UDC 517.946.9

ON THE SOLVABILITY OF CERTAIN BOUNDARY VALUE PROBLEMS FOR ELLIPTIC SYSTEMS WITH A DEGENERATION OF THE ORDER ON THE BOUNDARY

V. P. Didenko

Let \(D\) be a simply connected domain in the variables \((x_1, x_2)\), lying in the half-plane \(x_2>0\), whose smooth boundary \(\Gamma\) consists of the segment \(AB\) of the axis \(x_2=0\) and an arc \(\sigma\) with endpoints at the points \(A\) and \(B\). In the domain \(D\) we consider a system of linear partial differential equations of second order of the form

\[ x_2^m \sum_{i,j=1}^{2} A_{ij} u_{x_i x_j}+\sum_{i=1}^{2} a_i u_{x_i}+cu=h, \tag{1} \]

where \(m\) is a number satisfying the inequality \(0\le m<1\), \(A_{ij}\), \(a_i\) are square matrices of order \(n\), whose elements are real functions of \((x_1,x_2)\); \(h=(h_1,\ldots,h_n)\) is a given real vector; \(u=(u_1,\ldots,u_n)\) is the unknown vector.

It is assumed that \(A_{ij}\in C^{(2,\alpha)}(\overline D)\), \(a_i\in C^{(1,\alpha)}(\overline D)\), \(c,h\in C^{(0,\alpha)}(\overline D)\), and that the equation

\[ \det\left(\sum_{i,j=1}^{2} A_{ij}\lambda^{i+j-2}\right)=0 \tag{2} \]

has only complex roots \(\lambda\) in the closed domain \(\overline D\). The problem is posed as follows.

It is required to find a solution \(u\) of system (1), twice continuously differentiable in the domain \(D\), belonging to the class \(C^{(1,\alpha)}(\overline D)\), and satisfying the boundary condition

\[ \overline B(u)\equiv B_1u_{x_1}+B_2u_{x_2}+B_3u=f \quad \text{on } \Gamma, \tag{3} \]

where \(f=\{f_1,\ldots,f_n\}\) is a given real vector on \(\Gamma\) \(\bigl(f\in C^{(0,\alpha)}(\Gamma)\bigr)\), \(B_1,B_2,B_3\) are real square matrices given on the boundary \(\Gamma\) and in its neighborhood belonging to the domain \(D\), with \(B_1,B_2\in C^{(1,\alpha)}\), \(B_3\in C^{(0,\alpha)}\) on \(\Gamma\).

Alongside system (1) we shall consider the system of the following form:

\[ L^*(u)\equiv x_2^m\left[\sum_{i,j=1}^{2}(uA_{ij})_{x_i x_j} -\sum_{i=1}^{2}(x_2^{-m}ua_i)_{x_i} +x_2^m uc\right]=0; \tag{1*} \]

system \((1*)\) will be called the adjoint system to (1). Below we shall define, for system \((1*)\), a boundary value problem which we shall call adjoint to problem (1), (3).

In the present paper a method is given for reducing the problem to a system of singular integral equations of normal type in all cases when condition (14), formulated below, is satisfied.

It is also proved that, when condition (14) is satisfied, the homogeneous adjoint problem has a finite number of linearly independent solutions, and for the solvability of problem (1), (3) (for \(f=0\)) it is necessary and sufficient that the vector \(h\) be orthogonal, with weight \(x_2^{-m}\), to all linearly independent solutions of the homogeneous adjoint problem.

Let us construct the matrices \(v(x,y)\) and \(\overline v(x,y)\):

\[ v(x,y)=\frac{1}{2\pi^2}\operatorname{Re}\int_\gamma \Delta(y,\lambda)\ln [x_1-y_1+\lambda(x_2-y_2)]\,d\lambda, \]

\[ \overline v(x,y)=-\frac{1}{2\pi^2}\operatorname{Re}\int_{\gamma'} \Delta(x,\lambda)\{\ln [x_1-y_1+\lambda(x_2-y_2)]- \]

\[ -\ln [x_1-y_1+\lambda(x_2+y_2)]\}\,d\lambda, \tag{4} \]

where by \(\Delta(x,\lambda)\) is denoted the matrix inverse to the matrix
\[ \left(\sum_{i,j=1}^{2} A_{ij}\lambda^{i+j-2}\right), \]
and \(\gamma\) is a contour in the half-plane \(\operatorname{Im}\lambda>0\), enclosing all roots \(\lambda\) of equation (2) lying in this half-plane. The elements of the matrix (4) are single-valued functions with respect to \(x_1,x_2,y_1,y_2\) \((x\ne y)\),

\[ \ln(t+\lambda\tau)=\ln|t+\lambda\tau|+i\arg(t+\lambda\tau),\qquad -\pi\le \arg(t+\lambda\tau)<\pi, \]

where \(t,\tau\) are real numbers. It is clear that \(\ln(t+\lambda\tau)\) is a continuous function of \(t,\tau,\lambda\), \(\lambda\in\gamma\), except for the points \(\tau=0,\ t\le 0\). The continuity of the matrices (4) at the points \(t<0,\ \tau=0\) follows from the fact that

\[ \operatorname{Re}\, i\int_\gamma \Delta(x,\lambda)\,d\lambda=0. \tag{4'} \]

We shall seek the solution of problem (1), (3) in the form

\[ u(x)=\int_D y_2^{-m}\overline v(x,y)g(y)\,dy+ \int_\Gamma v(x,y)\varphi(y)\,dS_y, \tag{5} \]

where \(g(y)\), \(\varphi(y)\) are as yet unknown \(n\)-dimensional vector-functions of the classes \(C^{(0,\alpha)}(\overline D)\) and \(C^{(0,\alpha)}(\Gamma)\), respectively. Substituting expression (5) into system (1), we obtain

\[ g(x)+\int_D y_2^{-m}L_x[\overline v(x,y)]g(y)\,dy = -\int_\Gamma L_x[v(x,y)]\varphi(y)dS_y+h(x). \tag{6} \]

It is directly verified that the following uniform estimate holds

\[ \left|y_2^{-m}L_x[\overline v(x,y)]\right|\le \frac{\mathrm{const}}{r_{xy}^{1+m}}, \]

where \(r=[(y_1,x_1)^2+(y_2-x_2)^2]^{1/2}\). Therefore expression (6) may be regarded as a system of Fredholm integral equations of the second kind with non-

of the known vector-function \(g(y)\), if \(\varphi(y)\) is assumed known. The matrix \(y_2^{-m} L_x[\overline{v}(x,y)]\) can be represented in the form of a sum of square matrices of order \(n\) as follows:

\[ y_2^{-m} L_x[\overline{v}(x,y)] = -\|\alpha_{ij}(x,y)\| - \sum_{k=1}^{l}\|\beta_{ij}^{(k)}(y)\|\gamma_k(x), \]

where

\[ \max_{1\le i\le n}\int_D \sum_{j=1}^{n}|\alpha_{ij}(x,y)|\,dy<\mathrm{const}<1, \]

\(\gamma_1(x),\ldots,\gamma_l(x)\) are linearly independent functions in the domain \(D\), and the elements of the matrices \(\|\beta_{ij}^{(k)}(y)\|\) vanish in a strip near the axis \(y_2=0\).

Let us rewrite system (6) in the form

\[ g(x)-\int_D \|\alpha_{ij}(x,y)\|g(y)\,dy = H(x)+ \int_D \sum_{k=1}^{l}\|\beta_{ij}^{(k)}(y)\|\gamma_k(x)g(y)\,dy, \tag{7} \]

where \(H(x)\) is the right-hand side of expression (6).

Solving system (7) under the assumption that the right-hand side is known, and denoting by \(M(x,y)\) the resolvent of the system of Fredholm equations of the second kind with kernel \(\|\alpha_{ij}(x,y)\|\), we obtain

\[ g(x)=\sum_{k=1}^{l}N_k(x)C_k+\int_{\Gamma}\Omega(x,y)\varphi(y)\,dS_y+ \]

\[ +\int_D M(x,y)h(y)\,dy+h(x), \tag{8} \]

where

\[ \left. \begin{aligned} N_k(x)&=\gamma_k(x)+\int_D M(x,y)\gamma_k(y)\,dy,\\[4pt] \Omega(x,y)&=-L_x[v(x,y)]-\int_D M(x,t)L_t[v(t,y)]\,dt,\\[4pt] C_i&=\sum_{j=1}^{l}K_{ij}C_j+F_i\quad (i=1,\ldots,l),\\[4pt] K_{ij}&=\int_D \beta_i^*(y)\gamma_j(y)\,dy,\\[4pt] \beta_k^*(y)&=\|\beta_{ij}^{(k)}(y)\|+\int_D \|\beta_{ij}^{(k)}(t)\|M(t,y)\,dt,\\[4pt] F_i&=\int_D \beta_i^*(y)\left\{-\int_{\Gamma}L_y[v(y,t)]\varphi(t)\,dS_t+h(y)\right\}\,dy. \end{aligned} \right\} \tag{9} \]

Substituting expression (8) into (5), we obtain

\[ \begin{aligned} u(x)={}&\int_D \Omega_1(x,y)h(y)\,dy+\int_\Gamma \Omega_2(x,y)\varphi(y)\,dS_y+{}\\ &+\int_\Gamma v(x,y)\varphi(y)\,dS_y+\sum_{k=1}^{l}\overline{N}_k(x)C_k, \end{aligned} \tag{10} \]

where

\[ \Omega_1(x,y)=y_2^{-m}\overline{v}(x,y)+\int_D t_2^{-m}\overline{v}(x,t)M(t,y)\,dt, \]

\[ \Omega_2(x,y)=\int_D t_2^{-m}\overline{v}(x,t)\Omega(t,y)\,dt, \]

\[ \overline{N}_k(x)=\int_D y_2^{-m}\overline{v}(x,y)N_k(y)\,dy. \]

Substituting now expression (10) into the boundary condition (3) and assuming that the point \(x\in D\), we obtain

\[ \int_\Gamma \left[ B_1(x)\frac{\partial v(x,y)}{\partial x_1} +B_2(x)\frac{\partial v(x,y)}{\partial x_2} +B_3(x)v(x,y) \right]\varphi(y)\,dS_y = \]

\[ =\int_\Gamma K_1(x,y)\varphi(y)\,dS_y+\overline{f}(x), \tag{11} \]

where \(K_1(x,y)\) is a completely determined kernel whose singularity for \(x=y\) is less than one, and

\[ \overline{f}(x)=f(x)-\int_D \overline{B}_x[\Omega_1(x,y)]h(y)\,dy -\sum_{k=1}^{l}\overline{B}_x[\overline{N}_k(x)]C_k, \]

\(\overline{B}_x\) is the operator defined in (3); the \(x\) below means that the operator is applied with respect to the variable \(x\). Using (4), the right-hand side of equality (11) can be rewritten in the following form:

\[ \frac{1}{2\pi^2}\int_\Gamma \operatorname{Re}\int_{\gamma} \Delta(y,\lambda)\, \frac{B_1(x)+\lambda B_2(x)} {x_1-y_1+\lambda(x_2-y_2)}\,d\lambda\,\varphi(y)\,dS_y + \]

\[ +\int_\Gamma B_3(x)v(x,y)\varphi(y)\,dS_y \]

or

\[ \frac{1}{2\pi^2}\operatorname{Re}\int_{\gamma} \Delta(x,\lambda)\, \frac{B_1(x)+\lambda B_2(x)} {\cos(\widehat{S,x_1})+\lambda\cos(\widehat{S,x_2})} \,d\lambda \int_\Gamma \frac{\varphi(y)}{y_\lambda-x_\lambda}\,dy_\lambda + \]

\[ +\int_\Gamma K_2(x,y)\varphi(y)\,dS_y, \]

where \(K_2(x,y)\) is a kernel whose singularity for \(x=y\) is less than one, \(x_\lambda=x_1+\lambda x_2,\ y_\lambda=y_1+\lambda y_2\). Passing to the limit in equality (11), when \(x\to t\in\Gamma\), and using the Sokhotski–Plemelj formula, we obtain

\[ \operatorname{Re}\left[ \frac{\pi i}{2\pi^2}\int_\gamma \Delta(t,\lambda)\, \frac{B_1(t)+\lambda B_2(t)} {\cos\widehat{(S,t_1)}+\lambda\cos\widehat{(S,t_2)}}\,d\lambda\,\varphi(t) +\right. \]
\[ \left. +\frac{1}{2\pi^2}\int_\gamma \Delta(t,\lambda)\, \frac{B_1(t)+\lambda B_2(t)} {\cos\widehat{(S,t_1)}+\lambda\cos\widehat{(S,t_2)}}\,d\lambda \int_\Gamma \frac{\varphi(y)}{y_\lambda-t_\lambda}\,dy_\lambda \right] = \]
\[ =\int_\Gamma K_3(t,y)\varphi(y)\,dS_y+\bar f(t), \tag{12} \]

where \(K_3(t,y)\) is a kernel whose singularity for \(x=y\) is less than one. It is easy to see that

\[ \operatorname{Re}\left\{ \frac{1}{2\pi^2}\int_\gamma \Delta(t,\lambda)\, \frac{B_1(t)+\lambda B_2(t)} {\cos\widehat{(S,t_1)}+\lambda\cos\widehat{(S,t_2)}}\,d\lambda \int_\Gamma \frac{\varphi(y)}{y_\lambda-t_\lambda}\,dy_\lambda \right\} = \]
\[ =\operatorname{Re}\left\{ \frac{1}{2\pi^2}\int_\gamma \Delta(t,\lambda)\, \frac{B_1(t)+\lambda B_2(t)} {\cos\widehat{(S,t_1)}+\lambda\cos\widehat{(S,t_2)}}\,d\lambda \right\} \int_\Gamma \frac{\varphi(y)}{y-t}\,dy + \]
\[ +\int_\Gamma K_4(t,y)\varphi(y)\,dS_y, \]

where \(K_4(t,y)\) is a kernel whose singularity for \(x=y\) is less than one, \(x=x_1+ix_2,\ y=y_1+iy_2\). We can now rewrite equality (12) in the following form:

\[ -\frac{1}{2\pi}\operatorname{Im}\left\{ \int_\gamma \Delta(t,\lambda)\, \frac{B_1(t)+\lambda B_2(t)} {\cos\widehat{(S,t_1)}+\lambda\cos\widehat{(S,t_2)}}\,d\lambda \right\}\varphi(t) + \]
\[ +i\frac{1}{2\pi}\operatorname{Re}\left\{ \int_\gamma \Delta(t,\lambda)\, \frac{B_1(t)+\lambda B_2(t)} {\cos\widehat{(S,t_1)}+\lambda\cos\widehat{(S,t_2)}}\,d\lambda \right\} \frac{1}{\pi i}\int_\Gamma \frac{\varphi(y)}{y-t}\,dy = \]
\[ =\int_\Gamma K_5(t,y)\varphi(y)\,dS_y+\bar f(t), \tag{13} \]

where \(K_5(t,y)\) is a completely determined kernel whose singularity for \(t=y\) is less than one.

Equality (13) is a one-dimensional system of singular integral equations of normal type, provided the condition

\[ \det\int_\gamma \Delta(t,\lambda)\, \frac{B_1(t)+\lambda B_2(t)} {\cos\widehat{(S,t_1)}+\lambda\cos\widehat{(S,t_2)}}\,d\lambda \ne 0,\qquad t\in\Gamma. \tag{14} \]

is satisfied.

For the solvability of system (13) (under condition (14)), it is necessary and sufficient that a finite number of orthogonality conditions be satisfied, which, together with conditions (9), can be written in the following form:

\[ \int_D x_2^{-n}h(x)\omega_i(x)\,dx + \int_\Gamma f(S)\alpha_i(S)\,dS =0 \qquad (i=1,\ldots,\varkappa), \tag{15} \]

where \(\omega_i,\alpha_i\) are completely determined functions of the class \(C^{(1,\alpha)}(\overline D)\cap C^2(D)\) and \(C^{(0,\alpha)}(\Gamma)\), respectively.

Thus, we arrive at the following assertion.

Theorem 1. If the vectors \(h\) and \(f\) satisfy a finite number of conditions of the form (15) and condition (14) is fulfilled, then the solution of problem (1), (3) exists and is given by formula (5).

Denote by \(\hat\varphi\) the pair \(\{h,f\}\), where \(h\) is a vector defined in the domain \(D\), and \(f\) is a vector defined on \(\Gamma\). For elements \(\hat\varphi\) we introduce the scalar product in the following way:

\[ \left<\hat\varphi_1\cdot\hat\varphi_2\right> = \int_D x_2^{-m}h_1\cdot h_2\,dx + \int_\Gamma f_1\cdot f_2\,dS. \tag{*} \]

The completion, with respect to this scalar product, of the smooth elements \(\hat\varphi\) will be denoted by \(W\). It is easy to see that \(W\) is a Hilbert space.

By \(H_0\) denote the linear span with basis vectors \(\hat\psi_i=\{\omega_i,\alpha_i\}\) from (15). Since the number of vectors \(\hat\psi_i\) is finite, \(H_0\) is a closed subspace in \(W\). Among the elements \(\{h,f\}\in H_0\) there may be some for which problem (1), (3) is solvable. Denote this set by \(H_1\). It is clear that \(H_1\) is a closed subspace of the space \(W\). Thus, the space \(H_0\) can be represented in the form \(H_0=H_1\oplus H_2\), and the space \(W\) in the form \(W=H_1\oplus H_2\oplus H_3\).

Theorem 2. Under condition (14), for the solvability of problem (1), (3) it is necessary and sufficient that the vector \(\{h,f\}\) be orthogonal to the subspace \(H_2\).

Proof. Necessity. Suppose that for \(\hat\varphi=\{h,f\}\) \(\bigl(h\in C^{(0,\alpha)}(\overline D),\ f\in C^{(0,\alpha)}(\Gamma)\bigr)\) problem (1), (3) is solvable. The element \(\hat\varphi\) can be represented in the form

\[ \hat\varphi=\hat\varphi_1+\hat\varphi_2+\hat\varphi_3,\quad \text{where } \varphi_i\in H_i\quad (i=1,2,3). \]

For \(\hat\varphi_3\), problem (1), (3) is solvable by Theorem 1; for \(\hat\varphi_1\), problem (1), (3) is solvable by construction; hence it follows that for \(\hat\varphi_2\) problem (1), (3) must be solvable.

But, by construction, in \(H_2\) there are no such elements (except zero) for which problem (1), (3) is solvable; therefore, the element \(\hat\varphi_2\) is equal to zero in the sense of the scalar product (*). Consequently, \(\hat\varphi\) is orthogonal to the subspace \(H_2\).

Sufficiency. Suppose that \(\hat\varphi=\{h,f\}\in C^{(0,\alpha)}\) (this means that \(h\in C^{(0,\alpha)}(\overline D)\), \(f\in C^{(0,\alpha)}(\Gamma)\)) and that the element \(\hat\varphi\) is orthogonal to \(H_2\); then \(\hat\varphi\) can be represented as \(\hat\varphi=\hat\varphi_1+\hat\varphi_3\), where \(\hat\varphi_1\in H_1\), \(\hat\varphi_3\in H_3\). For \(\hat\varphi_3\), problem (1), (3) is solvable by Theorem 1; for \(\hat\varphi_1\), problem (1), (3) is solvable by the construction of \(H_1\); hence problem (1), (3) is solvable also for \(\hat\varphi\). The theorem is completely proved.

It is obvious that, by a change of variables, problem (1), (3) can be reduced to a problem with homogeneous boundary conditions:

\[ L(u)=h, \tag{16} \]

\[ \overline B u\big|_\Gamma=0. \tag{17} \]

In what follows, for simplicity of reasoning we shall consider problem (16), (17) instead of problem (1), (3).

Let us note that condition (15) for problem (16), (17) will have the form

\[ \int_D x_2^{-m} h(x)\omega_i(x)\,dx=0 \qquad (i=1,\ldots,\chi). \tag{18} \]

For the operators (1) and (1*) we write the following identity:

\[ \int_D x_2^{-m}\bigl[vL(u)-L^*(v)u\bigr]\,dx = \int_D\left\{ \sum_{i,j=1}^{2}\left[(vA_{ij}u_{x_i})_{x_j}-\bigl((vA_{ij})_{x_j}u\bigr)_{x_i}\right] -\sum_{i=1}^{2}(x_2^{-m}va_i u)_{x_i} \right\}dx. \tag{19} \]

The integral appearing on the right-hand side of (19) is transformed into an integral over the boundary \(\Gamma\), and we denote it by \(J(v,u)\). Denote by \(B^*\) the set of vectors \(v\in C^{(1,\alpha)}(\overline D)\) such that the equality

\[ J(v,u)=0 \tag{20} \]

holds for all \(u\in C^{(1,\alpha)}(\overline D)\) satisfying the boundary condition (17).

We now define the problem adjoint to problem (16), (17).

It is required to find a solution \(v\), twice continuously differentiable in the domain \(D\), of the system

\[ L^*(v)=0, \tag{21} \]

satisfying the boundary condition

\[ v\in B^*. \tag{22} \]

Theorem 3. If conditions (14) are fulfilled, then the number of linearly independent solutions of the homogeneous adjoint problem (21), (22) is finite.

Proof. Suppose that problem (21), (22) has an infinite number of linearly independent solutions \(v_i\). Then, for the solvability of problem (16), (17), it is necessary that an infinite number of conditions be fulfilled:

\[ \int_D x_2^{-m} h(x)v_i(x)\,dx=0 \qquad (i=1,2,\ldots,\infty). \]

This follows directly from identity (19) and the definition of the adjoint problem. But Theorem 2 asserts that, for the solvability of problem (16), (17), it is sufficient that only a finite number of orthogonality conditions of the type (18) be fulfilled. Hence problem (21), (22) has only a finite number of linearly independent solutions.

The theorem is completely proved.

Denote by \(H^*\) the set of linearly independent solutions of problem (21), (22).

Theorem 4. The spaces \(H_2\) (for \(f=0\)) and \(H^*\) coincide.

Proof. Let \(\varphi\in H^*\). Then, for the solvability of problem (16), (17), the condition

\[ \int_D x_2^{-m}h(x)\varphi(x)\,dx=0 \]

is necessary.

Hence, and from Theorem 2, it follows that \(\varphi \in H_2\). Consequently, \(H^* \subset H_2\).

Let \(\psi \in H_2\) (where \(\psi \in C^{(1,\alpha)}(\overline D)\cap C^{(2,0)}(D)\)); then, by Theorem 2, the condition

\[ \int_D x_2^{-m} h(x)\psi(x)\,dx=0 \tag{23} \]

must be satisfied for all \(h\) for which problem (1), (3) is solvable.

Take \(h=L(u_0)\), where \(u_0\) is an arbitrary twice continuously differentiable vector satisfying the condition

\[ \overline{B}u_0\big|_\Gamma=0. \]

For such a vector \(h\), problem (16), (17) is solvable. Consequently, according to (23), we have

\[ \int_D x_2^{-m} L(u_0)\psi(x)\,dx=0. \tag{24} \]

Using identity (19) and the fact that equality (24) is valid for any vector \(u_0\), \(\overline{B}u_0\big|_\Gamma=0\), we obtain that \(\psi \in H^*\); consequently, \(H_2 \subset H^*\).

The theorem is completely proved.

References

1. Bitsadze A. V. Equations of mixed type. Moscow, 1959.
2. Tovmasyan N. E. Dokl. Akad. Nauk SSSR, 160, No. 6, 1275–1278, 1965.
3. Muskhelishvili N. I. Singular integral equations. Moscow, 1962.

Received by the editors
March 1, 1966

Institute of Mathematics, Siberian Branch of the USSR Academy of Sciences