M. G. Dzhavadov

ASYMPTOTICS OF THE SOLUTION OF A BOUNDARY-VALUE PROBLEM FOR A SECOND-ORDER ELLIPTIC EQUATION IN DOMAINS IN WHICH ONE DIMENSION IS SUFFICIENTLY SMALL IN COMPARISON WITH THE OTHERS

(Presented by Academician I. N. Vekua on 11 VII 1964)

In the present paper we investigate the asymptotics of the solution of the following boundary-value problem.

Let \(Q\) be a cylinder of height \(h\), small in relation to the remaining dimensions. Denote the lateral surface of the cylinder by \(F\) and consider the following problem:

\[ \mathscr{L}u \equiv \sum_{i,j=1}^{n} a_{ij}\frac{\partial^2 u}{\partial x_i\partial x_j} + \sum_{i=1}^{n} a_i\frac{\partial u}{\partial x_i} - bu=0; \tag{1} \]

\[ a_{nn}\frac{\partial u}{\partial x_n} + \sum_{i=1}^{n-1} a_{in}\frac{\partial u}{\partial x_i} + \frac{a_n}{2}u = \begin{cases} P & \text{when } x_n=h,\\ 0 & \text{when } x_n=0; \end{cases} \tag{2} \]

\[ u\big|_{F}=\Phi(s,x_n), \tag{3} \]

where \(a_{ij}=a_{ji}\),

\[ \sum_{i,j=1}^{n} a_{ij}\xi_i\xi_j \ge \alpha \sum_{i=1}^{n}\xi_i^2, \quad \alpha=\mathrm{const}>0, \quad b\ge 0, \]

\(P(x_1,x_2,\ldots,x_{n-1})\) and \(\Phi(s,x_n)\) are prescribed smooth functions.

Make the change of variables \(x_n=th\). We have

\[ \mathscr{L}u \equiv \frac{a_{nn}}{h^2}\frac{\partial^2 u}{\partial t^2} + \frac{1}{h} \left( \sum_{i=1}^{n-1} a_{in}\frac{\partial^2 u}{\partial x_i\partial t} + a_n\frac{\partial u}{\partial t} \right) + \]

\[ + \sum_{i,j=1}^{n-1} a_{ij}\frac{\partial^2 u}{\partial x_i\partial x_j} + \sum_{i=1}^{n-1} a_i\frac{\partial u}{\partial x_i} - bu=0; \tag{1′} \]

\[ \frac{a_{nn}}{h}\frac{\partial u}{\partial t} + \sum_{i=1}^{n-1}\frac{\partial u}{\partial x_i} + \frac{a_n}{2}u = \begin{cases} P & \text{when } t=1,\\ 0 & \text{when } t=0; \end{cases} \tag{2′} \]

\[ u\big|_{F}=\Phi(s,th). \tag{3′} \]

We shall seek an asymptotic representation of the solution of the posed problem 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 residual term.

First iterative process. Splitting the operator \(\mathscr{L}\) in (1′) corresponds to a recurrent process, which is obtained if the approximate solution of equation (1′) is sought in the form

\[ \widetilde w_n=w_{-1}/h+w_0+hw_1+\ldots+h^n w_n+h^{n+1}w_{n+1}. \tag{4} \]

Substituting the expression for \(\widetilde w_n\) from (4) into (1′), (2′) and comparing terms with equal powers of \(h\), we obtain

\[ \partial^2 w_{-1}/\partial t^2=0; \quad \partial w_{-1}/\partial t=0 \quad \text{when } t=1; \quad \partial w_{-1}/\partial t=0 \quad \text{when } t=0; \tag{5} \]

\[ a_{nn}\frac{\partial^2 w_0}{\partial t^2} = -\sum_{i=1}^{n-1} a_{in}\frac{\partial^2 w_{-1}}{\partial x_i\partial t} - a_n\frac{\partial w_{-1}}{\partial t}; \tag{6} \]

\[ a_{nn}\frac{\partial w_0}{\partial t} + \sum_{i=1}^{n-1} a_{in}\frac{\partial w_{-1}}{\partial x_i} + \frac{a_n}{2}w_{-1} = \begin{cases} 0 & \text{when } t=1,\\ 0 & \text{when } t=0; \end{cases} \tag{7} \]

\[ a_{nn}\frac{\partial^2 w_1}{\partial t^2} = -\sum_{i=1}^{n-1} a_{in}\frac{\partial^2 w_0}{\partial x_i\partial t} - a_n\frac{\partial w_0}{\partial t} - \sum_{i,j=1}^{n-1} a_{ij}\frac{\partial^2 w_{-1}}{\partial x_i\partial x_j} - \sum_{i=1}^{n-1} a_i\frac{\partial w_{-1}}{\partial x_i} + bw_{-1}; \tag{8} \]

\[ a_{nn}\frac{\partial w_1}{\partial t}+\sum_{i=1}^{n-1} a_{in}\frac{\partial w_0}{\partial x_i}+\frac{a_n}{2}w_0 = \begin{cases} P & \text{for } t=1,\\ 0 & \text{for } t=0; \end{cases} \tag{9} \]

\[ a_{nn}\frac{\partial^2 w_k}{\partial t^2} = -\sum_{i=1}^{n-1} a_{in}\frac{\partial^2 w_{k-1}}{\partial x_i\partial t} -a_n\frac{\partial w_{k-1}}{\partial t} -\sum_{i,j=1}^{n-1} a_{ij}\frac{\partial^2 w_{k-2}}{\partial x_i\partial x_j} -\sum_{i=1}^{n-1} a_{ij}\frac{\partial w_{k-2}}{\partial x_i} +bw_{k-2}; \tag{10} \]

\[ a_{nn}\frac{\partial w_k}{\partial t} +\sum_{i=1}^{n-1} a_{in}\frac{\partial w_{k-1}}{\partial x_i} +\frac{a_n}{2}w_{k-1} = \begin{cases} 0 & \text{for } t=1,\\ 0 & \text{for } t=0, \end{cases} \qquad k=2,3,\ldots \tag{11} \]

From (5) it is clear that \(w_{-1}=\bar w_{-1}(x_1,\ldots,x_{n-1})\), i.e., the solution of problem (5) does not depend on \(t\) (on \(x_n\)).

Taking this into account, from (6) and (7) we find that

\[ w_0=\bar w_0 -t\left(\sum_{i=1}^{n-1}\frac{a_{in}}{a_{nn}}\frac{\partial w_{-1}}{\partial x_i} +\frac{a_n}{2a_{nn}}\bar w_{-1}\right), \qquad \text{where } \bar w_0=\bar w_0(x_1,\ldots,x_{n-1}). \]

We note that all the functions \(\bar w_k\) are to be determined subsequently. Substituting the expressions for \(w_{-1}\) and \(w_0\) into (8) and (9), we obtain

\[ a_{nn}\frac{\partial^2 w_1}{\partial t^2} = \sum_{i,j=1}^{n-1}\frac{a_{in}a_{jn}}{a_{nn}} \frac{\partial^2\bar w_{-1}}{\partial x_i\partial x_j} + \sum_{i=1}^{n-1}\frac{3}{2}\frac{a_{in}a_n}{a_{nn}} \frac{\partial\bar w_{-1}}{\partial x_i} +\frac{a_n^2}{2a_{nn}}\bar w_{-1} \]
\[ -\sum_{i,j=1}^{n-1}a_{ij}\frac{\partial^2\bar w_{-1}}{\partial x_i\partial x_j} -\sum_{i=1}^{n-1}a_i\frac{\partial\bar w_{-1}}{\partial x_i} +b\bar w_{-1}, \]

\[ a_{nn}\frac{\partial w_1}{\partial t} +\sum_{i=1}^{n-1} a_{in}\frac{\partial w_0}{\partial x_i} +\frac{a_n}{2}w_0 = \begin{cases} P & \text{for } t=1,\\ 0 & \text{for } t=0. \end{cases} \]

It is obvious that the adjoint homogeneous problem corresponding to problem (8) and (9) is a problem of the form (5), which has a nonzero solution independent of \(t\). Denote it by \(z_0\). Then the solvability condition for the last problem is written as follows:

\[ \int_0^1 a_{nn}\frac{\partial^2 w_1}{\partial t^2}z_0\,dt = a_{nn}\left.\frac{\partial w_1}{\partial t}z_0\right|_{t=0}^{t=1}. \]

Hence

\[ l_1(\bar w_{-1}) \equiv -\sum_{i,j=1}^{n-1}a_{ij}\frac{\partial^2\bar w_{-1}}{\partial x_i\partial x_j} -\sum_{i=1}^{n-1}\left(a_i-\frac{a_{in}a_n}{2a_{nn}}\right) \frac{\partial\bar w_{-1}}{\partial x_i} +\left(b-\frac{a_n^2}{4a_{nn}}\right)\bar w_{-1} =P. \]

Consequently,

\[ w_1=\bar w_1 -t\left(\sum_{i=1}^{n-1}a_{in}\frac{\partial\bar w_0}{\partial x_i} +\frac{a_n}{2}\bar w_0\right) + \]

\[ +\frac{t^2}{2!}\left( P+\sum_{i,j=1}^{n-1}\frac{a_{in}a_{jn}}{a_{nn}} \frac{\partial^2\bar w_{-1}}{\partial x_i\partial x_j} +\sum_{i=1}^{n-1}\frac{a_{in}a_n}{a_{nn}} \frac{\partial\bar w_{-1}}{\partial x_i} +\frac{a_n^2}{4a_{nn}}\bar w_{-1} \right). \]

Substituting the expressions for \(w_0\) and \(w_1\) into (10) and (11) for \(k=2\) and writing the solvability condition for the resulting problem, we find

\[ l_2(\bar w_0) \equiv -\sum_{i,j=1}^{n-1}a_{ij}\frac{\partial^2\bar w_0}{\partial x_i\partial x_j} -\sum_{i=1}^{n-1}\left(a_i-\frac{a_{in}a_n}{2}\right) \frac{\partial\bar w_0}{\partial x_i} +\left(b+\frac{a_n^2}{4}\right)\bar w_0 = \]

\[ = \frac{a_n}{4}P -\sum_{i,j,k=1}^{n-1}\frac{a_{ij}a_{kn}}{a_{nn}} \frac{\partial^3\bar w_{-1}}{\partial x_i\partial x_j\partial x_k} +\sum_{i,j=1}^{n-1} \frac{a_{in}a_{jn}a_n-a_{ij}a_n-2a_i a_{jn}}{4a_{nn}} \frac{\partial^2\bar w_{-1}}{\partial x_i\partial x_j} + \]

\[ +\sum_{i=1}^{n-1} \frac{a_{in}a_n^2-a_{in}a_n+2a_{in}b}{4a_{nn}} \frac{\partial\bar w_{-1}}{\partial x_i} +\frac{a_n b}{4a_{nn}}\bar w_{-1}. \]

Consequently, \(w_2=\overline w_2+f_2\), where \(f_2\) is a known function, if the functions \(\overline w_{-1}\), \(\overline w_0\), and \(\overline w_1\) have been determined.

Continuing the process in this way, we obtain that \(w_k=\overline w_k+f_k\), where \(f_k\) are known functions, and the functions \(\overline w_k\) are determined from the following problems:

\[ l_1(\overline w_{-1})=P,\qquad \overline w_{-1}\big|_{F_1}=0,\qquad l_2(\overline w_k)=\varphi_k,\qquad k=0,1,\ldots \tag{12} \]

where \(\varphi_k\) are known functions expressed in terms of previously determined functions.

The boundary conditions for the latter equations will be formulated below. Obviously, generally speaking, the functions \(w_k\) do not satisfy the boundary conditions on \(F\). Therefore, to \(w_k\) we add functions of boundary-layer type in such a way that the resulting sum satisfies all the boundary conditions. These functions are constructed by a second iteration process.

Second iteration process. In order to describe the second iteration process, in a sufficiently small neighborhood of \(F\) we introduce local coordinates \((\rho,y)\), where \(\rho\) is the distance along the normal to \(F\), \(y=y(t,y_1,\ldots,y_{n-1})\) is the coordinate of a point on \(F\), and make the change of variables \(\rho=ht\). In the new coordinates equation \((1')\) is written as

\[ \mathscr L v \equiv \frac{1}{h^2}M_0v+\frac{1}{h}M_1v+M_2v, \tag{1''} \]

where

\[ M_0\equiv a_{nn}\partial^2/\partial t^2 +A\partial^2/\partial t\,\partial \tau +B\partial^2/\partial \tau^2,\qquad M_1\equiv \sum_{i=1}^{n-2} a_{in}\frac{\partial^2}{\partial t\,\partial y_i} +\frac{\partial}{\partial t}+a\frac{\partial}{\partial \tau}, \]

\[ M_2\equiv \sum_{i,j=1}^{n-2}a_{ij}\frac{\partial^2}{\partial y_i\partial y_j} +\sum_{i=1}^{n-2}a_i\frac{\partial}{\partial y_i}-b. \]

We seek the boundary-layer solution of \((1'')\) in the form

\[ \widetilde v_n=v_{-1}/h+v_0+hv_1+\cdots+h^n v_n+h^{n+1}v_{n+1}, \tag{13} \]

so that

\[ a_{nn}\frac{\partial v_{-1}}{\partial t} +A\frac{\partial v_{-1}}{\partial \tau} = \begin{cases} 0 & \text{for } t=1,\\ 0 & \text{for } t=0; \end{cases} \]

\[ a_{nn}\frac{\partial v_k}{\partial t} +A\frac{\partial v_k}{\partial \tau} +\sum_{i=1}^{n-2}a_{in}\frac{\partial v_{k-1}}{\partial y_i} +\frac{a_n}{2}v_{k-1} = \begin{cases} 0 & \text{for } t=1,\\ 0 & \text{for } t=0; \end{cases} \tag{2''} \]

\[ v_k\big|_{t=0}=-w_k\big|_F+\Phi_k,\qquad k=-1,0,1,\ldots, \tag{3''} \]

where \(\Phi_k\) are the coefficients in the expansion of the function \(\Phi(s,th)\) in powers of \(h\).

Substituting the expression for \(\widetilde v_n\) from (13) into \((1'')\), comparing coefficients of equal powers of \(h\), and taking into account \((2'')\) and \((3'')\), we obtain

\[ M_0v_{-1}=0;\qquad a_{nn}\frac{\partial v_{-1}}{\partial t} +A\frac{\partial v_{-1}}{\partial \tau} = \begin{cases} 0 & \text{for } t=1,\\ 0 & \text{for } t=0; \end{cases} \qquad v_{-1}\big|_{\tau=0}=-w_{-1}\big|_F=0, \]

i.e. \(v_{-1}=0\);

\[ a_{nn}\frac{\partial^2 v_0}{\partial t^2} +A\frac{\partial^2 v_0}{\partial t\,\partial \tau} +B\frac{\partial^2 v_0}{\partial \tau^2}=0; \tag{14} \]

\[ a_{nn}\frac{\partial v_0}{\partial t} +A\frac{\partial v_0}{\partial \tau} = \begin{cases} 0 & \text{for } t=1,\\ 0 & \text{for } t=0; \end{cases} \tag{15} \]

\[ v_0\big|_{\tau=0}=-w_0\big|_F+\Phi_0=\Psi_0. \tag{16} \]

Finally,

\[ M_0v_k=-M_1v_{k-1}-M_2v_{k-2}; \tag{17} \]

\[ a_{nn}\frac{\partial v_k}{\partial t} +A\frac{\partial v_k}{\partial \tau} +\sum_{i=1}^{n-2}a_{in}\frac{\partial v_{k-1}}{\partial y_i} +\frac{a_n}{2}v_{k-1} = \begin{cases} 0 & \text{for } t=1,\\ 0 & \text{for } t=0; \end{cases} \tag{18} \]

\[ v_k\big|_{t=0}=\Phi_k-w_k\big|_F=\Psi_k,\qquad k=1,2,\ldots \tag{19} \]

The boundary conditions for determining the function \(\bar w_k\) are specified so that \(\Psi_k(0)=0\) and \(\Psi_k(1)=0,\ k=0,1,\ldots\).

Obviously, if \(\Psi_0\) is a smooth function, then the solution \(v_0\) of problem (14), (15), and (16) will also be a smooth function, except at the points \((0,0)\) and \((0,1)\). By methods of functional analysis it is proved that \(v_0\) has a first derivative from \(L_2(0,1)\).

Lemma 1. Let the function \(\Psi_0\) be such that \(\Psi(0)=0\) and \(\Psi(1)=0\). Then problem (14), (15), (16) has a unique solution, which is representable in the form
\[ v_0=\sum_{k=1}^{\infty} c_k e^{-\gamma_k\tau}\omega_k(t), \]
where \(\gamma>0\); \(\omega_{-k}(t)\) are the eigenfunctions of the problem
\[ a_{nn}\omega_k''(t)+A\lambda_k\omega_k'(t)+B\lambda_k^2\omega_k(t)=0, \]
\[ a_{nn}\omega_k'(0)+A\lambda_k\omega_k(0)=0,\qquad a_{nn}\omega_k'(1)+A\lambda_k\omega_k(1)=0, \]
which correspond to nonpositive values \(\lambda_k\).

Now let us find \(v_i\) \((i=1,2,\ldots)\). We define them as the solution of the equation \(M_0v_i=-M_1v_{i-1}-M_2v_{i-2}\) under the conditions that \(v_i\) is a function of boundary-layer type and satisfies conditions (18) and (19). Assuming that \(v_j\), for \(j<i\), is a function of boundary-layer type (this condition is fulfilled for \(j=0\)), we find that all the functions \(v_i\) are also functions of boundary-layer type.

Let us multiply \(v_i\) by smoothing functions and again denote the resulting functions by \(v_i\). Then for the solution of problem (1), (2), (3) we obtain the expansion
\[ u=\sum_{i=-1}^{n+1} h^i w_i+\sum_{j=0}^{n+1} h^i v_j+z_n. \tag{20} \]

Obviously, \({\mathcal L}z_n=h^{n+1}{\mathcal L}g=g_1\). Moreover,
\[ \frac{1}{h}a_{nn}\frac{\partial z_n}{\partial t} +\sum_{i=1}^{n-1}a_{in}\frac{\partial z_n}{\partial x_i} +\frac{a_n}{2}z_n = \begin{cases} 0 & \text{for } t=1,\\ 0 & \text{for } t=0, \end{cases} \qquad z_n\big|_F=0. \]

Lemma 2. If \(P(x_1,\ldots,x_{n-1})\) and \(\Phi(s,x_n)\) are sufficiently smooth functions, then the estimate
\[ \frac{1}{h^2}\left\|\frac{\partial z_n}{\partial t}\right\|^2 +\sum_{i=1}^{n-1}\left\|\frac{\partial z_n}{\partial x_i}\right\|^2 +\|z_n\|^2 \le c\|g_1\|^2, \]
holds, where \(c\) is a constant independent of \(h\), and the norm is understood in the sense of the metric of the space \(L_2\).

Thus, the following has been proved.

Theorem. Let \(P(x_1,\ldots,x_{n-1})\) and \(\Phi(s,x_n)\) be sufficiently smooth functions. Then problem (1), (2), (3) has a unique solution, the first and second iterative processes are feasible, for the solution of the problem the asymptotic expansion (20) holds, and the remainder tends to zero as \(h\to0\) like \(h^{n+1}\) in the metric \(L_2(Q)\).

The author expresses gratitude to L. A. Lyusternik and M. I. Vishik for formulating the problem and for advice in its solution.

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

Received
26 VI 1964

CITED LITERATURE

1. M. I. Vishik, L. A. Lyusternik, UMN, 12, issue 5 (77) (1957); UMN, 15, issue 3 (93) (1960); UMN, 15, issue 4 (1960).
2. A. L. Gol’denveizer, Prikl. matem. i mekh., 26, No. 4 (1962).
3. K. O. Friedrichs, R. F. Dressler, Comm. Pure and Appl. Math., 14, 1 (1961).
4. M. G. Dzhavadov, DAN, 159, No. 4 (1964).