MATHEMATICS

I. G. GLOBENKO

ON THE CONVERGENCE OF VARIATIONAL PROCESSES

(Presented by Academician S. L. Sobolev on 20 IV 1962)

In the work [1], V. P. Il’in studied in detail the question of the uniform convergence of a minimizing sequence to the solution of a variational problem, depending on the rate of its convergence in one or another integral metric. With respect to the domain in which a certain class of functions is considered, V. I. Il’in assumed that it satisfies the “cone condition.”

In the present work, on the basis of results obtained earlier by us [2], it is shown how the order of convergence in the integral metric changes depending on the magnitude of the zero angle of the domain.

We shall call a closed \(n\)-dimensional domain \(V_n\), bounded by surfaces whose equations in a given rectangular coordinate system have the form

\[ x_2^2+\cdots+x_n^2=\alpha_0^2 x_1^{2\lambda}, \qquad x_1=a \qquad (x_1 \ge 0,\ \lambda \ge 1,\ \alpha_0>0), \]

a conical body with parameters \(a,\alpha_0,\lambda\).

By \(W_p^{(l)}(\Omega)\) we shall denote the set of functions summable in the \(n\)-dimensional domain \(\Omega\), obtained by closing the set of \(C^{(l)}\) functions having continuous partial derivatives in \(\Omega\) up to order \(l\) inclusive, with respect to the norm

\[ \|f\|_{W_p^{(l)}(\Omega)}=\|f\|_{L_p(\Omega)}+\|\,|D^l f|\,\|_{L_p(\Omega)}, \]

where

\[ |D^l f|=\left( \sum_{i_1\ldots i_l=1}^{n} \left| \frac{\partial^l f}{\partial x_{i_1}\cdots \partial x_{i_l}} \right|^2 \right)^{1/2}. \]

With respect to the domain \(\Omega\), it is assumed that each point of its boundary is reached by a body congruent to the fixed conical body \(V_n\) and lying entirely in \(\overline{\Omega}\).

Theorem 1. Suppose that the function \(f\in W_p^{(l)}(\Omega)\) and, moreover, the following conditions hold:

a) \(lp \le \lambda(n-1)+1\);

b) all derivatives of order \(k\) are summable with exponent \(\nu>1\) on each section \(\Omega_t: x_{t+1}=\mathrm{const},\ldots,x_n=\mathrm{const}\) \((t\le n)\), and the inequality

\[ \|\,|D^k f|\,\|_{L_\nu(\Omega_t)} \le R \]

holds for any choice of \(\Omega_t\);

c) \(k\nu>\lambda(t-1)+1\).

Then at each point of the domain \(\overline{\Omega}\) the following estimates hold:

\[ |f(P)| \le C_1'\|f\|_{L_p(\Omega)} + C_1''\left( \|\,|D^l f|\,\|_{L_p(\Omega)}^{\,k\cdot\frac{\lambda(t-1)+1}{\nu}} R^{\frac{\lambda(n-1)+1}{p}-l} \right)^{1/\chi}, \tag{1} \]

if \(\|\,|D^l f|\,\|_{L_p(\Omega)}\le \alpha^\chi R,\quad lp<\lambda(n-1)+1;\)

\[ |f(P)| \le C_2'\|f\|_{L_p(\Omega)}+C_2\|\,|D^l f|\,\|_{L_p(\Omega)}, \tag{2} \]

if

\[ \|D^l f\|_{L_p(\Omega)} > a^\chi R,\quad lp < \lambda(n-1)+1, \]

or

\[ |f(P)| \leq C_3' \|f\|_{L_p(\Omega)} + C_3'' \|D^l f\|_{L_p(\Omega)} + C_3''' \|D^l f\|_{L_p(\Omega)} \left|\ln \frac{R}{\|D^l f\|_{L_p(\Omega)}}\right|^{\frac{1}{p'}}, \tag{3} \]

if

\[ \|D^l f\|_{L_p(\Omega)} \leq a^\chi R,\quad lp=\lambda(n-1)+1\quad \left(\frac1p+\frac1{p'}=1\right); \]

\[ |f(P)| \leq C_4' \|f\|_{L_p(\Omega)} + C_4'' \|D^l f\|_{L_p(\Omega)}, \tag{4} \]

if

\[ \|D^l f\|_{L_p(\Omega)} > a^\chi R,\quad lp=\lambda(n-1)+1, \]

where \(C\) with various indices are constants independent of \(f\); \(P\) is an arbitrary point of the domain \(\Omega\);

\[ \chi=\frac{\lambda(n-1)+1}{p}-l+k-\frac{\lambda(t-1)+1}{v}. \]

Corollary. From inequalities (1) and (2) the following estimates follow immediately:

\[ |f(P)| \leq C_5 \left( \|f\|_{W_p^{(l)}(\Omega)}^{\,k-\frac{\lambda(t-1)+1}{v}}\, R^{\frac{\lambda(n-1)+1}{p}-l} \right)^{\frac1\chi} \quad \text{when } \|f\|_{W_p^{(l)}(\Omega)} \leq a^\chi R; \tag{5} \]

\[ |f(P)| \leq C_6 \|f\|_{W_p^{(l)}(\Omega)} \quad \text{when } \|f\|_{W_p^{(l)}(\Omega)} > a^\chi R; \tag{6} \]

\(C_5, C_6\) are constants independent of the function \(f\).

Theorem 2. Under the assumptions of Theorem 1, the following estimate holds:

\[ |f(P)| \leq C_7 \left( \|f\|_{L_p(\Omega)}^{\,k-\frac{\lambda(t-1)+1}{v}}\, R^{\frac{\lambda(n-1)+1}{p}} \right)^{\frac{1}{\chi_1}}, \tag{7} \]

where \(\|f\|_{L_p(\Omega)} \leq a^{\chi_1}R\),

\[ \chi_1=k-\frac{\lambda(t-1)+1}{v} +\frac{\lambda(n-1)+1}{p}. \]

The constant \(C_7\) does not depend on the function \(f\).

Let us give an example showing that estimate (5), obtained above, is exact in order.

Example. Consider the function defined on the interval \([0,\infty)\):

\[ \psi(t)= \begin{cases} (1-t)^{l+1}(1+a_1t+\ldots+a_{l-1}t^{l-1}) =1+b_lt^l+\ldots+b_{2l}t^{2l}, & \text{for } 0\leq t\leq 1,\\ 0, & \text{for } t>1, \end{cases} \]

and the sequence of functions \(u_i(x_1,\ldots,x_n)=i^\beta \psi(ix_1)\), defined on the conic body

\[ V_n\{x_2^2+\ldots+x_n^2=a_0^2x_1^{2\lambda},\quad 0\leq x_1\leq 1,\quad \lambda\geq 1,\quad a_0>0\}; \]

\(\beta>1\) is a real number.

Elementary calculations show that the order of convergence of the right-hand side of inequality (5) for the sequence \(\{u_i\}\) coincides with the order of its uniform convergence.

Theorem 3. Let \(u\in W_p^{(l)}(\Omega)\), \(lp\leq \lambda(n-1)+1\); let \(\{u_i\}\in W_p^{(l)}(\Omega)\) be a sequence of functions of the form

\[ u_i(x_1,\ldots,x_n)=\sum_{j_1\cdots j_n=0}^{i} a_{j_1\cdots j_n}x_1^{j_1}\cdots x_n^{j_n}. \tag{8} \]

Suppose, further, that it is given that \(E_i\) decrease monotonically and

\[ \lim E_i i^{2\left[\frac{\lambda(n-1)+1}{p}-l\right]}(\ln i)^\alpha=0,\qquad lp<\lambda(n-1)+1; \]

\[ \lim E_i\left(\ln\frac{i}{E_i}\right)^{\frac{1}{p'}}(\ln i)^\alpha=0,\qquad lp=\lambda(n-1)+1; \]

\[ \alpha>1,\qquad E_i=\|u-u_i\|_{W_p^{(l)}(\Omega)}\qquad \left(\frac1p+\frac1{p'}=1\right). \]

Then \(u(x_1,\ldots,x_n)\) is equivalent in \(\overline{\Omega}\) to a continuous function \(\overline{u}(x_1,\ldots,x_n)\), to which the sequence \(\{u_i\}\) converges uniformly, and moreover

\[ |\overline{u}-u_i|\leq C_8E_i i^{2\left[\frac{\lambda(n-1)+1}{p}-l\right]} +C_8\sum_{s=r}^{\infty} E_{2^s}(2^s)^{2\left[\frac{\lambda(n-1)+1}{p}-l\right]}, \qquad lp<\lambda(n-1)+1; \]

\[ |\overline{u}-u_i|\leq C_9E_i\left(\ln\frac{i}{E_i}\right)^{\frac1{p'}} +C_9\sum_{s=r}^{\infty} E_{2^s}\left(\ln\frac{2^s}{E_{2^s}}\right)^{\frac1{p'}}, \qquad lp=\lambda(n-1)+1. \]

The constants \(C_8, C_9\) do not depend on the functions \(\overline{u}, u_i\); \(2^{r-1}<i\leq 2^r\).

If in the hypotheses of Theorem 3 we put \(l=1\), \(p=2\), then we obtain sufficiently precise conditions for uniform convergence of the minimizing sequence \(\{u_i\}\) to the solution of the variational problem corresponding to the problem of solving an elliptic equation of second order under homogeneous conditions (see (3)).

For \(p=2\), \(l=m\), an analogous result is obtained for a polyharmonic equation of order \(2m\).

Theorem 4. Let \(u\in W_2^{(1)}(\Omega)\), and let the functions \(u_i\) be polynomials of the form (8). Suppose further that

\[ E_i=\|u_i-u\|_{W_2^{(1)}(\Omega)}; \]

then, if

\[ \lim_{i\to\infty} E_i i^{\lambda(n-1)+1}(\ln i)^\alpha=0\qquad (\alpha>1), \]

the sequence of derivatives \(\{\partial u_i/\partial x_j\}_{j=1,2,\ldots}\) converges uniformly in \(\overline{\Omega}\) to \(\partial u/\partial x_j\), and moreover

\[ \left|\frac{\partial u}{\partial x_j}-\frac{\partial u_i}{\partial x_j}\right| \leq C_{10}E_i i^{\lambda(n-1)+1} +C_{10}\sum_{s=m}^{\infty} E_{2^s}2^{s[\lambda(n-1)+1]}, \]

where \(2^{m-1}<i\leq 2^m\), \(C_{10}\) does not depend on \(u,u_i\).

Theorem 5. Let \(u\in W_2^{(2)}(\Omega)\), and let the functions \(u_i\) be polynomials of the form (8). Suppose further that

\[ E_i=\|D^2(u-u_i)\|_{L_2(\Omega)}+\|D(u-u_i)\|_{L_2(\Omega)}+\|u-u_i\|_{L_2(\Omega)}. \]

Then, if the conditions

\[ \lim_{i\to\infty} E_i i^{\lambda(n-1)-1}(\ln i)^\alpha=0\quad(\alpha>1) \qquad \text{for } \lambda(n-1)>1, \]

\[ \lim_{i\to\infty} E_i\left(\ln \frac{i}{E_i}\right)^{1/2}(\ln i)^\alpha=0\quad(\alpha>1) \qquad \text{for } \lambda(n-1)=1, \]

are satisfied, then \(\{\partial u_i/\partial x_j\}\) converges uniformly to \(\partial u/\partial x_j\), and moreover

\[ \left|\frac{\partial u}{\partial x_j}-\frac{\partial u_i}{\partial x_j}\right| \leq C_{11}E_i i^{\lambda(n-1)-1} + C_{11}\sum_{s=m}^{\infty} E_{2^s}2^{s[\lambda(n-1)-1]} \qquad \text{for } \lambda(n-1)>1; \]

\[ \left|\frac{\partial u}{\partial x_j}-\frac{\partial u_i}{\partial x_j}\right| \leq C_{12}E_i\left(\ln\frac{i}{E_i}\right)^{1/2} + C_{12}\sum_{s=m}^{\infty} E_{2^s}\left(\ln\frac{2^s}{E_{2^s}}\right)^{1/2} \qquad \text{for } \lambda(n-1)=1, \]

where \(2^{m-1}<i\leq 2^m\); \(C_{11}, C_{12}\) do not depend on \(u, u_i\).

An analogous theorem can also be formulated for the uniform convergence of second derivatives.

In the theorems presented, the functions \(u_i\) were assumed to be polynomials in \(x_1,\ldots,x_n\); however, all the arguments in our theorems remain valid if, as functions of \(x_1,\ldots,x_n\), they are of some other definite type—what is important is only that estimates of the type of A. A. Markov’s theorem hold for them (see \((^4)\)). For example, if the \(u_i\) are trigonometric polynomials, then all the theorems proved remain valid, since for them an analogous theorem of S. N. Bernstein is valid (see \((^4)\)).

In \((^5)\) it is shown that if the functions \(u_i(x_1,\ldots,x_n)\) are functions of the form

\[ u_i(x_1,\ldots,x_n) = \omega(x_1,\ldots,x_n) \sum_{j_1\ldots j_n=0}^{i} a_{j_1\ldots j_n}x_1^{j_1}\cdots x_n^{j_n} = \omega P_i, \tag{9} \]

where \(\omega(x_1,\ldots,x_n)\) is a twice continuously differentiable function defined in an open domain containing \(\overline{\Omega}\), satisfying the conditions:

1) \(\omega(x_1,\ldots,x_n)\big|_S=0\);
2) \(\omega(x_1,\ldots,x_n)>0\) in \(\Omega\);
3) \(|\operatorname{grad}\omega|\big|_S>0\)

(\(S\) is the boundary of the domain \(\Omega\)), then for it an estimate of the type of A. A. Markov’s inequality holds:

\[ \max_{\overline{\Omega}} \left| \frac{\partial}{\partial x_j}(\omega P_i) \right| \leq A i^2 \max_{\overline{\Omega}} |\omega P_i|; \]

functions of this form may be used in solving boundary-value problems for differential equations for domains with zero corner points.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
12 IV 1962

References

\(^1\) V. P. Il’in, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 53 (1959).
\(^2\) I. G. Globenko, DAN, 132, No. 2 (1960).
\(^3\) S. G. Mikhlin, Direct methods in mathematical physics, 1950.
\(^4\) I. P. Natanson, Constructive function theory, Moscow–Leningrad, 1949.
\(^5\) I. Yu. Kharrik, DAN, 106, No. 2 (1956).