M. G. Dzhavadov

ASYMPTOTICS OF THE SOLUTION OF BOUNDARY-VALUE PROBLEMS FOR AN ELLIPTIC EQUATION IN THIN DOMAINS

(Presented by Academician I. N. Vekua, February 20, 1965)

In the work \((^3)\) we constructed the asymptotics of the solution of a boundary-value problem for a second-order elliptic equation with respect to a small parameter, where the small parameter enters into the geometry of the domain. The purpose of this note is to transfer the results obtained in \((^3)\) to the boundary-value problem for an elliptic equation of arbitrary order.

Let \(Q\) be a cylinder in \(n\)-dimensional space of height \(h\). We denote the lateral surface of this cylinder by \(F\). Suppose that the direction of the axis \(x_n\) coincides with \(h\), where \(h\) is sufficiently small in comparison with the other dimensions of \(Q\). In \(Q\) consider the problem

\[ Lu \equiv \partial^{2m}u / \partial x_n^{2m} + \mathcal{L}u = 0; \tag{1} \]

\[ \partial^m u/\partial x_n^m \big|_{x_n=0}=0,\quad \partial^m u/\partial x_n^m \big|_{x_n=h}=P,\quad \partial^k u/\partial x_n^k \big|_{x_n=0,\;x_n=h}=0, \tag{2} \]

\[ k=m+1,\ldots,2m-1; \]

\[ \partial^i u/\partial \nu^i\big|_F=0,\quad i=0,1,\ldots,m-1, \tag{3} \]

where \(P(x_1,\ldots,x_{n-1})\) is a given smooth function; \(\nu\) is the normal to the lateral surface of the cylinder \(Q\); \(L\) is an elliptic operator, and \(\mathcal{L}\) is an elliptic differential expression of order \(2m\) in the variables \(x_1,\ldots,x_{n-1}\).

Make the change of variables \(x_n=th\). In the new variables the problem (1), (2), and (3) is written as follows:

\[ L_1u \equiv h^{-2m}\partial^{2m}u/\partial t^{2m}+\mathcal{L}u=0; \tag{4} \]

\[ \partial^m u/\partial t^m \big|_{t=0}=0,\quad h^{-m}\partial^m u/\partial t^m \big|_{t=1}=P,\quad \partial^k u/\partial t^k \big|_{t=0,\;t=1}=0, \tag{5} \]

\[ k=m+1,\ldots,2m-1, \]

\[ \partial^i u/\partial \nu^i\big|_F=0,\quad i=0,1,\ldots,m-1. \tag{6} \]

We shall seek an asymptotic representation of the solution with respect to \(h\) in the form
\(u=\widetilde{w}_N+\widetilde{v}_N+z_N\), where the function \(\widetilde{w}_N\) will be determined by the first iterative process, \(\widetilde{v}_N\) by the second, and \(z_N\) is the remainder term.

First iterative process. To the splitting of the operator \(L\) in (4) there corresponds a recurrent process which is obtained if the approximate solution of equation (4) is sought in the form

\[ \widetilde{w}_N = h^{-m}w_{-m} + h^{-m+1}w_{-m+1} +\cdots+ h^{-1}w_{-1} + w_0 +\cdots+ h^Nw_N. \tag{7} \]

Substituting the expression for \(\widetilde{w}_N\) from (7) into (4) and (5) and comparing terms with equal powers of \(h\), we obtain

\[ \partial^{2m}w_{-m+i}/\partial t^{2m}=0,\quad \partial^k w_{-m+i}/\partial t^k \big|_{t=0,\;t=1}=0, \]

\[ i=0,1,\ldots,2m-1;\quad k=m,m+1,\ldots,2m-1; \tag{8} \]

\[ \partial^{2m}w_m/\partial t^{2m}=-\mathcal L w_{-m}; \tag{9} \]

\[ \partial^m w_m/\partial t^m\big|_{t=0}=0,\quad \partial^m w_m/\partial t^m\big|_{t=1}=P,\quad \partial^k w_m/\partial t^k\big|_{t=0,\ t=1}=0, \tag{10} \]

\[ k=m+1,\ldots,2m-1; \]

\[ \partial^{2m}w_{m+i}/\partial t^{2m}=-\mathcal L w_{-m+i},\quad \partial^k w_{m+i}/\partial t^k\big|_{t=0,\ t=1}=0, \tag{11} \]

\[ i=1,2,\ldots;\quad k=m,\ldots,2m-1. \]

From problem (8) we find that

\[ w_{-m+i}=\sum_{j=0}^{m-1} t^j w_{-m+i,j},\quad i=0,1,\ldots,2m-1; \tag{12} \]

here all the functions \(w_{-m+i,j}\) are to be determined below.

Taking (12), for \(i=0\), into account, from (9) and (10) we obtain

\[ \partial^{2m}w_m/\partial t^{2m} =-\sum_{j=0}^{m-1} t^j\mathcal L w_{-m,j}, \tag{13} \]

\[ \partial^m w_m/\partial t^m\big|_{t=0}=0,\quad \partial^m w_m/\partial t^m\big|_{t=1}=P,\quad \partial^k w_m/\partial t^k\big|_{t=0,\ t=1}=0, \tag{14} \]

\[ k=m+1,\ldots,2m-1. \]

Since the adjoint homogeneous problem corresponding to problem (13) and (14) has a nonzero solution of the form \(z=\sum_{j=0}^{m-1} c_j t^j\), in order for it to have a solution it is necessary and sufficient that

\[ \int_0^1 \left\{ \sum_{j=0}^{m-1} t^j\mathcal L w_{-m,j}\cdot \sum_{j=0}^{m-1} c_j t^j \right\}\,dt =(-1)^{m+1}c_{m-1}P. \]

Hence, expanding the integral and comparing the expressions with identical coefficients \(c_j\), we obtain

\[ \mathcal L w_{-m,0}+2^{-1}\mathcal L w_{-m,1}+\cdots+m^{-1}\mathcal L w_{-m,m-1}=0, \]

\[ 2^{-1}\mathcal L w_{-m,0}+3^{-1}\mathcal L w_{-m,1} +\cdots+(m+1)^{-1}\mathcal L w_{-m,m-1}=0, \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

\[ (m-1)^{-1}\mathcal L w_{-m,0}+m^{-1}\mathcal L w_{-m,1} +\cdots+(2m-2)^{-1}\mathcal L w_{-m,m-1}=0, \]

\[ m^{-1}\mathcal L w_{-m,0}+(m+1)^{-1}\mathcal L w_{-m,1}+\cdots \]

\[ \cdots+(2m-1)^{-1}\mathcal L w_{-m,m-1}=(-1)^{m+1}P. \]

From the fact that the determinant of this system is different from zero, we find

\[ \mathcal L w_{-m,j}=f_j(P),\quad j=0,1,\ldots,m-1, \tag{15} \]

where \(f_j(P)\) are known functions of the given function \(P\). Thus we find that the unknown functions \(w_{-m,j}\) must satisfy equations (15). Therefore,

\[ w_m=\sum_{j=0}^{m-1} t^j w_{m,j} +\sum_{k=2m}^{3m-1}\frac{(k-2m)!}{k!}\,t^k f_{k-2m}(P). \]

In an analogous manner, continuing the process, we find

\[ \mathcal L w_{-m+i,j}=0,\quad i=1,2,\ldots,4m-1;\quad j=0,1,\ldots,m-1. \]

Consequently,

\[ w_{-m+i}=\sum_{j=0}^{m-1} t^j w_{-m+i,j},\quad i=1,2,\ldots,4m-1. \]

At the next step we obtain

\[ \partial^{2m}w_{3m}/\partial t^{2m}=-\mathcal L w_m,\quad \partial^k w_{3m}/\partial t^k\big|_{t=0,\ t=1}=0, \]

\[ k=m,m+1,\ldots,2m-1. \]

From the solvability condition for the last problem we find

\[ \mathcal{L} w_{m,j}=\bar f_j,\qquad j=0,1,\ldots,m-1, \]

where \(\bar f_j\) are known functions of functions determined earlier. Hence,

\[ w_{3m}=\sum_{j=0}^{m-1} t^j w_{3m,j}+F_{3m}, \]

where \(F_{3m}\) is a known function. Summarizing what has been presented, we may write

\[ w_i=\sum_{j=0}^{m-1} t^j w_{i,j}+F_i,\qquad i=-m,\,-m+1,\ldots, \]

where each time \(F_i\) is a known function, and \(w_{ij}\) is determined from the following problems:

\[ \mathcal{L} w_{ij}=f_{ij},\qquad \partial^k w_{ij}/\partial \nu^k\big|_F=\varphi_{ijk}, \tag{16} \]

\[ i=-m,\,-m+1,\ldots;\qquad j,k=0,1,\ldots,m-1. \]

The conditions on \(\varphi_{ijk}\) under which the last problems are solvable will be formulated below.

Obviously, the functions \(w_i\) \((i=-m,\,-m+1,\ldots)\), generally speaking, do not satisfy the boundary condition (3). Therefore, to these functions we add boundary-layer functions \(v_i\), so that the resulting sum \(w_i+v_i\) satisfies all boundary conditions. These functions \(v_i\) are determined by a second iterative process.

Second iterative process. To carry out the second iterative process in a sufficiently small neighborhood of \(F\), we introduce local coordinates \((\rho,y)\), where \(\rho\) is the distance along the normal, and \(y=y(t,y_1,\ldots,y_{n-2})\) are the coordinates of a point on \(F\). Writing equation (4) in the new variables and making in the resulting equation the change of variables \(\rho=ht\), we obtain

\[ L_2u\equiv h^{-2m}\left(\partial^{2m}u/\partial t^{2m} +A\partial^{2m}u/\partial \tau^{2m}\right)+h^{-2m+1}(\ldots)+\ldots=0. \tag{17} \]

We seek an approximate solution of equation (17) in the form

\[ \widetilde v_N = h^{-m}v_{-m}+h^{-m+1}v_{-m+1}+\ldots+h^{-1}v_{-1}+v_0+\ldots+h^N v_N, \tag{18} \]

so that

\[ \partial^k v_i/\partial t^k\big|_{t=0,\,t=1}=0,\qquad \partial^j v_i/\partial \tau^j\big|_{t=0} = -\,\partial^j w_{i-j}/\partial \nu^j\big|_F, \tag{19} \]

\[ i=-m,\,-m+1,\ldots;\qquad j=0,1,\ldots,m-1;\qquad k=m,m+1,\ldots,2m-1; \]

here it is assumed that if \(i-j<-m\), then \(w_{i-j}\equiv0\).

Writing the expression for \(\widetilde v_N\) from (18) and (17), comparing the coefficients of equal powers of \(h\), and taking into account condition (19) for determining \(v_i\), we obtain the following problems:

\[ \partial^{2m}v_{-m}/\partial t^{2m} +A\frac{\partial^{2m}v_{-m}}{\partial \tau^{2m}}=0; \tag{20} \]

\[ \partial^k v_{-m}/\partial t^k\big|_{t=0,\,t=1}=0,\qquad v_{-m}\big|_{\tau=0}=-w_{-m}\big|_F,\qquad \partial^i v_{-m}/\partial \tau^i\big|_{\tau=0}=0, \tag{21} \]

\[ k=m,m+1,\ldots,2m-1;\qquad i=1,2,\ldots,m-1; \]

\[ \partial^{2m}v_i/\partial t^{2m} +A\left(\partial^{2m}v_i/\partial \tau^{2m}\right)=\mathcal{P}_i,\qquad i=-m+1,\,-m+2,\ldots; \tag{22} \]

\[ \partial^k v_i/\partial t^i\big|_{t=0,\,t=1}=0,\qquad \partial^j v_i/\partial \tau^j\big|_{\tau=0} =-\,\partial^j w_{i-j}/\partial \nu^j\big|_F, \tag{23} \]

\[ k=m,m+1,\ldots,2m-1;\qquad j=0,1,\ldots,m-1. \]

Before proceeding to find the function \(v_i\) of boundary-layer type, let us consider the problem

\[ \psi^{(2m)}(t)+A\lambda^{2m}\psi(t)=0; \tag{24} \]

\[ \psi^{(k)}(0)=0,\qquad \psi^{(k)}(1)=0,\qquad k=m,m+1,\ldots,2m-1, \tag{25} \]

where \(A>0\) and \(\lambda\) is a parameter.

Denote by \(\bar\lambda_{k,i}\) those eigenvalues of this problem for which \(\operatorname{Re}\lambda_{k,i}<0\), and by \(\bar\psi_k(t)\) the corresponding eigenfunctions.

Theorem 1. The system of eigenfunctions \(\{\bar\psi_k(t)\}\) constitutes an \(m\)-fold complete system, and the expansion theorem holds.

We note that the adjoint homogeneous problem corresponding to all the problems (20) and (21), (22) and (23) has a nonzero solution of the form
\(z=c_0\tau^m+c_1\tau^{m+1}+\cdots+c_{m-1}\tau^{2m-1}\). Taking this fact into account, in order that the problem (20) and (21) have a solution of boundary-layer type at \(\tau=0\), it is necessary and sufficient that

\[ \int_0^1 w_{-m}\big|_F\,dt=0, \]

i.e.

\[ \varphi_{-m,0,0}+\frac12\varphi_{-m,1,0}+\cdots+\frac1m\varphi_{-m,m-1,0}=0. \tag{26} \]

Consequently, we specify the boundary functions from problem (16) for \(i=-m\) so that (26) is satisfied. Then, on the basis of Theorem 1, we assert that problem (20) and (21) has a unique solution representable in the form

\[ v_{-m}=\sum_{k=1}^{\infty}\sum_{i=1}^{m} c_{k,i}e^{\bar\lambda_{ki}\tau}\bar\psi_k(t). \tag{27} \]

By the method of mathematical induction it is proved that all the remaining functions \(v_i\) are also functions of boundary-layer type. At each step, from the solvability condition for problem (22) and (23), we find conditions on the functions \(\varphi_{ijk}\) from the first iterative process. Multiplying all \(v_i\) by a smoothing function, and denoting the functions obtained again by \(v_i\), we thus obtain for the solution of the stated problem the representation

\[ u=\sum_{i=-m}^{N+2m}h^i(w_i+v_i)+z_N, \tag{28} \]

where \(z_N\) is the solution of the problem

\[ L_1z_N=g,\qquad \partial^k z_N/\partial t^k\big|_{t=0,t=1}=0,\qquad \partial z_N^i/\partial\nu^i\big|_F=0, \]

\[ k=m,\ldots,2m-1;\qquad i=0,1,\ldots,m-1, \]

where \(g=h^{N+1}g_1\), with \(g_1\) a known function. The estimate

\[ h^{-2m}\|\partial^m z_N/\partial t^m\|^2 +\sum_{i=1}^{m-1}\|\partial^m z_N/\partial x_i^m\|^2 +\|z_N\|^2\le c\|g\|^2, \]

is valid, where \(c\) does not depend on \(h\), and the norm is understood in the sense of the \(L_2\) metric.

Thus the following has been proved.

Theorem 2. Let \(P(x_1,\ldots,x_{n-1})\) be a sufficiently smooth function. Then for the solution of problem (1), (2), (3) there holds the asymptotic representation (27), where \(z_N\) tends to zero as \(h\to0\) like \(h^{N+1}\) in the \(L_2\) metric.

Taking this opportunity, I express my gratitude to L. A. Lyusternik and M. I. Vishik for formulating the problem and discussing the results.

Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR

Received
21 I 1965

CITED LITERATURE

1. M. I. Vishik, L. A. Lyusternik, UMN, 16, no. 3 (93) (1960).
2. M. G. Dzhavadov, DAN, 159, no. 4 (1964).
3. M. G. Dzhavadov, DAN, 160, no. 3 (1965).
4. M. G. Dzhavadov, DAN, 160, no. 4 (1965).