MATHEMATICS

A. I. Koshelev

INVOLUTIONAL TRANSFORMATIONS AND THE METHOD OF SUCCESSIVE APPROXIMATIONS FOR ELLIPTIC EQUATIONS

(Presented by Academician I. N. Vekua, July 10, 1962)

Consider, inside a bounded domain \(\Omega\) of \(n\)-dimensional Euclidean space, the quasilinear equation

\[ \sum_{i=1}^{n} \frac{\partial}{\partial x_i}\,[a_i(x,u,p_j)]-a_0(x,u,p_j)=0, \tag{1} \]

where the functions \(a_i\) \((i=0,1,\ldots,n)\) depend on \(x\in\Omega\), the unknown function \(u\), and its first derivatives

\[ p_j=\partial u/\partial x_j \quad (j=1,\ldots,n). \]

With respect to the functions \(a_i\) \((i=0,1,\ldots,n)\), we shall assume that the following conditions are satisfied:

1) For all \(x,u,p_j\) and arbitrary real \(\xi_0,\xi_1,\ldots,\xi_n\), the inequality

\[ \sum_{i,k=0}^{n}\frac{\partial a_i}{\partial p_k}\,\xi_i\xi_k \geqslant f(T)\sum_{i=0}^{n}\xi_i^2, \tag{2} \]

holds, where \(u=p_0\), \(T^2=\sum_{i=0}^{n}p_i^2\), and \(f(T)\) is a continuous, strictly positive function for \(T\geqslant 0\), having, as \(T\to+\infty\), order \(O(\ln^\gamma T)\) \((0<\gamma<1)\).

2) For all \(x,u,p_j\), the inequalities

\[ \left|\frac{\partial a_i}{\partial p_k}\right|<Af(T), \tag{3} \]

\[ |a_i|<Bf(T)T, \tag{4} \]

\[ \sum_{i=0}^{n}a_i^2 \geqslant C\sum_{i=0}^{n}p_i^2(1+\ln^{2\gamma}T), \tag{5} \]

hold, where \(A,B\), and \(C\) are certain positive constants.

3) For any function \(u\) continuous together with its first partial derivatives, the functions \(a_i\) \((i=0,1,\ldots,n)\) and their derivatives with respect to all arguments are functions summable over the domain \(\Omega\) with exponent \(p>n\).

We shall seek a generalized solution of the problem for equation (1) satisfying, on the sufficiently smooth boundary \(\Gamma\) of the domain \(\Omega\), the boundary condition

\[ u|_{\Gamma}=0. \tag{6} \]

By a generalized solution of problem (1)—(6) we mean such a func-

function \(u \in \overset{\circ}{W}{}_{2}^{(1)}\), for which the integral

\[ \int_{\Omega} T^{2}\left(1+\ln^{2\gamma} T\right)\,dx \qquad (0<\gamma<1) \tag{7} \]

is finite, and for every \(v \in \overset{\circ}{W}{}_{2}^{(1)}(\Omega)\) the relation

\[ \int_{\Omega}\left[ a_i(x,u,p_j)\frac{\partial v}{\partial x_i} + a_0(x,u,p_j)v \right]dx=0. \tag{8} \]

We shall now transform problem (1)—(6).

Equality (3) will be satisfied if there is found a vector-function
\(\vec{\lambda}\{\lambda_1,\ldots,\lambda_n\}\), possessing the generalized divergence

\[ \operatorname{div}\vec{\lambda}=\sum_{i=1}^{n}\frac{\partial \lambda_i}{\partial x_i}, \]

and satisfying the system of equations

\[ a_i(x,u,p_j)=\lambda_i \quad (i=1,2,\ldots,n);\qquad a_0(x,u,p_j)=\operatorname{div}\vec{\lambda}. \tag{9} \]

Since the Jacobian of system (9), by inequality (2), is different from zero, this system admits a unique solution

\[ p_i=p_i(x,\lambda_j,\operatorname{div}\vec{\lambda})\qquad (i=0,1,\ldots,n). \]

In order that the functions \(p_i(x,\lambda_j,\operatorname{div}\vec{\lambda})\) lead to a generalized solution of our problem, it is required that, for every sufficiently smooth vector function \(\mu\{\mu_1,\ldots,\mu_n\}\) vanishing in the boundary strip, the equality

\[ \sum_{i=1}^{n}\int_{\Omega} \left(\frac{\partial p_0}{\partial x_i}-p_i\right)\mu_i\,dx=0 \]

be fulfilled.

Integrating by parts, we arrive at the equality

\[ \int_{\Omega}\left( p_0\operatorname{div}\vec{\mu} + \sum_{i=1}^{n}p_i\mu_i \right)dx=0, \tag{10} \]

from which there must be found such a vector-function \(\vec{\lambda}\) that belongs to \(\mathscr L_2(\Omega)\) and on the boundary \(\Gamma\) satisfies the condition

\[ P_0\big|_{\Gamma}=0. \]

Suppose that from the equality \(a_0(x,u,p_j)=0\) it follows that \(u=0\). Then the preceding boundary condition may be replaced by the condition

\[ \operatorname{div}\vec{\lambda}\big|_{\Gamma}=0. \tag{11} \]

We shall show that problem (10)—(11) will have a solution to which the process of successive approximations converges.

In \((^1)\) we showed that, in the case when \(a_i(x,u,p_j)\) satisfy the so-called condition of restricted nonlinearity, the solution of problem (1)—(6) can be found by a process of successive approximations, which converges in the metric \(W_2^{(1)}\). At the same time, if the conditions of restricted nonlinearity are not fulfilled, then, as proved in our note \((^1)\), this process may diverge, despite the existence of a solution and its uniqueness.

It turns out that the application of the Friedrichs involution transformation (9) \((^2)\) to the boundary-value problem (1)—(2) leads to problem (10)—(11), for which the process of successive approximations may prove convergent. In the theory of plasticity this transformation transforms a problem expressed in terms of stress functions into an analogous problem expressed in terms of strain functions.

For the solution of problem (10)—(11) let us consider the successive approximations
\(\vec\lambda^{(k)}\{\lambda_1^{(k)},\ldots,\lambda_n^{(k)}\}\) \((k=0,1,\ldots)\), defined by the following scheme:

\[ \int\limits_{\Omega}\left(\operatorname{div}\vec\lambda^{(k+1)}\operatorname{div}\mu+\sum_{i=1}^{n}\lambda_i^{(k+1)}\mu_i\right)\,dx = \int\limits_{\Omega}\left(\operatorname{div}\vec\lambda^{(k)}\operatorname{div}\mu+\sum_{i=1}^{n}\lambda_i^{(k)}\mu_i\right)\,dx - \alpha\int\limits_{\Omega}\left[p_0\left(x,\operatorname{div}\vec\lambda^{(k)},\lambda_j^{(k)}\right)\operatorname{div}\mu +\sum_{i=1}^{n}p_i\left(x,\operatorname{div}\vec\lambda^{(k)},\lambda_j^{(k)}\right)\mu_i\right]\,dx, \tag{12} \]

\[ \operatorname{div}\vec\lambda^{(k+1)}\big|_{\Gamma}=0, \tag{13} \]

where as \(\vec\lambda^{(0)}\) one may take any vector function from \(W_p^{(2)}\) \((p>n)\), and \(\alpha\) is some sufficiently small positive constant. Denote
\(\lambda_0^{(k)}=\operatorname{div}\vec\lambda^{(k)}\) \((k=0,1,\ldots)\).

It can be proved, using equalities (12), that the successive approximations \(\vec\lambda^{(k)}\) satisfy the inequality

\[ \sum_{i=0}^{n}\int\limits_{\Omega}\left|\lambda_i^{(k+1)}-\lambda_i^{(k)}\right|^2 dx \le \left[1-\varphi\left(\Lambda^{(k)}\right)\right] \sum_{i=0}^{n}\int\limits_{\Omega}\left|\lambda_i^{(k)}-\lambda_i^{(k-1)}\right|^2 dx, \tag{14} \]

where

\[ \Lambda^{(k)}=\max_j\left(\left\|\lambda_j^{(k)}\right\|_C,\left\|\lambda_j^{(k-1)}\right\|_C\right) \]

and \(\varphi(t)\) is a continuous function, nonnegative for \(t\ge 0\), decreasing as \(t\to+\infty\), and having, as \(t\to+\infty\), order

\[ O\left(\frac{1}{\ln^\gamma t}\right)\qquad (0<\gamma<1). \]

It can also be proved on the basis of \((3)\) that the successive approximations \(\vec\lambda^{(k)}\) will be continuous in \(\Omega\), together with their first derivatives, and for them the estimate

\[ \left\|\vec\lambda^{(k)}\right\|_{C^{(1)}}\le (1+\beta)^k \]

holds, where \(\beta\) is some positive constant.

It then follows from inequality (14) that

\[ \left\|\vec\lambda^{(k+1)}-\vec\lambda^{(k)}\right\|_{\mathscr L_2(\Omega)} \le \left(1-\frac{\delta}{k^\gamma}\right) \left\|\vec\lambda^{(k)}-\vec\lambda^{(k-1)}\right\|_{\mathscr L_2(\Omega)}, \]

where \(\delta>0\) is a sufficiently small constant. The last inequality guarantees convergence of the process; moreover it is easy to show that \(\operatorname{div}\vec\lambda^{(k)}\) converges in the metric \(\stackrel{\circ}{W}_2^{(1)}\), so that the boundary condition (13) is fulfilled in the sense of S. L. Sobolev. The order of convergence coincides with the order of convergence of the series

\[ \sum_{n=1}^{+\infty}\frac{1}{n^r}, \]

where \(r\) is some positive constant greater than one. Thus, the following theorem is valid.

Theorem. If for equation (1) all the conditions listed above are satisfied, then problem (1)—(6), as well as problem (10)—(11), have generalized solutions; moreover the solution of problem (10)—(11) can be obtained by the method of successive approximations according to scheme (12).

Received
7 VII 1962

CITED LITERATURE

¹ A. I. Koshelev, DAN 142, No. 5 (1962). ² K. Friedrichs, Nachr. Ges. Wiss. Göttingen, 13 (1929). ³ A. I. Koshelev, UMN, 13, issue 4 (82), 29 (1958).