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Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations

  • Serhii V. Gryshchuk EMAIL logo and Sergiy A. Plaksa
Published/Copyright: April 1, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

We consider a commutative algebra 𝔹 over the field of complex numbers with a basis {e1, e2} satisfying the conditions (e12+e22)2=0,e12+e22≠0. Let D be a bounded simply-connected domain in ℝ2. We consider (1-4)-problem for monogenic 𝔹-valued functions Φ(xe1 + ye2) = U1(x, y)e1 + U2(x, y)i e1 + U3(x, y)e2 + U4(x, y)i e2 having the classic derivative in the domain Dζ = {xe1 + ye2 : (x, y) ∈ D}: to find a monogenic in Dζ function Φ, which is continuously extended to the boundary ∂Dζ, when values of two component-functions U1, U4 are given on the boundary ∂D. Using a hypercomplex analog of the Cauchy type integral, we reduce the (1-4)-problem to a system of integral equations on the real axes. We establish sufficient conditions under which this system has the Fredholm property and the unique solution. We prove that a displacements-type boundary value problem of 2-D isotropic elasticity theory is reduced to (1-4)-problem with appropriate boundary conditions.

Keywords: Biharmonic equation; Biharmonic algebra; Monogenic function; Schwarz-type boundary value problem; Displacements-type boundary value problem
MSC 2010: 30G35; 31A30; 74B05

1 Biharmonic monogenic functions and Schwarz-type boundary value problems for them

An associative commutative two-dimensional algebra 𝔹 with the unit 1 over the field of complex numbers ℂ is called biharmonic (see [1, 2]) if in 𝔹 there exists a basis {e1, e2} satisfying the conditions

(e12+e22)2=0,e12+e22≠0. (1)

Such a basis {e1, e2} is also called biharmonic.

In the paper [2] I. P. Melľnichenko proved that there exists the unique biharmonic algebra 𝔹, and he constructed all biharmonic bases in 𝔹. Note that the algebra 𝔹 is isomorphic to four-dimensional over the field of real numbers ℝ algebras considered by A. Douglis [3] and L. Sobrero [4].

In what follows, we consider a biharmonic basis {e1, e2} with the following multiplication table (see [1]):

e1=1,e22=e1+2ie2,

where i is the imaginary complex unit. We consider also a basis {1, ρ} (see [2]), where a nilpotent element ρ= 2e1 + 2i e2 satisfies the equality ρ2 = 0.

We use the euclidian norm ∥a∥:=|z1|2+|z2|2 in the algebra 𝔹, where a = z1e1 + z2e2 and z1, z2 ∈ ℂ. Consider a biharmonic plane μ{e1, e2} := {ζ = xe1 + ye2 : x, y ∈ ℝ} which is a linear span of the elements e1, e2 over the field of real numbers ℝ. With a domain D of the Cartesian plane xOy we associate the congruent domain Dζ := {ζ = xe1 + ye2 : (x, y) ∈ D} in the biharmonic plane μ{e1, e2} and the congruent domain DZ := {z = x + iy : (x, y) ∈ D} in the complex plane ℂ. Its boundaries are denoted by ∂D, ∂Dζ and ∂DZ, respectively. Let Dζ¯(orDZ¯) be the closure of domain Dζ (or DZ). In what follows, ζ = xe1 + ye2, z = x + iy and x, y ∈ ℝ.

We say that a function Φ : Dζ → 𝔹 is monogenic in a domain Dζ and denote Φ ∈ 𝓜𝔹(Dζ), if the derivative Φ′(ζ) exists at every point ζ ∈ Dζ:

Φ′(ζ):=limh→0,h∈μ{e1,e2}(Φ(ζ+h)−Φ(ζ))h−1.

Every Φ ∈ 𝓜𝔹(Dζ) has the derivative of any order in Dζ (cf., e.g., [5, 6]) and satisfies the equalities

(Δ2)2Φ(ζ)≡(∂4∂x4+2∂4∂x2∂y2+∂4∂y4)Φ(ζ)=Φ(4)(ζ)(e12+e22)2=0∀ζ∈Dζ.

due to the conditions (1).

Therefore, we shall also term such a function Φ by biharmonic monogenic function in Dζ. Any function Φ: Dζ → 𝔹 has an expansion of the type

Φ(ζ)=U1(x,y)e1+U2(x,y)ie1+U3(x,y)e2+U4(x,y)ie2, (2)

where Ul: D → ℝ, l = 1, 2, 3, 4, are real-valued component-functions. We shall use the following notation: Ul[Φ] := Ul, l = 1, 2, 3, 4.

All component-functions Ul, l = 1, 2, 3, 4, in the expansion (2) of any Φ ∈ 𝓜𝔹(Dζ) are biharmonic functions, i.e., satisfy the biharmonic equation (Δ2)2U(x, y)= 0 in D. At the same time, every biharmonic in a simply-connected domain D function U(x, y) is the first component U1 ≡ U in the expression (2) of a certain function Φ ∈ 𝓜𝔹(Dζ) and, moreover, all such functions Φ are found in [5, 6] in an explicit form.

It is proved in [1] that Φ ∈ 𝓜𝔹(Dζ) if and only if its each real-valued component-function in (2) is real differentiable in D and the following analog of the Cauchy – Riemann condition is fulfilled:

∂Φ(ζ)∂ye1=∂Φ(ζ)∂xe2∀ζ∈Dζ. (3)

Every Φ ∈ 𝓜𝔹(Dζ) is expressed via two corresponding analytic functions F : DZ → ℂ, F0: DZ → ℂ of the complex variable z in the form (cf., e.g., [5–7]):

Φ(ζ)=F(z)e1−(iy2F′(z)−F0(z))ρ∀ζ∈Dζ. (4)

The equality (4) establishes one-to-one correspondence between functions Φ ∈ 𝓜𝔹(Dζ) and pairs of complex-valued analytic functions F, F0 in DZ.

In what follows, we assume that the domain DZ is a bounded and simply-connected, and in this case we shall say that the domain Dζ is also bounded and simply connected.

V. F. Kovalev [8] considered the following boundary value problem for biharmonic monogenic functions: to find a function Φ ∈ 𝓜𝔹(Dζ) which is continuously extended onto the closure Dζ¯ when values of two component-functions in (2) are given on ∂Dζ, i.e., the following boundary conditions are satisfied:

Uk(x,y)=uk(ζ),Um(x,y)=um(ζ)∀ζ∈∂Dζ,1≤k<m≤4,

where uk and um are given functions.

We shall call such a problem by the (k-m)-problem. V. F. Kovalev [8] called it by a biharmonic Schwartz problem owing to its analogy with the classical Schwartz problem on finding an analytic function of the complex variable when values of its real part are given on the boundary of domain.

It was established in [8] that all biharmonic Schwartz problems are reduced to the main three problems: the (1-2)-problem or the (1-3)-problem or the (1-4)-problem.

It is shown (see [8–10]) that the fundamental biharmonic problem (cf, e.g., [11, p.146]) is reduced to the (1-3)- problem. In [9], we investigated the (1-3)-problem for cases where Dζ is either an upper half-plane or a unit disk in the biharmonic plane. Its solutions were found in explicit forms with using of some integrals analogous to the classic Schwarz integral. Similar results are obtained in [12, 13] for the (1-4)-problem.

In [10], using a hypercomplex analog of the Cauchy type integral, we reduced the (1-3)-problem to a system of integral equations and established sufficient conditions under which this system has the Fredholm property. It was made for the case where the boundary of domain belongs to a class being wider than the class of Lyapunov curves that was usually required in the plane elasticity theory (cf., e.g., [11, 14–16]).

In this paper we develop a method for reducing (1-4)-problem to a system of the Fredholm integral equations. The obtained results are appreciably similar to respective results for (1-3)-problem in [10], however, in contrast to (1-3)-problem, which is solvable in a general case if and only if a certain natural condition is satisfied, the (1-4)-problem is solvable unconditionally.

Let us underscore that (1-4)-problem is used in [17, 18] for solving a displacements-type boundary value problem on finding some partial derivatives of displacements by their limiting values in the case of 2-D isotropic elasticity theory.

Note that in the papers [4, 19–21] for investigations of biharmonic functions there are other approaches, which involve commutative finite-dimensional algebras over the field ℝ and suitable “analytic” hypercomplex functions.

2 Solving process of (1-4)-problem via analytic functions of the complex variable

A method for solving the (1-4)-problem by means of its reduction to classical Schwartz boundary value problems for analytic functions of the complex variable is delivered in [12, 13]. Let us formulate some results of such a kind.

For a continuous function u : ∂Dζ → ℝ, by u^ we denote the function defined on ∂DZ by the equality u^(z)=u(ζ)forallz=x+iy∈∂DZ.

The following theorem is proved in the papers [12, 13].

Theorem 2.1

Let ul: ∂Dζ → ℝ, l ∈{1, 4}, are continuous functions and, moreover, a function yF′(z) has a continuous extendibility to the boundary ∂DZ, where F is a solution of the classical Schwartz problem with the boundary condition

ReF(t)=u^1(t)−u^4(t)∀t∈∂DZ.

Then a solution of the (1-4)-problem is expressed by the formula (4), where a function F0 is a solution of the classical Schwartz problem with the boundary condition

ReF0(t)=12(u^4(t)−ImtImlimz→t,z∈DZF′(z))∀t∈∂DZ.

A particular case of Theorem 2.1 is the following theorem.

Theorem 2.2

A general solution of the homogeneous (1-4)-problem with u1 = u4 ≡ 0 is expressed by the formula Φ(ζ) = a1i e1 + a2e2, where a1 and a2 are arbitrary real constants.

Note, that Theorem 2.2 is proved in [12, 13] for an arbitrary (bounded or unbounded) simply connected domain Dζ.

3 Scheme for reducing the (1-4)-problem to a system of Fredholm integral equations

Let us develop a method for solving the inhomogeneous (1-4)-problem without an essential in Theorem 2.1 assumption that a function yF′(z) has a continuous extendibility to the boundary ∂DZ.

Let the boundary ∂Dζ be a closed smooth Jordan curve. We assume that every boundary function ul, l ∈{1, 4 }, satisfies the Dini condition

∫01ω(ul,η)ηdη<∞ (5)

with the modulus of continuity ω(ul,ϵ):=supζ1,ζ2∈∂Dζ:||ζ1−ζ2||≤ε∥ul(ζ1)−ul(ζ2)∥.

We seek solutions in a class of functions represented in the form

Φ(ζ)=12πi∫∂Dζ(φ1(τ)e1+φ4(τ)ie2)(τ−ζ)−1dτ∀ζ∈Dζ, (6)

where the functions φl:∂Dζ → ℝ, l ∈{1, 4}, satisfy Dini conditions of the type (5).

It is proved in Theorem 4.2 [10] that the integral (6) has limiting values Φ+(ζ0):=limζ→ζ0,ζ∈DζΦ(ζ) , Φ−(ζ0):=limζ→ζ0,ζ∈{μ{e1e2}∖Dζ¯Φ(ζ) in every point ζ0 ∈ ∂Dζ that are represented by the Sokhotski–Plemelj formulas:

Φ+(ζ0)=12φ(ζ0)+∫∂Dζφ(τ)(τ−ζ0)−1dτ, (7)

Φ−(ζ0)=−12φ(ζ0)+∫∂Dζφ(τ)(τ−ζ0)−1dτ, (8)

where φ(τ) :=φ1(τ)e1 + φ4(τ)i e2 and a singular integral is understood in the sense of its Cauchy principal value:

∫∂Dζφ(τ)(τ−ζ0)−1dτ:=limε→0∫{τ∈∂Dζ:||τ−ζ0||>ε}φ(τ)(τ−ζ0)−1dτ.

We use a conformal mapping z = τ(t) of the upper half-plane {t∈ ℂ : Im t > 0} onto the domain DZ. Denote τ1(t) :=Re τ(t), τ2(t) :=Im τ(t).

Inasmuch as the mentioned conformal mapping is continued to a homeomorphism between the closures of corresponding domains, the function

τ~(s):=τ1(s)e1+τ2(s)e2∀s∈R¯

generates a homeomorphic mapping of the extended real axis R¯:=R∪{∞} onto the curve ∂Dζ.

Introducing the function

g(s):=g1(s)e1+g4(s)ie2∀s∈R¯,

where gl(s):=φl(τ~(s)), l ∈ {1, 4}, we rewrite the equality (7) in the form (cf. [10])

Φ+(ζ0)=12g(t)+12πi∫−∞∞g(s)k(t,s)ds+12πi∫−∞∞g(s)1+st(s−t)(s2+1)ds,

where

k(t,s)=k1(t,s)e1+iρk2(t,s),k1(t,s):=τ′(s)τ(s)−τ(t)−1+st(s−t)(s2+1),k2(t,s):=τ′(s)(τ2(s)−τ2(t))2(τ(s)−τ(t))2−τ2′(s)2(τ(s)−τ(t)),

and a correspondence between the points ζ0 ∈ ∂Dζ ∖ {τ~(∞)} and t ∈ ℝ is given by the equality ζ0 = τ~(t).

Now, we single out components Ul[Φ+(ζ0)], l ∈{1, 4}, and after the substitution them into the boundary conditions of the (1-4)-problem, we shall obtain the following system of integral equations for finding the functions g1 and g4:

U1[Φ+(ζ0)]≡12g1(t)+12π∫−∞∞g1(s)(Imk1(t,s)+2Rek2(t,s))ds−1π∫−∞∞g4(s)Rek2(t,s)ds=u~1(t),U4[Φ+(ζ0)]≡12g4(t)+1π∫−∞∞g1(s)Rek2(t,s)ds++12π∫−∞∞g4(s)(Imk1(t,s)−2Rek2(t,s))ds=u~4(t)∀t∈R, (9)

where u~l(t):=ul(τ~(t)),l∈{1,4}.

Let C( R¯ ) denote the Banach space of functions g*: R¯ → ℂ that are continuous on the extended real axis R¯ with the norm ∥g∗∥C(R¯):=supt∈R|g∗(t)|.

In Theorem 6.13 [10] there are conditions which are sufficient for compactness of integral operators in equations of the system (9) in the space C( R¯ ).

To formulate such conditions, consider the conformal mapping σ(T) of the unit disk {T ∈ ℂ : |T|<1} onto the domain DZ such that τ(t)=σt−it+i for all t ∈{t ∈ ℂ : Im t > 0}.

Thus, it follows from Theorem 6.13 [10] that if the conformal mapping σ(T) have the nonvanishing continuous contour derivative σ′(T) on the unit circle Γ :={T ∈ ℂ : |T| = 1}, and its modulus of continuity

ωΓ(σ′,ε):=supT1,T2∈Γ,|T1−T2|≤ε|σ′(T1)−σ′(T2)|

satisfies a condition of the type (5), then the integral operators in the system (9) are compact in the space C( R¯ ).

Let 𝒟( R¯ ) denote the class of functions g*∈ C( R¯ ) whose moduli of continuity

ωR(g∗,ε)=supτ1,τ2∈R:|τ1−τ2|≤ε|g∗(τ1)−g∗(τ2)|,ωR,∞(g∗,ε)=supτ∈R:|τ|≥1/ε|g∗(τ)−g∗(∞)|

satisfy the Dini conditions

∫01ωR(g∗,η)ηdη<∞,∫01ωR,∞(g∗,η)ηdη<∞.

Since the functions φ1, φ4 in the expression (6) of a solution of the (1-4)-problem have to satisfy conditions of the type (5), it is necessary to require that corresponding functions g1, g4 satisfying the system (9) should belong to the class 𝒟( R¯ ). In the next theorem we state a condition on the conformal mapping σ(T), under which all solutions of the system (9) satisfy the mentioned requirement.

Theorem 3.1

Let every function ul:∂Dζ→ ℝ, l ∈{1, 4}, satisfy the condition (5). Let the conformal mapping σ(T) have the nonvanishing continuous contour derivative σ′(T) on the circle Γ, and its modulus of continuity satisfy the condition

∫02ωΓ(σ′,η)ηln3ηdη<∞.

Than the following assertions are true:

  1. the system of Fredholm integral equations (9) has the unique solution in C(R¯) ;

  2. all functions g1, g4 ∈ C(R¯) satisfying the system (9) belong to the class D(R¯) , and the corresponding functions φ1, φ4 in (6) satisfy Dini conditions of the type (5).

Proof

In such a way as in Theorem 7.2 [10], we establish that all functions g1, g4∈ C(R¯) satisfying the system (9) belong to the class D(R¯) . Thus, corresponding functions φ1, φ4 in (6) satisfy Dini conditions of the type (5).

To prove that the system (9) has the unique solution, consider the homogeneous system of equations (9) with u~1=u~4≡0.

Let g1=g10,g4=g40 be a solution of such a system. Consider the function

φ0(τ):=g10(s)e1+g40(s)ie2,τ=τ~(s),∀s∈R

corresponding to this solution.

Then the function, which is defined for all ζ∈ Dζ by the formula

Φ0(ζ):=12πi∫∂Dζφ0(τ)(τ−ζ)−1dτ, (10)

is a solution of the homogeneous (1-4)-problem with u1 = u4≡ 0 that is corresponding to the homogeneous system of equations (9). Taking into account Theorem 2.2, we have the equality Φ0(ζ) = a1i e1+a2e2 for all ζ∈ Dζ. Therefore, from the Sokhotski–Plemelj formulas (7), (8) we obtain the following relations:

φo(ζ0)=Φ0+(ζ0)−Φ0−(ζ0)=a1ie1+a2e2−Φ0−(ζ0)∀ζ0∈∂Dζ,

which imply the equality

Φ0−(ζ0)=−g10(t)e1+a1ie1+a2e2−g40(t)ie2,ζ0=τ~(t),∀t∈R. (11)

Thus, the function (10) is monogenic in μ{e1, e2}\ Dζ¯ and satisfies the equalities

U1[iΦ0−(ζ0)]=−a1,U4[iΦ0−(ζ0)]=a2∀ζ0∈∂Dζ.

Now, taking into account Theorem 2.2, which is undoubtedly true for unbounded domain μ{e1, e2}\ Dζ, one can easily conclude that Φ0 is constant in μ{e1, e2}\ Dζ. Moreover, Φ0(ζ)≡ 0 in μ{e1, e2}\ Dζ because the function (10) is vanishing at infinity. At last, after a substitution of 0 into the left-hand side of equality (11) we conclude that g10=g40≡0, i.e. the homogeneous system of equations (9) has only the trivial solution.

Therefore, by the Fredholm theory, the non-homogeneous system of equations (9) has a unique solution.  □

4 A displacements-type boundary value problem

A relation between the (1-4)-problem and a displacements-type boundary value problem of the plane elasticity theory is considered in [17, 18].

Consider the following displacements-type boundary value (ux, vy)-problem: to find in a domain D of the Cartesian plane xOy the first order partial derivatives V1:=∂u∂x,V2:=∂v∂y for displacements u = u(x, y), v = v(x, y) of an elastic isotropic body occupying D, when their limiting values are given on ∂D:

lim(x,y)→(x0,y0),(x,y)∈DVk(x,y)=hk(x0,y0)∀(x0,y0)∈∂D,k=1,2,

where hk:∂D→ ℝ, k = 1, 2, are given functions.

It is well-known the equalities (cf., e.g., [16, p. 5], [22, pp. 8–9]):

2μVk(x,y)=−Wk(x,y)+κ0(W1+W2)=:Ck(x,y)∀(x,y)∈D,k=1,2, (12)

where κ0:=λ+2μ2(λ+μ), μ and λ are Lamé constants (cf., e.g., [22, p. 2]) W1:=∂2W∂x2,W2:=∂2W∂y2, and W : D → ℝ is a biharmonic function in the domain D.

Taking into account the equalities (12), one can conclude that the (ux, vy)-problem is equivalent to finding the functions Ck(x, y), k = 1, 2, in the domain D, when their boundary values satisfy the conditions

Ck(x0,y0)=2μhk(x0,y0)∀(x0,y0)∈∂D,k=1,2. (13)

Assuming that W has continuous partial derivatives till the second order up to the boundary ∂D, we deduce that in this case (ux, vy)-problem is reduced to finding the functions 𝒲k, k = 1, 2, in the domain D, when their limiting values are given on ∂D.

Consider in the domain Dζ a function Φ*∈ 𝓜𝔹(Dζ) for which U1[Φ*] = W. Then it follows from the Cauchy–Riemann condition (3) that for the biharmonic monogenic function Φ:=Φ∗″ the following equalities hold:

U1[Φ(ζ)]=W1(x,y),U4[Φ(ζ)]=12(W1(x,y)−W2(x,y)).

Thus, in such a way (ux, vy)-problem is reduced to (1-4)-problem with appropriate boundary conditions, and Theorem 3.1 makes it possible to find the elastic equilibrium for an isotropic body occupying D. In this case, for finding stresses an additional assumption on the function W is required, but finding displacements is free of any assumptions (see [18]).

Acknowledgement

This research is partially supported by the Ministry of Education and Science of Ukraine (Project No. 0116U001528).

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Received: 2016-10-22
Accepted: 2017-2-12
Published Online: 2017-4-1

© 2017 Gryshchuk and Plaksa

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/math-2017-0025
Keywords for this article
Biharmonic equation; Biharmonic algebra; Monogenic function; Schwarz-type boundary value problem; Displacements-type boundary value problem
Creative Commons
BY-NC-ND 3.0

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