MATHEMATICS

S. G. MIKHLIN

ON THE RITZ METHOD IN NONLINEAR PROBLEMS

(Presented by Academician S. L. Sobolev on 3 X 1961)

In a number of works the existence of a solution of a variational problem for functionals of a fairly general form is established by proving the convergence of a minimizing sequence; among recent works on this topic we note the papers \((^{1,2})\). However, the important question of the actual construction of a minimizing sequence in the general case has remained open. For quadratic functionals bounded below, such a construction can be carried out by applying the Ritz method \((^3)\); I. I. Vorovich showed \((^4)\) that the Ritz method gives a minimizing sequence (whose convergence was proved by the same author) for functionals of the nonlinear theory of shells; an analogous result was obtained by A. Langenbach for some functionals of the theory of plasticity in the papers \((^2)\).

The purpose of the present article is to prove the following theorem.

Theorem. Let the functional \(\Phi(u)\) satisfy the following requirements: a) its domain of definition \(D(\Phi)\) is a linear set; b) on every finite-dimensional linear manifold in its domain of definition this functional is continuously differentiable; c) in some metric \(\rho\), defined on \(D(\Phi)\), the functional in question is increasing and lower semicontinuous.

Let, further, the coordinate sequence \(\{\varphi_n\}\) be subject to the usual conditions: 1) \(\varphi_n \in D(\Phi)\); 2) the elements \(\varphi_1, \varphi_2, \ldots, \varphi_n\) are linearly independent for every \(n\); 3) the coordinate sequence is complete in the metric \(\rho\).

Under the enumerated conditions, the Ritz method gives a minimizing sequence for the functional \(\Phi(u)\).

Proof. For any \(n\) one can construct the \(n\)-th approximate solution by Ritz’s method. Indeed, the expression \(\Phi\left(\sum_{k=1}^{n} a_k \varphi_k\right)\) is a function of the variables \(a_1, a_2, \ldots, a_n\), continuously differentiable for all values of these variables and tending to \(+\infty\) if \(a_1^2 + a_2^2 + \ldots + a_n^2 \to \infty\). This function attains its absolute minimum at least at one point, which is at a finite distance from the origin, and at this point

\[ \frac{\partial \Phi\left(\sum_{k=1}^{n} a_k \varphi_k\right)}{\partial a_j} = 0,\qquad j = 1, 2, \ldots, n. \]

We shall now prove that the approximate Ritz solutions form a minimizing sequence for the functional \(\Phi(u)\). Let \(\inf \Phi(u)=d\). Construct a minimizing sequence \(\{u^{(n)}\}\) such that

\[ \Phi\left(u^{(n)}\right) \leq d+\frac{1}{n}. \]

In view of condition 3), for each \(u^{(n)}\) one can choose such a linear combination

\[ v^{(N_n)}=\sum_{k=1}^{N_n} \alpha_k^{(n)}\varphi_k, \]

that \(\rho\bigl(u^{(n)}, v^{(N_n)}\bigr)<\delta_n\); choose the number \(\delta_n\) so small that, for any \(v\) satisfying the inequality \(\rho\bigl(u^{(n)}, v\bigr)<\delta_n\), one has \(\Phi\bigl(u^{(n)}\bigr)-\Phi(v)\geqslant -\frac{1}{n}\). Then

\[ \Phi\bigl(v^{(N_n)}\bigr)\leqslant \Phi\bigl(u^{(n)}\bigr)+\frac{1}{n} \]

and, a fortiori,

\[ \Phi\bigl(v^{(N_n)}\bigr)\leqslant d+\frac{2}{n}. \]

It follows that \(\{v^{(N_n)}\}\) is a minimizing sequence. Let

\[ u_p=\sum_{k=1}^{p} a_k\varphi_k \]

denote the \(p\)-th approximate solution by Ritz’s method. Then

\[ \Phi(u_{N_n})\leqslant \Phi\bigl(v^{(N_n)}\bigr)\leqslant d+\frac{2}{n}, \]

and \(\{\Phi(u_{N_n})\}\) is also a minimizing sequence. Finally, since \(\Phi(u_n)\) decreases monotonically as \(n\) increases, while the sequence \(\Phi(u_{N_n})\to d\), it follows that \(\Phi(u_n)\to d\), as was required to prove.

Received
28 IX 1961

REFERENCES

1. I. V. Gel’man, DAN, 120, No. 3 (1958); DAN, 122, No. 4 (1958); Uch. zap. Leningr. ped. inst. im. A. I. Gertsena, 166, 255 (1958); Izv. Vyssh. uchebn. zaved., Matematika, No. 4 (17) (1960).
2. A. Langenbach, DAN, 121, No. 2 (1958); Vestn. LGU, No. 1, ser. mat., mekh. i astr., issue 1, 38 (1961); A. Langenbach, Wiss. Zs. Humboldt Univ. Berlin, Math.-naturwiss. Reihe, 9 (1959–1960).
3. S. G. Mikhlin, Direct Methods in Mathematical Physics, 1950.
4. I. I. Vorovich, DAN, 105, No. 1 (1955); Izv. AN SSSR, ser. matem., 19, No. 4, 173 (1955).