GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 6, 1999, 525-536

ON THE PRANDTL EQUATION

R. DUDUCHAVA AND D. KAPANADZE

Abstract. The unique solvability of the airfoil (Prandtl) integrodifferential equation on the semi-axis ℝ^{+} = [0, ∞) is proved in the Sobolev space W_{p}^{1} and Bessel potential spaces H_{p}^{s} under certain restrictions on p and s.

§ 0. Introduction

The purpose of this paper is to investigate the integro-differential equation

Aν(t) = ν(t) − \frac{λ}{π} ∫_{0}^{∞} \frac{ν′(τ)}{τ − t} dτ = 0, t ∈ ℝ_{+}, λ = const > 0, (0.1)

which is known as the Prandtl equation.

Such equations occur, for instance, in elasticity theory (see [1] and § 2 below), hydrodynamics (aircraft wing motion, see [2]–[5]).

In elasticity theory, a solution ν(t) of (0.1) is sought for in the Sobolev space W_{p}^{1}(ℝ_{+}) and satisfies the boundary condition

ν(0) = c_{0} ≠ 0,  (0.2)

where the constant c_{0} is defined by elastic constants (see (2.12) below).

Theorem 0.1. Equation (0.1) with the boundary condition (0.2) has a unique solution in Sobolev spaces W_{p}^{1} if and only if 1 < p < 2.

The proof of the theorem is given in § 4.

We shall consider the nonhomogeneous equation corresponding to (0.1)

Aν(t) = f(t)  (0.3)

which will be treated as a pseudodifferential equation in the Bessel potential spaces, namely, A maps H̃_{p}^{s}(ℝ_{+}) into the space H_{p}^{s−1}(ℝ_{+}), s∈ℝ, 1<p<∞.

525 The necessary and sufficient conditions for equation (0.1) to be Fredholm are given and the index formula is derived (Theorem 3.1).

1991 Mathematics Subject Classification. 45J05, 45E05, 47B35.

Key words and phrases. Prandtl equation, Sobolev space, Fredholm operator, convolution.

1072-947X/99/1100-0525$16.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

In [1, § 32] the boundary value problem (0.1), (0.2) is solved by means of the Wiener–Hopf method. Applying the Fourier transform, equation (0.1) is reduced to a boundary value problem of function theory (BVPFTh) which is solved by standard procedures (see [3]).

As is known, an equivalent reduction of problem (0.1), (0.2) to the corresponding BVPFTh is possible only for Hilbert spaces H_{2}^{s}(ℝ_{+}), whereas for spaces H_{p}^{s}(ℝ_{+}), p ≠ 2, the BVPFTh should be considered in the complicated space FH_{p}^{s}(ℝ_{+}) that is not described exactly. Theorem 0.1 clearly implies that the case p = 2 is not suitable for considering problem (0.1), (0.2), whereas the case 1 < p < 2 can be treated directly, without applying the Fourier transform.

In this paper we develop a precise theory of the boundary value problem (0.1), (0.2) in the spaces H_{p}^{s}(ℝ_{+}) and W_{p}^{1}(ℝ_{+}) and suggest criteria (necessary and sufficient conditions) for its solvability.

Remark. By the results of [3], [6], the solution of equation (0.1) has the asymptotics

ν(t) = c_{0} + c_{1}t^{\frac{1}{2}} + o(t^{\frac{1}{2}} ), as t → 0, c_{1} = const ≠ 0.

A full asymptotic expression of the solution can be derived, but this makes the subject of a separate investigation.

§ 1. Basic Notation and Spaces

Let us recall some standard notation:

ℝ is the one-dimensional Euclidean space.

L_{p}(ℝ) (1 < p < ∞) is the Lebesgue space.

S(ℝ) is the Schwarz space of infinitely smooth functions rapidly vanishing at infinity.

S′(ℝ) is the dual Schwarz space of tempered distributions.

The Fourier transform

Fφ(ξ) = ∫_{ℝ} e^{iξx}φ(x) dx, x ∈ ℝ,  (1.1)

and the inverse Fourier transform

F^{−1}φ(x) = \frac{1}{2π} ∫_{ℝ} e^{−ixξ}φ(ξ) dξ, ξ ∈ ℝ,  (1.2)

are the bounded operators in both spaces S(ℝ) and S′(ℝ). Hence the convolution operator

a(D)φ = W_{a}^{0}φ := F^{−1}aFφ with a ∈ S′(ℝ), φ ∈ S(ℝ), (1.3)

is the bounded transformation from S(ℝ) into S′(ℝ) (see [7]).

The Bessel potential space H_{p}^{s}(ℝ) (s ∈ ℝ, 1 < p < ∞) is defined as a subset of S′(ℝ) endowed with the norm

∥φ| H_{p}^{s}(ℝ)∥ := ∥⟨D⟩^{s}φ| L_{p}(ℝ)∥, where ⟨ξ⟩^{s} = (1 + |ξ|^{2})^{\frac{s}{2}} . (1.4)

For a non-negative integer s ∈ N_{0} = {0, 1, … } the space H_{p}^{s}(ℝ) coincides with the Sobolev space W_{p}^{s}(ℝ), and in that case the equivalent norm is defined as follows:

∥φ| H_{p}^{s}(ℝ)∥ ≃ ∑_{k=0}^{s} ∥∂^{k}φ| L_{p}(ℝ)∥ provided s ∈ N_{0},  (1.5)

where ∂ denotes a (generalized) derivative.

The space H̃_{p}^{s}(ℝ_{+}) is defined as a subspace of H_{p}^{s}(ℝ) of the functions φ ∈ H_{p}^{s}(ℝ) supported in the half-space supp φ ⊂ \overline{ℝ}_{+}, where H_{p}^{s}(ℝ_{+}) denotes distributions φ on ℝ_{+} which admit an extension l_{+}φ ∈ H_{p}^{s}(ℝ). Therefore r_{+}H_{p}^{s}(ℝ) = H_{p}^{s}(ℝ_{+}).

If the convolution operator (1.3) has the bounded extension

W_{a}^{0} : L_{p}(ℝ) → L_{p}(ℝ),

then we write a ∈ M_{p}(ℝ). For μ ∈ ℝ let

M_{p}^{(μ)}(ℝ) = {⟨ξ⟩^{μ}a(ξ) : a ∈ M_{p}(ℝ)}.  (1.6)

The following fact is valid:

The operator

W_{a}^{0} : H_{p}^{s}(ℝ) → H_{p}^{s−μ}(ℝ)

is bounded if and only if a ∈ M_{p}^{(μ)}(ℝ).

P C_{p}(ℝ) will denote the closure of an algebra of piecewise-constant functions by the norm

∥a∥_{p}^{0} = ∥W_{a}^{0}| L_{p}∥.

Note that for 1 < p < ∞ all functions of bounded variation belong to P C_{p}(ℝ).

S_{ℝ} denotes the Cauchy singular integral operator

S_{ℝ}ν(t) = \frac{1}{πi} ∫_{ℝ} \frac{ν(τ)}{τ − t} dτ,  (1.7)

where the integral is understood in a sense of the Cauchy principal value:

S_{ℝ}ν(t) = \frac{1}{πi} lim_{N→∞} lim_{ε→0} ( ∫_{−N}^{t−ε} + ∫_{t+ε}^{N} )\frac{ν(τ)}{τ − t} dτ.

§ 2. Half-Plane with a Semi-Infinite Stringer along the Border [1]

Let us consider an elastic plate lying in the complex lower half-plane z = x + iy, y < 0. Superpose the stringer axis on the positive part of the real axis so that one stringer end would take its origin at 0 and the other would tend to infinity.

It is assumed that the stringer is an elastic line to which tensile force is applied. It is also assumed that stresses within the plate and the stringer are produced by a single axial force applied to the stringer origin 0 and directed along the negative x-axis.

Let E be the elastic constant of the plate, E_{0} the elastic modulus of the stringer, h the plate thickness, and S_{0} the stringer cross-section; h and S_{0} are assumed to be constant values.

According to the condition, the part of the half-plane border (on the lefthand side of the origin) is free from load. Therefore the boundary conditions are written as

σ_{y} = τ_{xy} = 0 for x < 0,  (2.1)

where σ_{x}, σ_{y}, σ_{xy} are the stress components. On the other part of the border, where the plate is reinforced by the stringer, forces are in the state of equilibrium and there is no bending moment, the boundary conditions read as

p_{0} − h ∫_{0}^{x} τ_{xy} dt + kσ_{x} = 0, −h ∫_{0}^{x} σ_{y} dt = 0 for x > 0, (2.2)

where k = \frac{E_{0}S_{0}}{E}. Combined together, the latter conditions acquire the form

p_{0} − h ∫_{0}^{x}(τ_{xy} + iσ_{y}) dt + kσ_{x} = 0 (x > 0).  (2.3)

Let us recall the well-known Kolosov–Muskhelishvili representation

σ_{x} + σ_{y} = 2[φ′(z) + \overline{φ′(z)}], σ_{y} − σ_{x} + 2iτ_{xy} = 2[\overline{z}φ′′(z) + ψ′(z)] (2.4)

(see [3]) and the Muskhelishvili formula

−i ∫_{0}^{t}(τ_{xy} + iσ_{y}) dτ = φ(t) + t\overline{φ′(t)} + \overline{ψ(t)} + const .  (2.5)

By virtue of (2.4) and (2.5) we can rewrite (2.1) and (2.3) as

\begin{matrix}φ(t) + t\overline{φ′(t)} + \overline{ψ(t)} = 0 (t < 0), \\ ip_{0} + h[φ(t) + t\overline{φ′(t)} + \overline{ψ(t)}] + \\ +ik Re [φ′(t) + \overline{φ′(t)} − t\overline{φ′′(t)} − \overline{ψ′(t)}] = 0 (t > 0), \end{matrix}  (2.6)

with some nonessential constants omitted.

To solve problem (2.6), we are to find a function w(t) = μ(t) + iν(t) on [0, ∞] which is related to the complex potentials φ(z), ψ(z) by the formulae

ψ(z) = −\overline{φ(z)} − zφ′(z), y < 0 (z = x + iy),  (2.7)
φ(z) = −\frac{p_{0}}{2πh} ln z + φ_{0}(z),  (2.8)
φ_{0}(z) = −\frac{1}{2πi} ∫_{0}^{∞} \frac{ω(τ)}{τ − z} dτ,  (2.9)

where under ln z we mean any fixed branch, say arg z = 0 when x > 0, y = 0.

For the function w(t) we assume that w(t) ∈ L_{p}(ℝ_{+}) for some p > 1, w′(t) ∈ L_{1}(ℝ_{+}).

For the function w(t) we get

μ(t) = 0,  (2.10)
ν(t) − \frac{λ}{π} ∫_{0}^{∞} \frac{ν′(τ)}{τ − t} dτ = 0 (t > 0)  (2.11)

where

λ = \frac{2E_{0}S_{0}}{Eh}

and

ν(0) = −\frac{p_{0}}{h} .  (2.12)

Thus for the density of integral (2.9) we have obtained the Prandtl equation (2.11) and the boundary condition (2.12).

One can readily obtain equation (2.11) by considering the problem of an infinite plane with a half-infinite stringer attached along the half-axes ℝ_{+}.

§ 3. A Nonhomogeneous Equation in Bessel Potential Spaces

Lemma 3.1. The Prandtl operator

Aν(t) = ν(t) − \frac{λ}{π} ∫_{0}^{∞} \frac{ν′(τ)}{τ − t} , λ > 0,  (3.1)

emerging in equation (2.11) is a convolution operator

Aν(t) = F^{−1}(1 + λ|x|)Fν(t)

with the symbol 1 + λ|x| ∈ M_{p}^{(1)}(ℝ) of first order (see [8]).

Proof. Note that

FS_{ℝ}ν(t) = F(\frac{1}{πi} ∫_{ℝ} \frac{ν(τ)}{τ − t} dτ) = − sgn xFν(t)

[8, § 1] and

F(ν′(t)) = −ixFν(t).

Therefore

FAν(t) = (1 − iλ(−ix)(− sgn x))Fν(t) = (1 + λ|x|)Fν(t)

and the operator

r_{+}A : H̃_{p}^{s}(ℝ_{+}) → H_{p}^{s−1}(ℝ_{+}), 1 < p < ∞, s ∈ ℝ,  (3.2)

is bounded [8, § 5].

Let us investigate operator (3.1).

Theorem 3.1. Let s ∈ ℝ and s = [s] + {s}, [s] = 0, ±1, ±2, …, 0 ≤ {s}<1, be the decomposition of s into the integer part and the fractional one. The operator r_{+}A in (3.2) is Fredholm if and only if |\{s\}− \frac{1}{p}|≠ \frac{1}{2}. When the latter condition is fulfilled, the operator r_{+}A is invertible, invertible from the left or invertible from the right provided that κ is zero, positive or negative, respectively.

Here

\begin{aligned}κ & = [s] if |\{s\} − \frac{1}{p}| < \frac{1}{2} , \\ κ & = [s] + 1 if \{s\} − \frac{1}{p} > \frac{1}{2} , \\ κ & = [s] − 1 if \{s\} − \frac{1}{p} < −\frac{1}{2}, \end{aligned}  (3.3)

and

Ind r_{+}A = −κ.

We need the following lemma from [8, § 5].

Lemma 3.2. The operators

Λ_{+}^{s} = (D + i)^{s}l_{+}, Λ_{−}^{s} = r_{+}(D − i)^{s},
(D ± i)^{±s}φ = F^{−1}(x ± i)^{±s}Fφ, φ ∈ C_{0}^{∞}(ℝ_{+}),

arrange the isomorphisms of the spaces

Λ_{+}^{s} : H̃_{p}^{s}(ℝ_{+}) → L_{p}(ℝ_{+}), Λ_{−}^{−s} : L_{p}(ℝ_{+}) → H_{p}^{s}(ℝ_{+}).  (3.4)

Proof of Theorem 3.1. Consider the lifted operator B = Λ_{−}^{s−1}r_{+}AΛ_{+}^{−s}

\begin{matrix}H̃_{p}^{s}(ℝ_{+}) & ⟶^{r_{+}A} & H_{p}^{s−1}(ℝ_{+}) \\ ↓^{Λ_{+}^{−s}} & & ↓^{Λ_{−}^{s−1}} \\ L_{p}(ℝ_{+}) & ⟶^{B} & L_{p}(ℝ_{+}) \end{matrix}.  (3.5)

Due to Lemma 3.2 the operators r_{+}A and B are isometrically equivalent and therefore it suffices to study the operator B in the space L_{p}(ℝ_{+}) (see diagram (3.5)).

The presymbol b(x) of B equals

b(x) = \frac{1 + λ|x|}{(x + i)^{s}} (x − i)^{s−1} = (\frac{x − i}{x + i})^{s} \frac{1 + λ|x|}{x − i}  (3.6)

belonging to the class P C_{p}(ℝ), 1 < p < ∞ [8].

The corresponding p-symbol reads as

b_{p}(x, ξ) = \frac{1}{2} [b̃(x − 0) + b̃(x + 0)] +
+ \frac{1}{2} [b̃(x − 0) − b̃(x + 0)] coth π(\frac{i}{p} + ξ)  (3.7)

[8, § 4], where b̃(x ± 0) = b(x ± 0), x ∈ ℝ, b̃(∞ ± 0) = b(±∞).

When s is not an integer, s ≠ 0, ±1, …, the function (\frac{x−i}{x+i})^{s} has a jump

on ℝ^{∘} = ℝ ∪ {∞} and we fix this jump at infinity, i. e., b(−∞) ≠ b(+∞).

Since s = [s] + {s}, where [s] = 0, ±1, ±2, …, 0 ≤ {s} < 1, we can rewrite b(x) as follows:

b(x) = (\frac{x − i}{x + i})^{[s]}(\frac{x − i}{x + i})^{\{s\}} \frac{1 + λ|x|}{x − i} =
= −(\frac{x − i}{x + i})^{[s]} \frac{1 + λ|x|}{(x^{2} + 1)^{\frac{1}{2}}} (\frac{x − i}{x + i})^{\{s\}− \frac{1}{2}} ≡ g(x)b^{0}(x),  (3.8)

where

g(x) = −(\frac{x − i}{x + i})^{[s]} \frac{1 + λ|x|}{(x^{2} + 1)^{\frac{1}{2}}} , b^{0}(x) = (\frac{x − i}{x + i})^{\{s\}− \frac{1}{2}} ,  (3.9)

g(x) is a continuous function and ind g = [s].

Now let us investigate the p-symbol of b^{0}(x). We shall consider three cases.

I. \{s\} = \frac{1}{2} . It is easy to show that b^{0}(x) is the continuous function b^{0}(−∞) = b^{0}(+∞) and ind b_{p}^{0} = 0.

II. 0 ≤ \{s\} < \frac{1}{2} . This situation is shown in Fig. 1, where the image of b(x) is plotted on the complex plane and the answer depends on the connecting function coth π(\frac{i}{p} + ξ) (coth z = \frac{e^{z}+e^{−z}}{e^{z}−e^{−z}}) which fills up the gap between b^{0}(±∞).

[figure: b^{0}(x); e^{2π(\{s\}− \frac{1}{2} )i}; 1] Fig. 1

Let us define the image Im b_{p}^{0}(∞, 0).

Since

b_{p}^{0}(∞, 0) = \frac{1}{2}(1 + e^{2π(\{s\}− \frac{1}{2} )i}) − \frac{1}{2}(1 − e^{2π(\{s\}− \frac{1}{2} )i})i ctg \frac{π}{ρ} ,

we obtain

Im b_{p}^{0}(∞, 0) = i[ sin(2π\{s\} − π) − ctg \frac{π}{p} + ctg \frac{π}{p} cos(2π\{s\} − π)] =

= [ − sin 2π\{s\} − ctg \frac{π}{p} − cos \frac{π}{p} cos 2π\{s\}]i =

= −2 cos π\{s\}[ sin π\{s\} + ctg \frac{π}{p} cos π\{s\}]i =

= −\frac{2 cos π\{s\}}{sin \frac{π}{p}} cos (\frac{π}{p} − π\{s\})i.

If −\frac{2 cos π\{s\}}{sin \frac{π}{p}} cos(\frac{π}{p} − π\{s\}) > 0, then ind b_{p}^{0} = −1 and this inequality implies

cos (\frac{π}{p} − π\{s\})<0 ⟹ 2πk + \frac{π}{2} < \frac{π}{p} − π\{s\}< \frac{3π}{2} + 2πk ⟹ \frac{1}{2} < \frac{1}{p} − \{s\}

because cos π\{s\} > 0, 0 ≤ \{s\} < \frac{1}{2} , −\frac{1}{2} < \frac{1}{p} − \{s\} < 1.

In a similar manner, \frac{1}{i} Im b_{p}^{0}(∞, 0) < 0 implies \frac{1}{p} − \{s\} < \frac{1}{2}, ind b_{p}^{0} = 0 and if \frac{1}{p} − \{s\} = \frac{1}{2}, then inf |b_{p}^{0}| = 0.

III. When \frac{1}{2} < \{s\} < 1, we can proceed as in the foregoing case and obtain

ind b_{p}^{0} = 1 if \frac{1}{p} − \{s\} < −\frac{1}{2} , ind b_{p}^{0} = 0 if \frac{1}{p} − \{s\} > −\frac{1}{2},

and inf |b_{p}^{0}| = 0 if \frac{1}{p} − \{s\} = −\frac{1}{2} .

ON THE PRANDTL EQUATION 533

Hence by virtue of the equality ind b_{p} = ind g + ind b_{p}^{0} we have

ind b_{p}^{0} = [s] − 1 if \{s\} − \frac{1}{p} < −\frac{1}{2} ,

inf |b_{p}| = 0 if |\{s\} − \frac{1}{p}| = \frac{1}{2} ,

ind |b_{p}| = [s] if |\{s\} − \frac{1}{p}| < \frac{1}{2},

and ind b_{p} = [s] + 1 if \{s\} − \frac{1}{p} > \frac{1}{2} .

(3.10)

Lemma 3.3. The function b (see (3.6)) has the p′-factorization

b(ξ) = b_{−}(ξ)(\frac{ξ − i}{ξ + i})^{κ}b_{+}(ξ)

(see Definition 1.22 and Theorem 4.4 in [8]). Here

I. κ = [s], b_{±}(ξ) = g_{±}(ξ)(\frac{−2i}{ξ ∓ i})^{∓(\{s\}− \frac{1}{2} )}, when |\{s\} − \frac{1}{p}| < \frac{1}{2} ,
II. κ = [s] + 1, b_{±}(ξ) = g_{±}(ξ)(\frac{−2i}{ξ ∓ i})^{∓(\{s\}− \frac{3}{2} )}, when \{s\} − \frac{1}{p} > \frac{1}{2} ,
III. κ = [s] − 1, b_{±}(ξ) = g_{±}(ξ)(\frac{−2i}{ξ ∓ i})^{∓(\{s\}+ \frac{1}{2} )}, when \{s\} − \frac{1}{p} < \frac{1}{2} ,

where

g_{±}(ξ) = ± exp \frac{1}{2}(I ± S_{ℝ}) ln \frac{1 + λ|ξ|}{(ξ^{2} + 1)^{\frac{1}{2}}} .

Proof. This fact is valid since g(ξ) = −(\frac{ξ−i}{ξ+i})^{[s]} \frac{1+λ|ξ|}{(ξ^{2}+1)^{\frac{1}{2}}} is a nonvanishing continuous function and has the following general p′-factorization which is the same for all 1 < p < ∞:

g(ξ) = g_{−}(ξ)(\frac{ξ − i}{ξ + i})^{[s]}g_{+}(ξ)

with

g_{±}(ξ) = ± exp \frac{1}{2} (I ± S_{ℝ}) ln \frac{1 + λ|ξ|}{(ξ^{2} + 1)^{\frac{1}{2}}} .

For b^{0}(ξ) we have

b^{0}(ξ) = (\frac{−2i}{ξ + i})^{\{s\}− \frac{1}{2}} (\frac{−2i}{ξ − i})^{\frac{1}{2} −\{s\}},

− \frac{1}{p} < \frac{1}{2} − \{s\} < 1 − \frac{1}{p} or |\{s\} − \frac{1}{p}| < \frac{1}{2} ,

b^{0}(ξ) = (\frac{−2i}{ξ + i})^{\{s\}− \frac{3}{2}} (\frac{ξ − i}{ξ + i})(\frac{−2i}{ξ − i})^{\frac{3}{2} −\{s\}},

− \frac{1}{p} < \frac{3}{2} − \{s\} < 1 − \frac{1}{p} or \{s\} − \frac{1}{p} > \frac{1}{2} ,

b^{0}(ξ) = (\frac{−2i}{ξ + i})^{\{s\}+ \frac{1}{2}} (\frac{ξ − i}{ξ + i})^{−1}(\frac{−2i}{ξ − i})^{− \frac{1}{2} −\{s\}},

− \frac{1}{p} < −\frac{1}{2} − \{s\} < 1 − \frac{1}{p} or \{s\} − \frac{1}{p} < −\frac{1}{2} .

§ 4. A Homogeneous Equation in the Bessel Potential Spaces

Theorem 4.1. Let

1 ≤ s < \frac{1}{p} + \frac{1}{2}  (4.1)

and A be the operator defined by equation (3.1). Then the operator

r_{+}A : H_{p}^{s}(ℝ_{+}) → H_{p}^{s−1}(ℝ_{+})  (4.2)

is Fredholm and

Ind r_{+}A = 1.  (4.3)

Proof. Since 0 ≤ s − 1 < \frac{1}{p}, the spaces H̃_{p}^{s−1}(ℝ_{+}) and H_{p}^{s−1}(ℝ_{+}) can be identified (see [9, Theorem 2.10.3c]); thus

∂ : H_{p}^{s}(ℝ_{+}) → H_{p}^{s−1}(ℝ_{+}) = H̃_{p}^{s−1}(ℝ_{+}), ∂u(x) := \frac{du(x)}{dx} ,

is a bounded operator. Now

r_{+}A = I − 2iλS_{ℝ_{+}}∂ : H_{p}^{s}(ℝ_{+}) → H_{p}^{s−1}(ℝ_{+}), S_{ℝ_{+}}u(x) = \frac{1}{πi} ∫_{0}^{∞} \frac{u(y) dy}{y − x}

is bounded because S_{ℝ_{+}} : H̃_{p}^{θ}(ℝ_{+}) → H_{p}^{θ}(ℝ_{+}) is bounded for arbitrary θ ∈ ℝ [8, § 5] and the embedding H_{p}^{s}(ℝ_{+}) ⊂ H_{p}^{s−1}(ℝ_{+}) is continuous [9, § 2.8].

Next we have to show that dim Ker r_{+}A = 1.

Let us fix arbitrary u_{0} ∈ H_{p}^{s}(ℝ_{+}) with u_{0}(0) = 1 (note that u(0) exists due to the embedding H_{p}^{s}(ℝ_{+}) ⊂ C(ℝ_{+}) [9, § 2.8]). Then

H_{p}^{s}(ℝ_{+}) = H̃_{p}^{s}(ℝ_{+}) + {λu_{0}}_{λ∈ℂ}  (4.4)

because an arbitrary function v ∈ H_{p}^{s}(ℝ_{+}) can be represented as

v = v_{0} + v(0)u_{0}, v_{0} = v − v(0)u_{0} ∈ H̃_{p}^{s}(ℝ_{+}).

Since u_{0} ∈ H_{p}^{s}(ℝ_{+}), we have r_{+}Au_{0} ∈ H_{p}^{s−1}(ℝ_{+}) and due to Theorem 3.1 (r_{+}A is invertible) there exists a function φ_{0} ∈ H̃_{p}^{s}(ℝ_{+}) such that φ_{0} = −r_{+}A(r_{+}Au_{0}). Then φ_{0} + u_{0} ∈ Ker r_{+}A because r_{+}A(φ_{0} + u_{0}) = 0.

Now let v_{1}, v_{2} ∈ Ker r_{+}A. Due to Theorem 3.1 v_{k}(0) ≠ 0 because if v_{k} ∈ H̃_{p}^{s}(ℝ_{+}) ∩ Ker r_{+}A, then v_{k} = 0 (k = 1, 2). For the same reason v = v_{1}− \frac{v_{1}(0)}{v_{2}(0)} v_{2} = 0, because v(0) = 0 and v ∈ Ker r_{+}A. Thus dim Ker r_{+}A = 1.

From (4.4) we obtain H_{p}^{s}(ℝ_{+}) = H̃_{p}^{s}(ℝ_{+}) + Ker r_{+}A and by Theorem 3.1 we conclude that r_{+}AH_{p}^{s}(ℝ_{+}) = H_{p}^{s−1}(ℝ_{+}), i.e., dim Co Ker r_{+}A = 0. The results obtained imply that (4.2) is Fredholm and (4.3) holds.

Proof of Theorem 0.1. We know that

H_{p}^{s}(ℝ_{+}) ⊂ W_{p_{1}}^{1}(ℝ_{+}) provided that 1<p≤p_{1} <∞, s − \frac{1}{p} ≥1 − \frac{1}{p_{1}} (4.5)

[9, § 2.8]. On the other hand, for any 1 < p_{1} < 2 we can find s and p which satisfy conditions (4.1) and (4.5). Therefore by Theorem 4.1 the solutions of the Prandtl homogeneous equations can be written as

v = v_{0} + c_{0}u_{0}, v_{0} ∈ H̃_{p}^{s}(ℝ_{+}).  (4.6)

Obviously, v(0) = c_{0} (see (0.2)) while v_{0} in (4.6) is the unique solution of the equation r_{+}Av_{0} = −c_{0}r_{+}Au_{0} ∈ H_{p}^{s−1}(ℝ_{+}) provided that 1 < p < 2 (see Theorem 3.1).

If p_{1} ≥ 2, from (4.5) we obtain

s − 1/p ≥ 1/2 .  (4.7)

Note that for \frac{1}{p} < s < \frac{1}{p} +1 operator (4.2) is bounded and representation (4.4) holds (see the proof of Theorem 4.1). Therefore for

1/p + 1/2 ≤ s < 1/p + 1  (4.8)

we obtain either [s] = 0 and 1/2 ≤ {s} − 1/p < 1, or [s] = 1 and −1/2 ≤ {s} − 1/p < 0.

By Theorem 3.1 we conclude that Ind r_{+}A = −1 provided that |\{s\}− \frac{1}{p}| ≠ \frac{1}{2} (for |\{s\} − \frac{1}{p}| = \frac{1}{2} operator (3.2) is not normally solvable as proved in [8, § 4]). Hence dim Ker r_{+}A = 0 (including the case |\{s\} − \frac{1}{p}| = \frac{1}{2}) and dim Co Ker r_{+}A = 1.

Now let us show that operator (4.2) has the trivial kernel Ker r_{+}A = {0}. For this we consider the function u_{0} ∈ H_{p}^{s}(ℝ_{+}), u_{0}(0) = 1, from the proof of Theorem 4.1. Then we cannot find φ ∈ H̃_{p}^{s}(ℝ_{+}) such that r_{+}Aφ = −c_{0}r_{+}Au_{0}, c_{0} ≠ 0. Be it otherwise, we would have

c_{0}u_{0}+φ=I(c_{0}u_{0}+φ)=2λiS_{ℝ_{+}}∂(c_{0}u_{0} + φ)∈H_{p}^{s−1}(ℝ_{+})=H̃_{p}^{s−1}(ℝ_{+}), (4.9)

since the spaces H_{p}^{s−1}(ℝ_{+}) and H̃_{p}^{s−1}(ℝ_{+}) can be identified for \frac{1}{p} − 1 < s − 1 < \frac{1}{p} . Thus c_{0} = c_{0}u_{0} + φ(0) = 0, which is a contradiction. Therefore under condition (4.8) operator (4.2) is invertible and equation (0.1) would have only a trivial solution in W_{p}^{1}(ℝ_{+}) (p ≥ 2).

If s > 1 + \frac{1}{p}, operator (4.2) is unbounded, since there exists a function u ∈ H_{p}^{s}(ℝ_{+}) with the property u′(0) ≠ 0, u′ ∈ H_{p}^{s−1}(ℝ_{+}) ⊂ C(\overline{ℝ}_{+}). Thus S_{ℝ}u′ has a logarithmic singularity at 0 and the inclusion u′ ∈ H_{p}^{s−1}(ℝ_{+}) ⊂ C(\overline{ℝ}_{+}) fails to hold.

If s = 1 + \frac{1}{p}, operator (4.2) is unbounded. Otherwise, due to the boundedness, for s = 1, 1 < p_{0} < 2, the complex interpolation theorem will imply that operator (4.2) is bounded for \frac{1}{p} + \frac{1}{2} ≤ s < \frac{1}{p} + 1, which contradict the proved part of the theorem.

References

1. A. I. Kalandiya, Mathematical methods of two-dimensional elasticity. (Russian) Mir, Moscow, 1973.

2. J. Weissinger, Über eine Erweiterung der Prandtlschen Theorie der tragenden Linie. Math. Nachr. 2(1949), H. 1/2, 46–106.

3. N. I. Muskhelishvili, Singular integral equations. (Translated from Russian) P. Noordhoff, Groningen, 1953.

4. I. N. Vekua, On the integro-differential equation of Prandtl. (Russian) Prikl. Mat. Mekh. 9(1945), No. 2, 143–150.

5. L. G. Magnaradze, On some system of singular integro-differential equations and on a linear Riemann boundary value problem. (Russian) Soobshch. Akad. Nauk Gruzin. SSR IV(1943), No. 1, 3–9.

6. O. Chkadua and R. Duduchava, Pseudodifferential equations on manifolds with boundary: the Fredholm property and asymptotics. Math. Nachr. (to appear).

7. R. V. Duduchava, On multidimensional singular integral operators, I, II. J. Operator Theory 11(1984), 41–76, 199–214.

8. R. V. Duduchava, Integral equations with fixed singularities. Teubner, Leipzig, 1979.

9. H. Triebel, Interpolation theory, function spaces, differential operators. North-Holland, Amsterdam, 1978.

(Received 20.07.1998)

Authors’ address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Aleksidze St., Tbilisi 380093

Georgia