MATHEMATICS

Yu. S. NIKOL’SKII

ON THE THEORY OF WEIGHT CLASSES OF DIFFERENTIABLE FUNCTIONS OF SEVERAL VARIABLES AND ITS APPLICATIONS TO BOUNDARY-VALUE PROBLEMS FOR ELLIPTIC EQUATIONS

(Presented by Academician I. M. Vinogradov on 8 XII 1964)

The present investigations are close to the works of L. D. Kudryavtsev on weight classes of differentiable functions of several variables in unbounded domains; among these works we mention \((^{1-3})\).

Let \(E_n\) be the \(n\)-dimensional space of points \(x=(x_1,\ldots,x_n)\) with real coordinates, and let \(\Omega_1\) be a bounded domain containing the origin \(O=(0,\ldots,0)\), whose boundary is an \((n-1)\)-dimensional surface \(\Gamma\) of class \(C^{(r)}\). Let \(\Omega\) be the set of all points \(x\) lying outside \(\Gamma\). We define on \(\overline{\Omega}\) a continuous function \(\varphi=\varphi(\rho)\) \(\left(\rho=\sqrt{x_1^2+\cdots+x_n^2}\right)\) such that, for some positive \(d\), the inequality \(\varphi(\rho)\ge d\) holds for all \(x\in\overline{\Omega}\). We consider the class \(W_{p,\varphi}^{(r)}(\Omega)\) of functions \(f=f(x)\), defined on \(\Omega\) together with their generalized derivatives up to order \(r\) inclusive, for which the finite norm

\[ \|f\|_{W_{p,\varphi}^{(r)}(\Omega)} = \|f\|_{L_p(\Omega^*)} + \sum_{|k|=r} \left\| \frac{f^{(k)}(x)}{\varphi(\rho)} \right\|_{L_p(\Omega)} \]

has meaning, where \(r\) is a nonnegative integer; \(1<p<\infty\); \(\Omega^*\) is the part of \(\Omega\) belonging to a ball with center at the origin and containing \(\Gamma\) strictly inside it. As usual, we assume that

\[ \|f\|_{L_p(\Omega)} = \left(\int_{\Omega}|f|^p\,d\Omega\right)^{1/p}, \qquad f^{(k)}(x) = \frac{\partial^{|k|}f}{\partial x_1^{k_1}\cdots \partial x_n^{k_n}}, \]

where \(k=(k_1,\ldots,k_n)\) is an integer vector \((k_j\ge 0,\ j=1,2,\ldots,n)\), \(|k|=k_1+\cdots+k_n\).

We restrict ourselves to the consideration of functions \(\varphi\) representable in the form
\(\varphi=\rho^{\alpha_0}\lambda(\rho)\) \(\left(\alpha_0=(n-p)/p\right)\), where

\[ \int_a^\infty \frac{dz}{z\varkappa(z)}<\infty, \qquad \varkappa(z)= \left\{ \min_{a\le \rho<\infty} \left[\frac{\lambda(z\rho)}{\lambda(\rho)}\right]^p \right\}^{1/p}, \qquad a=\max_{x\in\Gamma}\rho. \tag{1} \]

For such classes we prove the validity of the following embeddings*:

Theorem 1. The following embeddings hold:

\[ W_{p,\varphi}^{(r)}(\Omega)\to W_{p,\varphi\rho^k}^{(r-k)k}(\Omega), \qquad k=0,1,\ldots,r; \tag{2} \]

\[ W_{p,\varphi}^{(r)}(\Omega)\to W_{p,\varphi\rho^{\,k(1-1/p)}}^{(r-k)}(\Lambda_{n-k}), \tag{3} \]

where \(k=0,1,\ldots,r,\ k<n,\ \Lambda_{n-k}=\Omega\cdot E_{n-k}\).

* As is customary, we assume that if \(B_1,B_2\) are normed spaces, then \(B_1\to B_2\) means that \(B_1\subseteq B_2\) and there exists a constant \(c>0\), independent of \(x\in B_1\), such that \(\|x\|_{B_2}\le c\|x\|_{B_1}\).

We note that if \(\lambda(\rho)=\rho^\varepsilon(\varepsilon>0)\), the embedding (2) and the embedding (3) for \(k=1\) were proved by L. D. Kudryavtsev \((^1)\). From condition (1) it is clear that the growth of the function \(\varphi\) at infinity in such theorems may be arbitrarily large, provided only that it is bounded below by a certain condition. Condition (1) is sufficient for the validity of the indicated embeddings; however, one can give examples of such weight functions \(\varphi\) for which the integral (1) diverges and at the same time the embeddings (2), (3) no longer hold.

Consider the variational problem in which embedding (2) is applied. Let

\[ E(f,g)=\int_\Omega \sum_{|k|,|l|\le r} a_{kl}(x) f^{(k)}(x) g^{(l)}(x)\,d\Omega,\qquad E(f)=E(f,f), \]

where the sum is extended over all possible pairs of integer nonnegative vectors
\(k=(k_1,\ldots,k_n)\), \(l=(l_1,\ldots,l_n)\), for which
\(|k|, |l|\le r\), and \(a_{kl}(x)=a_{lk}(x)\) are functions measurable on \(\Omega\), for which the inequalities

\[ |a_{kl}(x)|\le M^2/[\rho^{\,r-\min(|k|,|l|)}\varphi(\rho)]^2 \]

hold.

It is assumed that there exists a number \(\lambda>0\), independent of \(x\in\Omega\), such that

\[ \sum_{|k|,|l|\le r} a_{kl}(x)\xi_k\xi_l \ge \frac{\lambda}{(\varphi(\rho))^2}\sum_{|k|=r}\xi_k^2, \]

where \(\xi_k\) are variables corresponding to the vectors \(k\). On the boundary \(\Gamma\) of the domain \(\Omega\), prescribe functions
\(\psi_s\in W_2^{2(r-s-1/2)}(\Gamma)\) \((s=0,1,\ldots,r-1)\), where
\(W_2^{(\alpha)}(\Gamma)\) (\(\alpha\) non-integer) denote the usual fractional Sobolev classes (see, for example, \((^4)\)). Let \(\mathfrak{M}\) be the class of functions
\(f\in W_{2,\varphi}^{(r)}(\Omega)\) satisfying the boundary conditions

\[ \partial^s f/\partial n^s\big|_\Gamma=\psi_s,\qquad s=0,1,\ldots,r-1. \]

In the paper it is proved that the class \(\mathfrak{M}\) is nonempty.

Let us also prescribe on \(\Omega\) a function \(F=F(x)\) possessing the property that

\[ |(F,v)|=\left|\int_\Omega F(x)v(x)\,d\Omega\right| \le c_F \|v\|_{W_{2,\varphi}^{(r)}(\Omega)}, \tag{4} \]

where \(c_F\) is a constant depending on \(F\), but not depending on
\(v\in W_{2,\varphi}^{(r)}(\Omega)\). Obviously, if

\[ \int_\Omega \rho^{2r}\varphi^2 F^2\,d\Omega<\infty, \]

then condition (4) is fulfilled. In the paper it is shown what more precise sufficient condition the function \(F\) must satisfy in order that condition (4) hold. Consider also the functional

\[ K(f)=E(f)-2(F,f). \]

Under the assumptions made, we prove the following theorems.

Theorem 2. In the class \(\mathfrak{M}\) the functional \(K(f)\) is bounded from below; moreover, there exists, and is unique, a function \(u\in\mathfrak{M}\) giving the minimum of the functional \(K(f)\) in the class \(\mathfrak{M}\).

This theorem is equivalent to the following theorem.

Theorem 3. In the class \(\mathfrak{M}\) there exists a unique generalized solution of the equation

\[ \sum_{|k|,|l|\le r}(-1)^{|l|}\frac{\partial^l}{\partial x^l} \bigl(a_{kl}(x)u^{(k)}(x)\bigr)=F(x), \tag{5} \]

which is the Euler equation of the functional \(K(f)\).

Theorem 4. There exists a constant \(c>0\) such that the generalized solution \(u\) of the boundary-value problem under consideration (in \(\Omega\)) satisfies the inequality

\[ \|u\|_{W^{(r)}_{2,\varphi}(\Omega)} \leq c\left\{ \sum_{s=0}^{r-1}\|\psi_s\|_{W^{(r-s-1/2)}_2(\Gamma)}+c_F \right\}, \]

where

\[ c_F=\sup_{\|v\|_{W^{(r)}_{2,\varphi}(\Omega)}\leq 1}|(F,v)|. \]

If one requires that the functions \(a_{kl}(x)\) and \(F(x)\) be sufficiently smooth, then it follows from known results (see, for example, \((^{5-8})\)) that the solution \(u\) has continuous partial derivatives on \(\Omega\) up to order \(2r\) inclusive, and \(u\) becomes a classical solution of equation (5).

Theorem 5. The classical solution of equation (5) in the class \(\mathfrak M\) is unique.

The proof of the last theorem is based essentially on the fact that it is shown that the class \(\mathfrak M_{00}\) is everywhere dense in the class \(\mathfrak M_0\) in the sense of the metric

\[ D_\Omega(f)=\int_\Omega \sum_{|k|=r}\left[\frac{f^{(k)}(x)}{\varphi(\rho)}\right]^2\,d\Omega, \]

where \(\mathfrak M_0\) is the class of functions \(f\in W^{(r)}_{2,\varphi}(\Omega)\) having zero boundary values on \(\Gamma\) \((\psi_s=0,\ s=0,1,\ldots,r-1)\), and \(\mathfrak M_{00}\) is the class of finite functions \(f\in\mathfrak M_0\).

We note that these investigations develop the works of L. D. Kudryavtsev \((^{2,3})\), in which the variational method for solving the first boundary-value problem was considered, in the case of an unbounded domain, for self-adjoint elliptic equations of second order in a somewhat different formulation.

In conclusion I express my deep gratitude to L. D. Kudryavtsev for formulating the problem and for his constant attention.

Moscow Institute of Physics and Technology

Received
30 XI 1964

CITED LITERATURE

\(^{1}\) L. D. Kudryavtsev, DAN, 153, No. 3, 530 (1963).
\(^{2}\) L. D. Kudryavtsev, On the first boundary-value problem for elliptic equations with coefficients decreasing at infinity, Soviet-American Symposium on Partial Differential Equations, Novosibirsk, 1963.
\(^{3}\) L. D. Kudryavtsev, DAN, 157, No. 1, 45 (1964).
\(^{4}\) L. I. Slobodetskii, DAN, 118, No. 2, 243 (1958).
\(^{5}\) P. I. Lizorkin, DAN, 134, No. 4, 761 (1960).
\(^{6}\) O. A. Ladyzhenskaya, DAN, 79, No. 5, 723 (1951).
\(^{7}\) O. A. Ladyzhenskaya, DAN, 120, No. 5, 956 (1958).
\(^{8}\) S. Agmon, A. Douglis, and L. Nirenberg, Estimates for solutions of elliptic equations near the boundary under general boundary conditions, IL, 1962.