|cosψx, xd, x2 x2d |x−xd|sinψx, xd. 6.1
We assume thatψx, xd∈αd, αd 2πifd a1n, andψx, xd∈αd−π, αd π ifd b1n. We recall thatαsis the angle between the direction of theOx1axis and the tangent vectorτxtoΓ1at the pointxs.
Hence,αd αa1n0ifd a1n, andαd αb1n−0ifd b1n.
Thus, the angleψx, xdvaries continuously in the neighbourhood of the endpoint xd, cut alongΓ1.
Let μs be a solution of the integral equation ensured by Theorem 4 in 2 or by Theorem 4.4 in3or by Theorem 4 in4. The integral representation for the solutionvx
of the Problem D is constructed in2–4on the basis of the functionμsthat is a solution of the certain integral equation.
Alternatively, one can assume that μs is an element of a solution to equations ensured by Corollary 3.2 in5or by Theorem 4 in6. The solutionvxof the Problem D is constructed in5,6with the help of the functionμs, which is an element of a solution to certain equations.
We will use the notationμ1s μs|s−d|1/2, and putμ1d μ1a1n μ1a1n0if d a1n, and μ1d μ1bn1 μ1b1n−0ifd b1n.
At first, we study the behaviour of the gradient of a solutionvxof the problem D at the tips of the cracks. Using the representation of the derivatives of harmonic potentials in terms of Cauchy type integralssee10and using the properties of these integrals near the endpoints of the integration line, presented in7, we can prove the following assertion.
Theorem 6.1. Let vx be a solution of the problem D ensured by Theorem 4.1. Let xd be an arbitrary endpoint of the cracksΓ1, that is,xd∈Xandd a1nord b1nfor somen 1, . . . , N1. Then the derivatives of the solution of the problem D in the neighbourhood ofxdhave the following asymptotic behaviour.
Ifd a1n, then
∂
∂x1vx μ1
a1n 2x−xa1n1/2sin
ψ x, x
a1n α
a1n 2
O1,
∂
∂x2vx − μ1
a1n
2x−xa1n1/2cos ψ
x, x a1n
α a1n 2
O1,
6.2
Ifd b1n, then
∂
∂x1vx − μ1
b1n
2x−xb1n1/2 cos ψ
x, x bn1
α b1n 2
O1,
∂
∂x2vx − μ1
b1n 2x−xb1n1/2sin
ψ x, x
b1n α
b1n 2
O1.
6.3
ByO1we denote functions which are continuous at the endpointxd. Moreover, the functions denoted as O1 are continuous in the neighbourhood of the endpoint xd cut along Γ1 and are continuously extensible toΓ1and toΓ1−from this neighbourhood.
The formulas of the theorem demonstrate the following curious fact. In the general case, the derivatives of the solution of problem D near the endpointxdof cracksΓ1behave asO|x−xd|−1/2. However, ifμ1d 0, then∇vxis bounded and even continuous at the endpointxd∈X.
On the basis of Theorem 3.1, Theorem 6.1 and formula 4.1, we may study the behaviour of the gradient of a solutionuxof the problem D1at the tips of the cracks. Using notations introduced above and notations fromSection 3, we arrive at the following assertion.
