UDC 517.881 : 517.43

THE FUNCTION SPACE \(W^{m}_{p,\alpha_1,\ldots,\alpha_s}(T,\Omega)\) AND COMPLETELY CONTINUOUS OPERATORS IN WEIGHTED SPACES

D. F. Kalinichenko

In the theory of partial differential equations, and in particular in the investigation of various kinds of boundary value problems, one or another functional space is used to a considerable extent. A large role in these questions is played, for example, by the spaces \(W_p^m\), which were studied by S. L. Sobolev [1], V. I. Kondrashov [2, 3], L. N. Slobodetskii [4], and others; the spaces \(H_p^r\) of S. M. Nikol’skii [5, 6], weighted spaces [7—12], the general theory of which was created by L. D. Kudryavtsev [13], the spaces introduced by O. V. Besov [14], and others.

In the present work the functional space \(W^{m}_{p,\alpha_1,\ldots,\alpha_s}(T,\Omega)\) is defined and studied, in which distributive self-adjoint operators are introduced in a corresponding way. It is proved that these operators are bounded and completely continuous. Some spectral properties of the introduced operators are also indicated. In particular, it is shown that the eigenfunctions of such operators satisfy the corresponding differential equation and are, in a certain sense, a solution of a boundary value problem.

§ 1. THE SPACE \(W^{m}_{p,\alpha_1,\ldots,\alpha_s}(T,\Omega)\) AND ITS PROPERTIES

1. Let \(\Omega\) be a bounded domain of the \(n\)-dimensional Euclidean space \(E_n\) of points \(x=(x_1,\ldots,x_n)\), bounded by a sufficiently smooth\(^*\) \((n-1)\)-dimensional manifold \(\Gamma\).

Assume that the domain \(\Omega\) is a Lipschitz image of a cube\(^ {**}\). On the boundary \(\Gamma\) choose a finite number \(s\) of pieces \(\Gamma_1,\ldots,\Gamma_s\), which pairwise are at a finite \((>0)\) distance from one another, and on \(\Omega\) define a system of functions \(\sigma_j(x)\) \((j=1,\ldots,s)\), satisfying the inequalities

\[ a_{1,j}\rho_j(x)\leq \sigma_j(x)\leq a_{2,j}\rho_j(x), \]

where \(\rho_j(x)=\rho(x,\Gamma_j)\) is the distance from the point \(x\in\Omega\) to the piece \(\Gamma_j\), and \(a_{1,j}, a_{2,j}\) are positive constants.

We shall consider on the domain \(\Omega\) functions \(u(x)\in W^{m}_{p,\alpha_1,\ldots,\alpha_s}(\Omega)\). By definition, a function \(u\equiv u(x)\in L_p(\Omega)\) belongs to the space \(W^{m}_{p,\alpha_1,\ldots,\alpha_s}(\Omega)\), \(1<p<\infty\), \(-1<\alpha_j<p-1\), \(j=1,\ldots,s\), if \(u\)

\(^*\) The smoothness of the boundary ensures the existence of boundary values (according to embedding theorems) for a function and all its partial derivatives up to order \(m-1\) inclusive.

\(^ {**}\) All star-shaped domains are Lipschitz images of a cube (see [15]).

has on \(\Omega\) generalized (in the sense of S. L. Sobolev) derivatives up to order \(m\) inclusive, and if the expression

\[ \int_{\Omega} \Xi \sum_{|k|=m} |D^k u|^p\, d\Omega < \infty, \]

is finite, where \(\Xi=\prod_{j=1}^s \sigma_j^{\alpha_j}\), \(k=(k_1,\ldots,k_n)\) is a multi-index; \(k_i \ge 0\) are integers,

\[ |k|=\sum_i k_i,\qquad D^k=D_{x_1}^{k_1}\cdots D_{x_n}^{k_n},\qquad D_i^{k_i}=\frac{\partial^{k_i}}{\partial x_i^{k_i}},\qquad D_i=\frac{\partial}{\partial x_i}. \]

Suppose further that on the boundary \(\Gamma\) of the domain \(\Omega\) there is given some system of functions \(T=\{\tau_k\}\), \(\tau_k=\tau_k(x)\), \(0\le |k|\le m-1\), such that \(\tau_k\in L_q(\Gamma)\) if \(|k|\le m-2\), and \(\tau_k\in L_\gamma(\Gamma)\) if \(|k|=m-1\), with

\[ \int_{\Gamma}\tau_k\,d\Gamma\ne 0 \]

for all admissible \(k\). The values of \(q\) and \(\gamma\) will be determined below.

Definition. A function \(u(x)\in W^m_{p,\alpha_1,\ldots,\alpha_s}(\Omega)\), \(1<p<\infty\), \(\frac1p+\frac1q=1\), \(-1<\alpha_j<p-1-\frac{p}{\gamma}\), \(j=1,\ldots,s\), belongs to the space \(W^m_{p,\alpha_1,\ldots,\alpha_s}(T,\Omega)\) if \(u(x)\) satisfies the relations\(^*\)

\[ \int_{\Gamma}\tau_k D^k u\,d\Gamma=0,\qquad \tau_k\in T,\qquad 0\le |k|\le m-1. \tag{1} \]

Thus the space defined is linear.

1. In this subsection, for the reader’s convenience, we formulate two already known propositions, to which we shall refer later.

Proposition A (see [17]). Let \(\Omega\) be a bounded domain, bounded by an \((n-1)\)-dimensional manifold \(\Gamma\) of smoothness \(\eta\ge 2\), and let it be the Lipschitz image of a cube. If \(1<p<\infty\), \(-1<\alpha_j<p-1-\frac{p}{\gamma}\), \(\frac1p+\frac1q=1\) \((j=1,\ldots,s)\), \(\varkappa(x)\in L_\gamma(\Gamma)\), and

\[ \int_{\Gamma}\varkappa(x)\,d\Gamma\ne 0, \]

then there exists a constant \(C=C(\varkappa)\), independent of the function \(u(x)\), such that for functions \(u(x)\in W^1_{p,\alpha_1,\ldots,\alpha_s}(\Omega)\) the inequality

\[ \int_{\Omega}|u|^p\,d\Omega \le C(\varkappa) \left( \left|\int_{\Gamma}\varkappa u\,d\Gamma\right|^p + \int_{\Omega}\left(\Xi\sum_{i=1}^n |D_i u|^p\,d\Omega\right) \right). \tag{2} \]

holds.

Remark. If \(\Xi\equiv 1\), then \(\varkappa(x)\in L_q(\Gamma)\).

Proposition\(^{{**}}\) B (see [18]). a) A set of functions \(u(x)\), bounded in \(W^m_{p,\alpha_1,\ldots,\alpha_s}(\Omega)\), is compact in the spaces into which it is embedded, on manifolds of dimension \(\nu>n-p|k|+\alpha\), \(\alpha=\max_{1\le j\le s}\alpha_j\):

1) in \(W^{m-|k|}_q\), where

\[ q<q^*=\frac{\nu p}{\,n-p|k|+\alpha\,}, \]

if \(n>p|k|-\alpha\);

2) in \(W^{m-|k|}_{q^{**}}\), where \(q^{**}\) is any number \(>1\), when \(n=p|k|-\alpha\);

\(^*\) See also the author’s notes [16].
\(^{{**}}\) For a special case see [19].

3) in \(C^{m-|k|}=C^{m-\left[\frac{n+\alpha}{p}\right]-1}\) for \(n<p|k|-\alpha,\quad k=(k_1,\ldots,k_n)\).

b) The limiting functions \(u(x)\) for the sequence \(u_j(x)\), selected according to the compactness just noted, belong to \(W_{p,\alpha_1,\ldots,\alpha_s}^{m}(\Omega)\).

1. Below, in order to estimate derivatives, we shall need

Lemma 1. If the function \(u\in W_{p,\alpha_1,\ldots,\alpha_s}^{m}(T,\Omega)\), then for all \(k,\ 0\le |k|\le m-1\),

\[ \|D^k u\|_{L_p(\Omega)}^p \le C\sum_{i=1}^{n} \left\|\left(\Xi^{\delta_{m-1}^{|k|}}\right)^{\frac1p}D_iD^k u\right\|_{L_p(\Omega)}^p, \]

\(\delta_{m-1}^{|k|}\) is the Kronecker symbol.

Proof. We apply inequality (2) to the derivatives \(D^k u,\ 0\le |k|\le m-1\). Then we obtain

\[ \|D^k u\|_{L_p(\Omega)}^p \le C\left( \left|\int_{\Gamma}\varkappa_k D^k u\,d\Gamma\right|^p + \sum_{i=1}^{n} \left\|\left(\Xi^{\delta_{m-1}^{|k|}}\right)^{\frac1p}D_iD^k u\right\|_{L_p(\Omega)}^p \right). \tag{3} \]

If in inequality (3) (see the definition of the space \(W_{p,\alpha_1,\ldots,\alpha_s}^{m}(T,\Omega)\)) we take \(\varkappa_k=\tau_k\in T\), then all surface integrals in (3) vanish.

The lemma is proved.

Most often, in the space \(W_{p,\alpha_1,\ldots,\alpha_s}^{m}(\Omega)\) the norm is defined in the following way:

\[ \|u\|_{W_{p,\alpha_1,\ldots,\alpha_s}^{m}(\Omega)} = \left( \|u\|_{L_p(\Omega)}^p + \sum_{|k|\le m} \left\| \left(\Xi^{\delta_m^{|k|}}\right)^{\frac1p} D^k u \right\|_{L_p(\Omega)}^p \right)^{\frac1p}, \tag{4} \]

or

\[ \|u\|_{W_{p,\alpha_1,\ldots,\alpha_s}^{m}(\Omega)} = \left( \|u\|_{L_p(\Omega)}^p + \sum_{|k|=m} \left\| \Xi^{\frac1p}D^k u \right\|_{L_p(\Omega)}^p \right)^{\frac1p}. \tag{5} \]

These norms are equivalent. On the basis of Proposition \(A\), we define in \(W_{p,\alpha_1,\ldots,\alpha_s}^{m}(\Omega)\) the norm

\[ \|u\|= \left( \sum_{|k|\le m-1} \left| \int_{\Gamma}\tau_k D^k u\,d\Gamma \right|^p + \sum_{|k|=m} \left\| \Xi^{\frac1p}D^k u \right\|_{L_p(\Omega)}^p \right)^{\frac1p}, \tag{6} \]

which we shall call the principal norm. Norm (6) is equivalent to the norms (4) and (5). Indeed, norm (4) can be estimated from above by norm (6), if Proposition \(A\) is applied to each derivative (up to order \((m-1)\) inclusive). The reverse estimate is obtained if, to the right-hand side of the expression

\[ \left| \int_{\Gamma}\tau_k D^k u\,d\Gamma \right|^p \le \|\tau_k\|_{L_q(\Gamma)}^{\frac{p}{q}} \|D^k u\|_{L_p(\Gamma)}^p \]

one applies, in accordance with the embedding theorems \([1,20,10,18]\), the inequality

\[ \|D^k u\|_{L_p(\Gamma)}^p \le C\left( \|D^k u\|_{L_p(\Omega)}^p + \sum_{i=1}^{n} \left\| \left(\Xi^{\delta_{m-1}^{|k|}}\right)^{\frac1p} D_iD^k u \right\|_{L_p(\Omega)}^p \right). \tag{7} \]

1. Theorem 1. The space \(W^m_{p,\alpha_1,\ldots,\alpha_s}(T,\Omega)\) is complete.

Proof. Let \(\{u_l\}\) \((l=1,2,\ldots)\) be a fundamental sequence of elements \(u_l \in W^m_{p,\alpha_1,\ldots,\alpha_s}(T,\Omega)\):

\[ \lim_{\lambda,\mu\to\infty}\|u_\lambda-u_\mu\|=0,\qquad u_\lambda,u_\mu\in\{u_l\}. \tag{8} \]

If we agree that \(\sigma \equiv \sigma(x)>0\) is some function defined on \(\Omega\), and \(L_{p,\sigma}(\Omega)\) is the space of functions \(u\) with norm

\[ \|u\|_{L_{p,\sigma}(\Omega)}=\|\sigma^{1/p}u\|_{L_p(\Omega)}, \]

then from (8), Lemma 1, and the completeness of \(L_p(\Omega)\) and \(L_{p,\sigma}(\Omega)\) there follows the existence of functions \(u^{(k)}\), \(k=(k_1,\ldots,k_n)\), \(0\le |k|\le m\), \((u^{(0)}\equiv u)\), such that

\[ \lim_{l\to\infty}\|D^k u_l-u^{(k)}\|_{L_{p,\sigma^{(k)}(\Omega)}}=0,\qquad 0\le |k|\le m, \]

where \(\sigma^{(k)}\equiv 1\), if \(0\le |k|\le m-1\), and \(\sigma^{(k)}=\Xi\), if \(|k|=m\).

Let \(\Omega_\delta\) be the set of all points \(x\in\Omega\) for which \(\rho(x,\Gamma)>\delta\). Then, since \(u_l\in W_p^m(\Omega_\delta)\) \((l=1,2,\ldots)\), we have

\[ \|D^k u^{(0)}-u^{(k)}\|_{L_p(\Omega_\delta)}=0,\qquad |k|\le m, \]

for any sufficiently small \(\delta>0\), and for those \(k\) for which \(0\le |k|\le m-1\), even

\[ \|D^k u^{(0)}-u^{(k)}\|_{L_p(\Omega)}=0. \]

Hence

\[ \|D^k u^{(0)}-u^{(k)}\|_{L_{p,\Xi}(\Omega_\delta)}=0,\qquad |k|=m. \]

Passing in this equality to the limit (as \(\delta\to 0\)), we obtain \(u^{(0)}\in W^m_{p,\alpha_1,\ldots,\alpha_s}(\Omega)\).

Below we shall denote \(u^{(0)}\equiv u\). It remains to prove, for the limiting function, the validity of condition (1). For this we use inequality (7). We obtain

\[ \left|\int_\Gamma \tau_k D^k u\,d\Gamma\right| = \left|\int_\Gamma \tau_k D^k(u-u_l)\,d\Gamma\right| \le \|\tau_k\|_{L_q(\Gamma)}\|D^k(u-u_l)\|_{L_p(\Gamma)} \le \]

\[ \le C\left( \|D^k(u-u_l)\|^p_{L_p(\Omega)} + \sum_{i=1}^{n} \left\| \left(\Xi^{\frac{\delta_i|k|}{m}-1}\right)^{1/p} D_iD^k(u-u_l) \right\|^p_{L_p(\Omega)} \right)^{1/p}. \]

Passing to the limit (as \(l\to\infty\)), we obtain relation (1) for the limiting function \(u\).

From the results of [21] there follows the separability of the spaces
\(W^m_{p,\alpha_1,\ldots,\alpha_s}(\Omega)\) and
\(W^m_{p,\alpha_1,\ldots,\alpha_s}(T,\Omega)\).

§ 2. Other norms equivalent to the basic norm

In this section we shall operate in parallel with functions from the space
\(W^m_{p,\alpha_1,\ldots,\alpha_s}(\Omega)\) and its subspace
\(W^m_{p,\alpha_1,\ldots,\alpha_s}(T,\Omega)\), and in both cases the basic norm will serve as the norm.

1. From the embedding theorems for weighted spaces (see [18, 10, 22]), and also Proposition B (§ 1, item 2), it follows that if the function \(u(x)\in W_{p,\alpha_1,\ldots,\alpha_s}^m(\Omega)\), then on manifolds \(\Gamma_\nu\) of dimension \(\nu>n-(m-|k|)+\alpha\geq 0\) the derivative \(D^k u(x)\) (of order \(k\), \(|k|=0,1,\ldots,m-1\)) satisfies the following summability conditions:

a) \(D^k u(x)\in L_q(\Gamma_\nu)\), where

\[ q<\frac{\nu p}{\,n-(m-|k|)p+\alpha\,}, \]

if \(n>(m-|k|)p-\alpha\);

b) \(D^k u(x)\in L_q(\Gamma_\nu)\), where \(q\) is any number \(>1\), if \(n=(m-|k|)p-\alpha\);

c) \(D^k u(x)\in C^{|k|}(\Gamma_\nu)=C^{\left[\frac{n+\alpha}{p}\right]-1}(\Gamma_\nu)\), if \(n<(m-|k|)p-\alpha\).

In our case \(\nu=n\), if the domain \(u\) is considered, and \(\nu=n-1\), if the boundary \(\Gamma\) is considered. In both cases

\[ \nu>n-(m-|k|)p+\alpha,\qquad \alpha=\max_{0\leq j\leq s}\alpha_j . \]

Let us consider the worst case (in the sense of summability), case a). Let \(\nu=n\). Then

\[ q=\frac{np}{\,n-(m-|k|)p+\alpha\,} =p+\frac{p[(m-|k|)p-\alpha]}{\,n-(m-|k|)p+\alpha\,}. \]

Here the term

\[ \frac{p[(m-|k|)p-\alpha]}{\,n-(m-|k|)p+\alpha\,}>0, \]

since \(-1<\alpha<p-1\). Consequently (for \(0\leq |k|\leq m-1\)),

\[ D^k u\in L_{p+\varepsilon_k}(\Omega), \]

where

\[ \varepsilon_k<\frac{p[(m-|k|)p-\alpha]}{\,n-(m-|k|)p+\alpha\,}. \]

Similarly, for \(\nu=n-1\), we obtain \(D^k u\in L_{p+\zeta_k}(\Gamma)\), where

\[ \zeta_k<\frac{p[(m-|k|)p-1-\alpha]}{\,n-(m-|k|)p+\alpha\,}. \]

1. Let, in the domain \(\Omega\) on which the functions \(u\in W_{p,\alpha_1,\ldots,\alpha_s}^m(\Omega)\) are defined, a system of functions be given

\[ \Psi=\{\psi_k(x)\},\qquad \psi_k(x)\in L_{\frac{p+\varepsilon_k}{\varepsilon_k}}(\Omega), \]

and on the boundary \(\Gamma\) of the domain \(\Omega\) let a system of functions be given

\[ \Phi=\{\varphi_k(x)\},\qquad \varphi_k(x)\in L_{\frac{p+\zeta_k}{\zeta_k}}(\Gamma),\qquad k=(k_1,\ldots,k_n),\quad 0\leq |k|\leq m-1 . \]

Lemma 2. If \(\mathfrak M\) is a set of functions \(u\in W_{p,\alpha_1,\ldots,\alpha_s}^m(\Omega)\), for which \(\|u\|\leq 1\), then for every function \(u\in\mathfrak M\) the inequalities

\[ \left\||\psi_k|^{\frac1p}D^k u\right\|_{L_p(\Omega)}^p\leq C,\qquad \left\||\varphi_k|^{\frac1p}D^k u\right\|_{L_p(\Gamma)}^p\leq C, \tag{9} \]

hold, where \(C>0\) is a constant independent of the functions \(u\in\mathfrak M\), \(k=(k_1,\ldots,k_n)\), \(0\leq |k|\leq m-1\).

Proof. Since (see item 1 of the present paragraph) \(D^k u \in L_{p+\zeta_k}(\Gamma)\), we have

\[ \left\| |\varphi_k|^{\frac{1}{p}} D^k u \right\|_{L_p(\Gamma)}^p \le \left\| \varphi_k \right\|_{L_{p+\zeta_k}(\Gamma)}\, \left\| D^k u \right\|_{L_{p+\zeta_k}(\Gamma)}^p . \tag{10} \]

Let us estimate the second factor on the right-hand side of (10). From the embedding theorems and the condition of the lemma \((\|u\| \le 1)\), for all functions \(u \in \mathfrak M\) and for fixed \(k\) \((0 \le |k| \le m-1)\) we have

\[ \left(\|D^k u\|_{L_p(\Omega)}^p + \sum_{i=1}^{n} \left\| \Xi^{\delta_m^{|k|}-1} \right\|^{\frac{1}{p}} D_iD^k u \right\|_{L_p(\Omega)}^p \right)^{\frac{1}{p}} \le C . \]

But then from Proposition B (see item 2, § 1) it follows that the set under consideration \(\{D^k u\}\) is compact at least in \(L_{p+\zeta_k}(\Gamma)\), and therefore is bounded in the norm of this space. Thus the second inequality in (9) is proved, and the first is proved analogously.

Corollary. For every function \(u(x) \in W^m_{p,\alpha_1,\ldots,\alpha_s}(\Omega)\) the inequalities

\[ \left\| |\psi_k|^{\frac{1}{p}} D^k u \right\|_{L_p(\Omega)}^p \le C\|u\|^p,\qquad \left\| |\varphi_k|^{\frac{1}{p}} D^k u \right\|_{L_p(\Gamma)}^p \le C\|u\|^p, \tag{11} \]

hold, where \(C>0\) is a constant independent of \(u\), \(0\le |k|\le m-1\).

The proof of the corollary follows immediately from Lemma 2 if, in the inequalities (9), instead of the function \(u\) one takes the function \(\dfrac{u}{\|u\|}\).

1. Define in the space \(W^m_{p,\alpha_1,\ldots,\alpha_s}(\Omega)\) one more norm \(\langle u\rangle\) as follows:

\[ \langle u\rangle = \left( \|u\|^p + \sum_{|k|\le m-1} \left\| |\psi_k|^{\frac{1}{p}}D^k u \right\|_{L_p(\Omega)}^p + \sum_{|k|\le m-1} \left\| |\varphi_k|^{\frac{1}{p}}D^k u \right\|_{L_p(\Gamma)}^p \right)^{\frac{1}{p}} . \tag{12} \]

From Lemma 2 and the corollary to it, it follows immediately that

Theorem 2. The norms \(\|u\|\) and \(\langle u\rangle\) are equivalent in the space
\(W^m_{p,\alpha_1,\ldots,\alpha_s}(\Omega)\).

§ 3. BILINEAR FUNCTIONALS

1. In the space \(W^m_{2,\alpha_1,\ldots,\alpha_s}(T,\Omega)\), for arbitrary functions \(u(x)\) and \(v(x)\), define the bilinear functional

\[ (u,v) = \int_{\Omega} \sum_{|k|,\ |l|\le m} \Xi_{k,l}^{(m)} A_{k,l}D^k uD^l v\,d\Omega , \tag{13} \]

where

\[ \Xi_{k,l}^{(m)} = \Xi^{\frac{\delta_m^{|k|}+\delta_m^{|l|}}{2}}, \]

\(\delta_m^{|k|}\), \(\delta_m^{|l|}\) are Kronecker symbols, and the coefficients \(A_{k,l}\equiv A_{k,l}(x)\) satisfy the conditions: a) if \(|k|=|l|=m\), then \(A_{k,l}\) is measur-

measurable and bounded in \(\Omega\); b) \(A_{k,l}\in L_{q_{k,l}}(\Omega)\), where

\[ q_{k,l}>\frac{2n}{2(m-|l|)-\alpha}, \]

if \(|k|=m\) or \(|l|=m\),

\[ q_{k,l}>\frac{n}{2m-(|k|+|l|)-\alpha}, \]

if \(|k|, |l|=0, 1,\ldots,m-1\); c) \(A_{k,l}=A_{l,k}\). We also require that

\[ (u,u)\ge C\sum_{|k|=m}\|\rho\|^{1/2}D^{k}u\|_{L_2(\Omega)}^{2}. \tag{14} \]

It is not hard to see that the bilinear functional (13) satisfies the properties of an inner product. With the aid of the inner product (13) we define, in the usual way, a metric in the space \(W^m_{2,\alpha_1,\ldots,\alpha_s}(T,\Omega)\). The norm \(|u|\) of an element \(u\) is defined by the equality

\[ |u|^{2}=(u,u). \tag{15} \]

Using Lemma 1 (see §1, item 3) and taking into account inequality (14), it is easy to establish that there exist constants \(C_1>0\) and \(C_2>0\), independent of the function \(u\), for which\(^*\)

\[ C_1\|u\|\le |u|\le C_2\|u\|. \tag{16} \]

Since the space \(W^m_{2,\alpha_1,\ldots,\alpha_s}(T,\Omega)\) is complete and separable in the metric \(\|u\|\), it follows from (16) that it is a complete separable Hilbert space also in the metric \(|u|\).

1. In the space \(W^m_{2,\alpha_1,\ldots,\alpha_s}(T,\Omega)\) we define one more bilinear functional

\[ [u,v]=\int_{\Omega}\sum_{|k|,\,|l|\le m-1}\psi_{k,l}D^k uD^l v\,d\Omega+ \]

\[ +\int_{\Gamma}\sum_{|k|,\,|l|\le m-1}\varphi_{k,l}D^k uD^l v\,d\Gamma, \tag{17} \]

where

\[ \psi_{k,l}\equiv\psi_{k,l}(x)\in L_{\chi_{k,l}}(\Omega),\qquad \chi_{k,l}=\max\left(\frac{p+\varepsilon_k}{\varepsilon_k},\frac{p+\varepsilon_l}{\varepsilon_l}\right); \]

\[ \varphi_{k,l}\equiv\varphi_{k,l}(x)\in L_{\eta_{k,l}}(\Gamma),\qquad \eta_{k,l}=\max\left(\frac{p+\zeta_k}{\zeta_k},\frac{p+\zeta_l}{\zeta_l}\right). \]

Theorem 3. The bilinear functional \([u,v]\) is bounded in the norm \(|\cdot|\).

Proof. To obtain the estimate

\[ |[u,v]|\le C|u|\,|v|, \tag{18} \]

it is enough to obtain the corresponding estimate for each term in (17). All terms in (17) are estimated by the same method. Therefore, as an example, we estimate one term from the first integral in the right-hand side of (17):

\[ \left|\int_{\Omega}\psi_{k,l}D^k uD^l v\,d\Omega\right| \le \left[\int_{\Omega}|\psi_{k,l}|(D^k u)^2\,d\Omega\right]^{\frac12} \left[\int_{\Omega}|\psi_{k,l}|(D^l v)^2\,d\Omega\right]^{\frac12}. \]

\(^*\) Here \(\|u\|\) is the basic norm (see §1, item 3).

If to each factor in the right-hand side of this inequality we apply the consequence of Lemma 2 (see §2, item 2) and take into account the equivalence of the norms (see (16)), then we arrive at the required inequality (18).

§ 4. Completely continuous operators in \(W^{m}_{2,\alpha_1,\ldots,\alpha_s}(T,\Omega)\)

In this paragraph we introduce certain linear self-adjoint operators and consider some of their properties. Let us note at once that operators so defined generally depend on the metric of the space and on the form of the functional defining them. This distinction is especially apparent in the formulation of boundary value problems satisfied by the eigenfunctions of the corresponding operators. Therefore, by combining different metrics of the space with different bilinear functionals, one can obtain a definite set of linear operators and the corresponding boundary value problems for eigenvalues.

For definiteness we shall consider the space \(W^{m}_{2,\alpha_1,\ldots,\alpha_s}(T,\Omega)\) with the norm \(|\cdot|\). By a well-known theorem of functional analysis (see [23], p. 401), the bilinear functional (17) defines in the space under consideration a bounded linear operator \(G_{\psi,\varphi}\), which acts from \(W^{m}_{2,\alpha_1,\ldots,\alpha_s}(T,\Omega)\) into \(W^{m}_{2,\alpha_1,\ldots,\alpha_s}(T,\Omega)\). This operator is defined by the relation

\[ [u,v]=(G_{\psi,\varphi}u,v) \tag{19} \]

for all \(u,\ v\in W^{m}_{2,\alpha_1,\ldots,\alpha_s}(T,\Omega)\). Below we shall assume that \(\psi_{k,l}=\psi_{l,k}\), \(\varphi_{k,l}=\varphi_{l,k}\), which ensures the symmetry of the functional \([u,v]\) and the self-adjointness of the operator \(G_{\psi,\varphi}\).

Theorem 4. The self-adjoint operator \(G_{\psi,\varphi}\) is completely continuous.

Proof. Since the operator \(G_{\psi,\varphi}\) is self-adjoint and bounded, in order to prove its complete continuity it suffices to show (see [24], p. 223) that from every bounded sequence \(\{u_l\}\) \((l=1,2,\ldots)\) of elements \(u_l\in W^{m}_{2,\alpha_1,\ldots,\alpha_s}(T,\Omega)\) one can extract a subsequence \(\{u_{l'}\}\) for which

\[ \lim_{p',q'\to\infty} \bigl(G_{\psi,\varphi}(u_{p'}-u_{q'}),\,u_{p'}-u_{q'}\bigr)=0. \tag{20} \]

Suppose that the sequence \(\{u_l\}\) is bounded, i.e.

\[ |u_l|\le C \qquad (l=1,2,\ldots). \tag{21} \]

Applying Lemma 1 (§1, item 3) to the elements of the sequence \(\{u_l\}\), we obtain

\[ \|D^{k}u_l\|_{L_2(\Omega)}\le C\|u_l\|,\qquad 0\le |k|\le m-1, \]

where \(C>0\) is a constant independent of \(u_l\). Hence, taking (21) into account, we have

\[ \|D^{k}u_l\|_{L_2(\Omega)}\le C \qquad (l=0,1,\ldots). \tag{22} \]

From inequalities (22) and (21) it follows that, for fixed \(k\), \(|k|=0,1,\ldots,m-1\), the sequence \(\{D^k u_l\}\) is bounded in the norm \(W^{m-|k|}_{2,\alpha_1,\ldots,\alpha_s}(\Omega)\), i.e.

\[ \|D^{k}u_l\|^{2}_{L_2(\Omega)} + \sum_{|\chi|=m-|k|} \|\Xi^{1/2}D^{k+\chi}u_l\|^{2}_{L_2(\Omega)} \le C. \tag{23} \]

Therefore, in accordance with the theorem on the complete continuity of the embedding operator from \(W_{2,\alpha_1,\ldots,\alpha_s}^{m-|k|}(\Omega)\) into \(L_{2+\varepsilon_k}(\Omega)\) and \(L_{2+\zeta_k}(\Gamma)\) (see [18]), for a fixed \(k\), \(0\le |k|\le m-1\), the sequence \(\{D^k u_l\}\) \((l=1,2,\ldots)\) is compact in \(L_{2+\varepsilon_k}(\Omega)\) and \(L_{2+\zeta_k}(\Gamma)\), where

\[ 0<\varepsilon_k< \frac{2[2(m-|k|)-\alpha]}{\,n-2(m-|k|)+\alpha\,} \quad\text{and}\quad 0<\zeta_k< \frac{2[2(m-|k|)-1-\alpha]}{\,n-2(m-|k|)+\alpha\,}. \]

This means that from the sequence \(\{u_l\}\) one can extract a subsequence \(\{u_{l_\nu}\}\) for which the corresponding subsequence \(\{D^k u_{l_\nu}\}\) converges in \(L_{2+\varepsilon_k}(\Omega)\) and in \(L_{2+\zeta_k}(\Gamma)\). From the subsequence \(\{u_{l_\nu}\}\) thus extracted, for another fixed \(k\), \(0\le |k|\le m-1\), one can in turn extract a subsequence \(\{u_{l_{\nu\mu}}\}\) such that, for the chosen \(k\), the corresponding subsequence \(\{D^k u_{l_{\nu\mu}}\}\) also converges in \(L_{2+\varepsilon_k}(\Omega)\) and in \(L_{2+\zeta_k}(\Gamma)\). Continuing this process (obviously a finite number of times), we obtain a subsequence \(\{u_{l'}\}\subset\{u_l\}\) which, for all \(k\) \((0\le |k|\le m-1)\), satisfies the conditions

\[ \lim_{p',q'\to\infty}\int_\Omega \left|D^k(u_{p'}-u_{q'})\right|^{p+\varepsilon_k}\,d\Omega=0, \tag{24} \]

\[ \lim_{p',q'\to\infty}\int_\Gamma \left|D^k(u_{p'}-u_{q'})\right|^{p+\zeta_k}\,d\Gamma=0. \tag{25} \]

Any elements \(u_{p'}\) and \(u_{q'}\) of the extracted sequence \(\{u_{l'}\}\) satisfy the equality (see (19))

\[ \left|(G_{\varphi,\varphi}(u_{p'}-u_{q'}),\,u_{p'}-u_{q'})\right| = \left|[u_{p'}-u_{q'},\,u_{p'}-u_{q'}]\right|. \tag{26} \]

In order to obtain relation (20), one must first estimate each term in the functional \([u_{p'}-u_{q'},\,u_{p'}-u_{q'}]\) in the proper way, and then (taking this estimate into account) pass in (26) to the limit as \(p',q'\to\infty\). Since the estimate of all terms on the right-hand side of (26) is carried out in the same way, we shall, for example, restrict ourselves to estimating one term:

\[ \left| \int_\Omega \psi_{k,l}D^k(u_{p'}-u_{q'})D^l(u_{p'}-u_{q'})\,d\Omega \right| \le \]

\[ \le \left\|\left|\psi_{k,l}\right|^{1/2}D^k(u_{p'}-u_{q'})\right\|_{L_2(\Omega)} \left\|\left|\psi_{k,l}\right|^{1/2}D^l(u_{p'}-u_{q'})\right\|_{L_2(\Omega)} \le \]

\[ \le \left\|\left|\psi_{k,l}\right|^{1/2}\right\|_{L_{\frac{2+\varepsilon_k}{\varepsilon_k}}(\Omega)} \left\|D^k(u_{p'}-u_{q'})\right\|_{L_{2+\varepsilon_k}(\Omega)} \times \]

\[ \times \left\|\left|\psi_{k,l}\right|^{1/2}\right\|_{L_{\frac{2+\varepsilon_l}{\varepsilon_l}}(\Omega)} \left\|D^l(u_{p'}-u_{q'})\right\|_{L_{2+\varepsilon_l}(\Omega)}. \]

The theorem is proved.

§ 5. GENERALIZED SOLUTIONS

Definition. Every function \(u(x)\in W_{2,\alpha_1,\ldots,\alpha_s}^{m}(T,\Omega)\) is called a generalized solution of the equation (with parameter \(\lambda\))

\[ Lu \equiv \lambda \sum_{|k|,\ |l|\le m} (-1)^{|l|} D^l\bigl(\Xi_{k,l}^{(m)} A_{k,l}D^k u\bigr) - \]

\[ {}- \sum_{|k|,\ |l|\le m-1} (-1)^{|l|} D^l\bigl(\psi_{k,l}D^k u\bigr)=0, \tag{27} \]

if it satisfies the relation

\[ [u,v]=\lambda (u,v) \tag{28} \]

for every infinitely differentiable function \(v\) finite in \(\Omega\).

From the properties of the functional \([u,v]\) and the definition of the operator \(G_{\psi,\varphi}\) and of a generalized solution of equation (27), an important consequence follows.

Corollary. In order that a function \(u(x)\in W_{2,\alpha_1,\ldots,\alpha_s}^{m}(T,\Omega)\) be a generalized solution of equation (27), it is sufficient that it be an eigenfunction of the corresponding operator \(G_{\psi,\varphi}\).

From relation (28) for the eigenfunctions of the operator \(G_{\psi,\varphi}\) there follows the criterion—

Theorem 5. In order that two eigenfunctions \(u\) and \(v\) of the operator \(G_{\psi,\varphi}\), at least one of which corresponds to a nonzero eigenvalue, be orthogonal in the metric \((u,v)\), it is necessary and sufficient that \([u,v]=0\).

Remark. Since in the functional \([u,v]\) (see (17), §3) some terms may be absent (because some of the functions \(\psi_{k,l}\) and \(\varphi_{k,l}\) are equal to zero), the eigenfunctions of the corresponding partial operators \(C_{\psi,\varphi}\), or their derivatives (generally speaking, of different orders), are orthogonal in a definite sense, respectively, in the domain \(\Omega\), on the boundary \(\Gamma\) of the domain \(\Omega\), or even on certain parts of the boundary \(\Gamma\).

We now consider the question of the completeness of the system of eigenfunctions of the operators studied here which correspond to eigenvalues different from zero. For the operator \(G_{\psi,\varphi}\), in many cases the completeness of these functions holds. For example, if the functional \([u,v]\) has the form

\[ [u,v]=\int_{\Omega}\sum_{|k|\le m-1}\psi_k D^k u D^k v\,d\Omega +\int_{\Gamma}\sum_{|k|\le m-1}\varphi_k D^k u D^k v\,d\Gamma \]

and the volume integral of this functional contains all partial derivatives of some order \(\varkappa\), \(|k|=\varkappa\), \(0\le \varkappa\le m-1\), the coefficients of which \(\psi_k\) are positive (negative), while all the remaining coefficients are respectively nonnegative (nonpositive), then the following is valid.

Theorem 6. The eigenfunctions of the completely continuous self-adjoint operator \(G_{\psi,\varphi}\), corresponding to eigenvalues different from zero, form a complete system.

Proof. Let \(\lambda=0\) be an eigenvalue, and let \(u(x)\) be the corresponding eigenfunction of the operator \(G_{\psi,\varphi}\). Then, in particular, \([u,u]=0\). Hence it follows that all partial derivatives of order \(\varkappa\) are equal to zero almost everywhere in \(\Omega\), and, consequently, all partial derivatives of order \(\varkappa-1\) are constants: \(D^k u=C_k,\ |k|=\varkappa-1\). If at least one of these constants \(C_k\ne0\), then, by the definition of the space \(W_{2,\alpha_1,\ldots,\alpha_s}^{m}(T,\Omega)\), for it we must have

\[ C_k\int_{\Gamma}\tau_k\,d\Gamma=\int_{\Gamma}\tau_k D^k u\,d\Gamma=0, \tag{29} \]

which contradicts the condition \(\int_\Gamma \tau_k\,d\Gamma \ne 0\) (see §1, item 1). If all \(C_k=0\), \(|k|=\varkappa-1\), then we write (29) for derivatives of order \(\varkappa-2\), and so on. The theorem is proved.

§ 6. DIFFERENTIAL PROPERTIES OF GENERALIZED SOLUTIONS

We shall say, as usual, that a function \(f(x)\in C^{(\varkappa,\alpha)}(g)\), if all its ordinary partial derivatives of order \(\varkappa\) satisfy in \(g\) the Hölder condition with exponent \(\alpha\), \(0<\alpha<1\),

\[ |f^{(\varkappa)}(x)-f^{(\varkappa)}(y)|<C|x-y|^\alpha,\qquad x,y\in g. \]

Theorem 7. Let (in the previous notation) the operator

\[ \mathscr L_m u \equiv (-1)^m \sum_{|k|,\ |l|=m} D^k\bigl(\Xi A_{k,l}D^l u\bigr) \]

be of elliptic type and admit a fundamental solution. If the coefficients
\(A_{k,l}(x)\in C^{(\max(|k|,|l|),\alpha)}(g)\), \(|k|,|l|=0,1,\ldots,m\),
\(\psi_{k,l}(x)\in C^{(\max(|k|,|l|),\alpha)}(g)\), \(|k|,|l|=0,1,\ldots,m-1\), and the weight functions
\(\sigma_j(x)\in C^{(2m,\alpha)}(g)\) \((j=1,2,\ldots,s)\) on every domain \(g\), \(\bar g\subset\Omega\), then every eigenfunction \(u\) of the operator \(G_{\psi,\varphi}\), corresponding to an eigenvalue \(\lambda\ne0\), being a generalized solution of equation (27), belongs to \(C^{(2m)}(g)\) and almost everywhere in the domain \(\Omega\) is an ordinary (classical) solution of equation (27).

Proof. For the sake of simplifying the exposition, Theorem 7 will be proved for the case when the operator \(\mathscr L_m u\) has the form

\[ \mathscr L_m u \equiv (-1)^m \sum_{|k|=m}\frac{m!}{k_1!\cdots k_n!}\,D^k(\Xi D^k u). \]

Such a form of the operator \(\mathscr L_m u\) corresponds to the scalar product

\[ D(u,v)=\int_\Omega \Xi \sum_{|k|=m}\frac{m!}{k_1!\cdots k_n!}\,D^k uD^k v\,d\Omega +\int_\Omega \sum_{\substack{|k|,|l|\le m\\ |k|+|l|<2m}}\Xi^{(m)}_{k,l}A_{k,l}D^k uD^l v\,d\Omega \tag{30} \]

instead of \((u,v)\) (see (13)). As before (see (19), §4), the relation

\[ [u,v]=D(G_{\psi,\varphi}u,v) \]

defines a completely continuous self-adjoint operator \(G_{\psi,\varphi}\).

Let \(u\) be an eigenfunction, and let \(\lambda\ne0\) be the corresponding eigenvalue of the operator \(G_{\psi,\varphi}\). Then

\[ [u,v]=\lambda D(u,v). \tag{31} \]

We shall prove that this function \(u\in C^{(2m)}(g)\) and almost everywhere in \(\Omega\) is a classical solution of the equation

\[ \lambda\left[(-1)^m \sum_{|k|=m}\frac{m!}{k_1!\cdots k_n!}D^k\left(\Xi D^k u\right)+ \right. \]
\[ \left. +\sum_{\substack{|k|,|l|\le m\\ |k|+|l|\le 2m}}(-1)^{|l|}D^l\left(\Xi A_{k,l}D^k u\right)\right] = \sum_{\substack{|k|,|l|\le m-1}}(-1)^{|l|}D^l\left(\psi_{k,l}D^k u\right). \tag{32} \]

For an arbitrary function \(v\), differentiable a sufficient number of times and finite in \(\Omega\), expression (31) takes the form

\[ \int_\Omega \sum_{|k|,|l|\le m-1}\psi_{k,l}D^k uD^l v\,d\Omega =\lambda D(u,v). \tag{33} \]

First of all, we note that the right-hand side of equation (33) contains no terms in which the sum of the orders of the derivatives of the functions \(u(x)\) and \(v(x)\) is equal to \(2m-1\). Indeed, such terms can be represented in the form

\[ \int_\Omega \Xi_{m-1,m}^{(m)} A_{i_1,\ldots,i_m}^{j_1,\ldots,j_{m-1}} (D_{j_1}\cdots D_{j_{m-1}}u)(D_{i_1}\cdots D_{i_m}v)\,d\Omega = \]
\[ = -\sum_{\gamma=1}^{m}\int_\Omega D_{i_\gamma}\left(\Xi_{m-1,m}^{(m)} A_{i_1,\ldots,i_m}^{j_1,\ldots,j_{m-1}}\right) (D_{i_1}\cdots D_{i_{\gamma-1}}D_{j_\gamma}\cdots D_{j_{m-1}}u)\times \]
\[ \times (D_{j_1}\cdots D_{j_{\gamma-1}}D_{i_{\gamma+1}}\cdots D_{i_m}v)\,d\Omega + \]
\[ +\sum_{\gamma=1}^{m-1}\int_\Omega D_{i_\gamma}\left(\Xi_{m-1,m}^{(m)} A_{i_1,\ldots,i_m}^{j_1,\ldots,j_{m-1}}\right) (D_{i_1}\cdots D_{i_\gamma}D_{j_{\gamma+1}}\cdots D_{j_{m-1}}u)\times \]
\[ \times (D_{j_1}\cdots D_{j_{\gamma-1}}D_{i_{\gamma+1}}\cdots D_{i_m}v)\,d\Omega - \]
\[ -\int_\Omega \Xi_{m-1,m}^{(m)} A_{i_1,\ldots,i_m}^{j_1,\ldots,j_{m-1}} (D_{i_1}\cdots D_{i_m}u)(D_{j_1}\cdots D_{j_{m-1}}v)\,d\Omega, \tag{34} \]

where \(D_{l_\eta}=\dfrac{\partial}{\partial x_{l_\eta}}\) \((l_\eta=1,2,\ldots,n)\). By virtue of the symmetry of the bilinear functional \(D(u,v)\), the corresponding terms cancel. Thus it can be shown that (33) contains no terms in which the sum of the orders of the derivatives of the functions \(u\) and \(v\) is an odd number. However, taking into account the subsequent change of functions, as a result of which such terms appear again, there is no point in dwelling on this. Substituting (34) into (33) and, in an obvious way, relabeling the coefficients, we obtain

\[ \lambda\left[ \int_\Omega \Xi \sum_{|k|=m}\frac{m!}{k_1!\cdots k_n!}D^k uD^k v\,d\Omega + \right. \]
\[ \left. +\int_\Omega \sum_{\substack{|k|,|l|\le m\\ |k|+|l|<2m-1}} B_{k,l}D^k uD^l v\,d\Omega \right] = \int_\Omega \sum_{|k|,|l|\le m-1} \psi_{k,l}D^k uD^l v\,d\Omega. \tag{35} \]

If now in (35) we make the substitution (see [26, 13])

\[ \bar u=\Xi^{1/2}u,\qquad \bar v=\Xi^{1/2}v, \tag{36} \]

then we arrive at the equality

\[ \lambda\left[ \int_{\Omega}\sum_{|k|=m}\frac{m!}{k_1!\cdots k_n!}\,D^k \overline{u}\,D^k\overline{v}\,d\Omega+ \right. \]

\[ \left. +\int_{\Omega}\sum_{\substack{|k|,\,|l|\le m\\ |k|+|l|<2m-1}} C_{k,l}D^k\overline{u}\,D^l\overline{v}\,d\Omega \right] = \int_{\Omega}\sum_{|k|,\,|l|\le m-1} \Psi_{k,l}D^k\overline{u}\,D^l\overline{v}\,d\Omega, \tag{37} \]

where the notation has been introduced

\[ C_{k,l}= \sum_{\substack{|k|\le |\chi|\le m\\ |l|\le |\eta|\le m\\ \chi\supset k,\ \eta\supset l}} P_{\chi}(D^{\chi-k}\Xi^{1/2})P_{\eta}(D^{\eta-l}\Xi^{1/2})B_{\chi,\eta}, \]

\[ \Psi_{k,l}= \sum_{\substack{|k|\le |\chi|\le m\\ |l|\le |\eta|\le m\\ \chi\supset k,\ \eta\supset l}} P_{\chi}(D^{\chi-k}\Xi^{1/2})P_{\eta}(D^{\eta-l}\Xi^{1/2})\psi_{\chi,\eta}, \]

\(P_{\chi},\ P_{\eta}\) are polynomials, respectively, of degrees \(\chi\) and \(\eta\).

Now, on the basis of equation (37), by the Sobolev–Kondrashov method (see [1, 25]) we shall show that the proper function \(u\) of the operator \(G_{\psi,\varphi}\) coincides almost everywhere in \(\Omega\) with a certain function \(u_0\in C^{(2m)}(\Omega)\), which satisfies in \(\Omega\) the differential equation (32).

Let \(\delta>0\) be a sufficiently small number, and let \(M_0\) be some interior point of the domain \(\Omega\), for which \(\rho(M_0,\Gamma)=\delta\). We denote by \(r\) the distance to \(M_0\) from any point of the domain. Using the fundamental solution

\[ g(r)= \begin{cases} r^{2m-n}, & \text{if } 2m-n \text{ is odd or negative},\\ r^{2m-n}\ln r, & \text{if } 2m-n\ge 0 \text{ and is even}, \end{cases} \]

for the polyharmonic equation \(\Delta^m u=0\), just as in [1], p. 18, we construct the function

\[ \xi=g(r)\left[\psi\left(\frac{r}{h_1}\right)-\psi\left(\frac{r}{h_2}\right)\right], \tag{38} \]

where \(0<h_1<h_2<\delta\), and \(\psi(t)\) is a monotonically decreasing function, infinitely differentiable on the entire real axis and such that \(\psi(t)=1\) for \(t\le \frac12\) and \(\psi(t)=0\) for \(t\ge 1\). It is easy to see that if the point from which \(r\) is measured belongs to \(\Omega_\delta\), then \(\xi\) and all its partial derivatives vanish on the boundary of the domain \(\Omega\). Consequently, for \(v=\xi\) equation (37) holds; using the definition of generalized derivative, it may be rewritten in the form

\[ \lambda\left[ (-1)^m\int_{\Omega}\overline{u}\,\Delta^m\xi\,d\Omega+ \int_{\Omega}\sum_{\substack{|k|,\,|l|\le m\\ |k|+|l|<2m-1}} C_{k,l}D^k\overline{u}\,D^l\xi\,d\Omega \right] \]

\[ = \int_{\Omega}\sum_{|k|,\,|l|\le m-1} \Psi_{k,l}D^k\overline{u}\,D^l\xi\,d\Omega. \tag{39} \]

But

\[ \Delta^m\xi = \Delta^m\left[g(r)\psi(r/h_1)\right] - \Delta^m\left[g(r)\psi(r/h_2)\right]. \]

is the difference of two functions differentiable arbitrarily many times,

\[ \omega_{h_i}=\Delta^m[g(r)\psi(r/h_i)]\quad (i=1,2), \tag{40} \]

which, as shown in [1], p. 119, may be taken as averaging kernels with radii \(h_i\), respectively.

Consider the functions \(\chi_\eta(x)=D^\eta \xi,\ 0\leqslant |\eta|\leqslant m\). They have continuous derivatives of any order in the whole space and are equal to zero (just as \(\xi\) is) outside the ball of radius \(h_2\) with center at some point \(M_0\in\Omega_\delta\). Therefore, in equation (39), instead of the function \(\xi\) one may take the function \(\chi_\eta(x)\), \(0\leqslant |\eta|\leqslant m\), since it possesses the properties of the function \(\xi\) (finiteness, infinite differentiability). Then we obtain

\[ \lambda\left[(-1)^{m+|\eta|}\int_\Omega D^\eta \bar u\,\Delta^m\xi\,d\Omega +\int_\Omega \sum_{\substack{|k|,\ |l|\leqslant m\\ |k|+|l|<2m-1}} C_{k,l}D^k\bar u\,D^l\xi\,d\Omega\right] = \]

\[ =\int_\Omega \sum_{|k|,\ |l|\leqslant m-1} \Psi_{k,l}D^k\bar u\,D^{\,l+\eta}\xi\,d\Omega . \tag{41} \]

If in equality (41) \(|\eta|=m\) or \(|\eta|=m-1\), then in some terms the number \(|l+\eta|\) may be greater than \(2m-2\). In this case, using the definition of the generalized derivative in the corresponding terms, we transfer part of the derivatives from the function \(\xi\) to the multiplier accompanying it, so that in all terms of the sums in equality (41) the maximal order of the derivative of the function \(\xi\) does not exceed \(2m-2\). In order to mark in some way the indicated transformation of some terms in (41), we shall mark by the sign “\(\wedge\)” the coefficients \(C_{k,l}\) and \(\Psi_{k,l}\) in the transformed terms. For example, if \(|\eta|=|l|=m\), then for all \(k,\ |k|=0,1,\ldots,m-2\),

\[ \int_\Omega \widehat C_{j_1,\ldots,j_m}^{\,i_1,\ldots,i_{|k|}} (D_{i_1}\ldots D_{i_{|k|}}\bar u) (D_{j_1}\ldots D_{j_m}D_{\gamma_1}\ldots D_{\gamma_m}\xi)\,d\Omega = \]

\[ =\int_\Omega \left[ D_{\gamma_1}D_{\gamma_2} \left( C_{j_1,\ldots,j_m}^{\,i_1,\ldots,i_{|k|}} D_{i_1}\ldots D_{i_{|k|}}\bar u \right) \right] D_{j_1}\ldots D_{j_m}D_{\gamma_3}\ldots D_{\gamma_m}\xi\,d\Omega, \]

\(i_\chi,\ j_\lambda,\ \gamma_\mu=1,2,\ldots,n\). Then, also taking into account notation (40), we rewrite equality (41) in the form

\[ \int_\Omega \omega_{h_1}D^\eta \bar u\,d\Omega +\int_\Omega \sum_{\substack{|k|,\ |l|\leqslant m\\ |k|+|l|<2m-1}} \widehat C_{k,l}(D^k\bar u)\, D^{\,l+\eta}[g(r)\psi(r/h_1)]\,d\Omega - \]

\[ -\frac{(-1)^{m+|\eta|}}{\lambda} \int_\Omega \sum_{|k|,\ |l|\leqslant m-1} \widehat\Psi_{k,l}(D^k\bar u)\, D^{\,l+\eta}[g(r)\psi(r/h_1)]\,d\Omega = \]

\[ =\int_\Omega \omega_{h_2}D^\eta \bar u\,d\Omega +\int_\Omega \sum_{\substack{|k|,\ |l|\leqslant m\\ |k|+|l|<2m-1}} \widehat C_{k,l}(D^k\bar u)\, D^{\,l+\eta}[g(r)\psi(r/h_2)]\,d\Omega - \]

\[ -\frac{(-1)^{m+|\eta|}}{\lambda} \int_\Omega \sum_{|k|,\ |l|\leqslant m-1} \widehat\Psi_{k,l}(D^k\bar u)\, D^{\,l+\eta} \left[ g(r)\psi\left(\frac{r}{h_2}\right) \right]\,d\Omega, \tag{42} \]

\[ 0\leqslant |\eta|\leqslant m . \]

We show that writing the integrals in (42) separately (in comparison with (41)) is legitimate. For this it is enough to show that integrals of the type of potentials of the form have meaning

\[ J_{k,l}^{(\eta)} = \int_{\Omega} \widehat{\Psi}_{k,l}(D^k \bar u)\, D^{l+\eta} \left[ g(r)\psi\left(\frac{r}{h_i}\right) \right]\,d\Omega, \tag{43} \]

\[ |k|,\ |l|=0,\ 1,\ \ldots,\ m-1 \qquad (i=1,2). \]

\[ Y_{k,l}^{(\eta)} \int_{\Omega} \widehat{C}_{k,l}(D^k \bar u)\, D^{l+\eta} \left[ g(r)\psi\left(\frac{r}{h_i}\right) \right]\,d\Omega, \tag{44} \]

\[ |k|,\ |l|=0,\ 1,\ \ldots,\ m;\quad |k|+|l|<2m-1 \qquad (i=1,2). \]

For definiteness we put \(2m-n<0\). Then

\[ g(r)\psi\left(\frac{r}{h_i}\right)=O(r^{2m-n}), \tag{45} \]

\[ D^{l+\eta} \left[ g(r)\psi\left(\frac{r}{h_i}\right) \right] = O(r^{2m-n-|l|-|\eta|}),\qquad r\to 0. \tag{46} \]

In both cases (45) and (46) (taking account of the sign “\(\widehat{\ }\)”) \(0\le |l|+|\eta|\le 2m-2\). Therefore in all cases
\[ n-2m\le n-2m+|l|+|\eta|\le n-2. \]
Then, taking into account that at least \(D^k\bar u\in L_2(g)\), and, consequently, the kernels of the integrals of potential type (43) and (44) in any case belong to \(L_2(g)\). The maximal degree of singularity \(r^{-1}\) for \(0\le |\eta|\le m\) in all the integrals (43) and (44) does not exceed \(n-2\). But integrals of potential type with such a singularity exist (see [1], pp. 48–50, and [13], pp. 149–151). The mean functions

\[ (D^\eta \bar u)_{h_i} = \int_{\Omega} \omega_{h_i}D^\eta \bar u\,d\Omega \qquad (i=1,2), \tag{47} \]

as is known (see [1]), are defined and infinitely differentiable in the whole space. Thus, the separation of the integrals in (42) is justified. We now proceed to obtaining an integral representation for the function \(\bar u\) and its derivatives up to order \(m\) inclusive.

In equality (42) each term in the sums occurring there (as was already noted) is an integral of potential type.

We note also that, since \(\psi(r/h_i)=0\) outside the ball of radius \(r<h_i\) \((i=1,2)\), the integration in (42) is actually carried out not over \(\Omega\), but over the ball of radius \(r\le h_i\), belonging (together with its closure) to the domain \(\Omega\). From the theorems on integrals of potential type (see [1], pp. 48–49) it follows that, as \(h_i\to 0\), the integrals of potential type tend to zero in the metric \(L_2\), while the mean functions (47), as \(h_i\to 0\), tend to \(D^\eta\bar u\) in the \(L_2\)-norm. Thus, the left- and right-hand sides of equality (42) tend, as \(h_i\to 0\), in \(L_2(\Omega_\delta)\), to the function \(D^\eta u\), \(0\le |\eta|\le m\), almost everywhere in \(\Omega_\delta\) for any \(\delta>0\), and, moreover, do not depend on \(h_i\). This means that almost everywhere in \(\Omega\) the representation holds

\[ D^\eta \bar u = \int_{\Omega} \omega_h D^\eta \bar u\,d\Omega + \]

\[ + \int\limits_{\Omega} \sum_{\substack{|k|,\, |l| \le m \\ |k|+|l|<2m-1}} \widehat{C}_{k,l}\,(D^k \bar u)\,D^{l+\eta} \left[g(r)\psi\left(\frac{r}{h}\right)\right]\,d\Omega - \]

\[ - \frac{(-1)^{m+|\eta|}}{\lambda} \int\limits_{\Omega}\sum_{|k|,\,|l|\le m-1} \widehat{\Psi}_{k,l}\,(D^k \bar u)\,D^{l+\eta} \left[g(r)\psi\left(\frac{r}{h}\right)\right]\,d\Omega, \tag{48} \]

\(0 \le |\eta| \le m\). Then, using representation (48), we shall refine the differentiability properties of the function \(\bar u\) itself. But first let us show that every term on the right in (48) is a continuous function for all \(\eta\), \(0 \le |\eta| \le m\).

Let, for definiteness, \(|\eta|=m\). From the theorems on integrals of potential type (see [1], pp. 48–50, and [13], pp. 149–151) it follows that each term \(J_{k,l}^{(|\eta|=m)}\) and \(Y_{k,l}^{(|\eta|=m)}\) (for the notation see (43) and (44)) in (48) is continuous and bounded in \(\Omega_\delta\), if \(n<4\), and belongs to \(W_{p_1}^m\), where \(p_1 \le \dfrac{2n}{n-4}\), if \(n \ge 4\). To verify this, it suffices to investigate only one (the worst of all) term \(J_{k,l}^{(\eta)}\) for \(|\eta|=|k|=m\), \(|l|=m-2\). Then from (48) it follows that, for \(|k|=m\),

\[ \text{a}_1)\quad D^k\bar u \in C(\Omega_{\delta_1}), \quad \text{if } n<4; \]

\[ \text{b}_1)\quad D^k\bar u \in W_{p_1}^m(\Omega_{\delta_1}), \quad p_1 \le \frac{2n}{n-4}, \quad \text{if } n \ge 4,\quad \delta_1>0 \text{ arbitrary.} \]

Having this additional information about the \(m\)-th generalized derivatives, we apply once more, to each term of the sums on the right-hand side of (48), the theorems on integrals of potential type. As a result we obtain new information (assuming \(|k|=m\)):

\[ \text{a}_2)\quad D^k\bar u \in C(\overline{\Omega}_{\delta_2}), \quad \text{if } n<8; \]

\[ \text{b}_2)\quad D^k\bar u \in W_{p_2}^m(\Omega_{\delta_2}), \quad p_2 \le \frac{2n}{n-8}, \quad \text{if } n \ge 8, \]

and so on. After the \(\mu\)-th step we shall have \((|k|=m)\):

\[ \text{a}_\mu)\quad D^k\bar u \in C(\overline{\Omega}_{\delta_\mu}), \quad \text{if } n<4\mu; \]

\[ \text{b}_\mu)\quad D^k\bar u \in W_{p_\mu}^m(\Omega_{\delta_\mu}), \quad p_\mu \le \frac{2n}{n-4\mu}, \quad \text{if } n \ge 4\mu. \]

Since the \(n\)-dimensionality of the space is fixed, there will be such an (obviously finite) number \(\mu\) for which \(n<4\mu\). This means that after the \(\mu\)-th step we arrive at the conclusion: \(D^k\bar u \in C(\Omega_{\delta_\mu})\), \(|k|=m\), and arbitrary \(\delta_\mu\) \((\delta_\mu>\delta_{\mu-1}>\cdots>\delta_1>0)\). In exactly the same way one can show that \(D^k\bar u \in C(\overline{\Omega}_{\delta_\mu})\) for \(|k|=0,1,\ldots,m\) and arbitrary \(\delta_\mu>0\). But then each of the integrals—terms of the sums on the right-hand side of (48)—is an ordinary potential with continuous density in \(n\)-dimensional space. Consequently, each of the functions \(D^k\bar u\) for \(0\le |k|\le m\) has in \(\Omega\) all continuous partial derivatives of first order. But this means that \(\bar u \in C^{(m)}(\Omega_{\delta_\mu})\) for any \(\delta_\mu>0\), since the classical derivatives (when they exist) coincide with the generalized derivatives under consideration. Moreover, the function \(\bar u\) has in \(\Omega\) all continuous partial derivatives of order \(m+1\). And then one can obtain an integral representation for any \((m+1)\)-st classical derivative of \(u\). For this it is necessary, in relation (39), instead of the function \(\xi\) (see (38)) to take the function \(\chi_k(x)=D^k\xi\) for \(|k|=m+1\) and carry out the same arguments that led us from relation (39) to the integral

representation (48) for the generalized derivatives \(D^\eta \bar u\). Then we arrive at the representation

\[ D_{\gamma_1}\ldots D_{\gamma_{m+1}}\bar u = \int_\Omega \omega_h D_{\gamma_1}\ldots D_{\gamma_{m+1}}\bar u\,d\Omega + \]

\[ +\int_\Omega \sum_{\substack{|k|,\ |l|\le m\\ |k|+|l|<2m-1}} \widehat C_{k,l}(D^k\bar u)D^lD_{\gamma_1}\ldots D_{\gamma_{m+1}} \left[g(r)\psi\left(\frac rh\right)\right]\,d\Omega - \]

\[ -\frac1\lambda\int_\Omega \sum_{|k|,\ |l|\le m-1} \Psi_{k,l}(D^k\bar u)D^lD_{\gamma_1}\ldots D_{\gamma_{m+1}} \left[g(r)\psi\left(\frac rh\right)\right]\,d\Omega. \tag{49} \]

The sign “\(\sim\)” here has the same meaning as before (see (42)). In representation (49) all terms \(J_{k,l}^{(\eta)}\) and \(Y_{k,l}^{(\eta)}\) (for the notation see (43) and (44)) with \(|\eta|=m+1\) are ordinary volume potentials whose densities in any case satisfy the Hölder condition with exponent \(\alpha,\ 0<\alpha<1\).

But then the function \(\bar u\) in any case has in \(\Omega\) all continuous partial derivatives of order \(m+2\), etc. If we carry out \(m-2\) times the arguments that led us from relation (39) to expression (49), we arrive at the representation

\[ D_{\gamma_1}\ldots D_{\gamma_{2m-2}}\bar u = \int_\Omega \omega_h D_{\gamma_1}\ldots D_{\gamma_{2m-2}}\bar u\,d\Omega + \]

\[ +\int_\Omega \sum_{\substack{|k|,\ |l|\le m\\ |k|+|l|<2m-1}} \left[D^l(C_{k,l}D^k\bar u)\right] D_{\gamma_1}\ldots D_{\gamma_{2m-2}} \left[g(r)\psi\left(\frac rh\right)\right]\,d\Omega - \]

\[ -\frac{(-1)^{3m}}{\lambda} \int_\Omega \sum_{|k|,\ |l|\le m-1} \left[D^l(\Psi_{k,l}D^k\bar u)\right] D_{\gamma_1}\ldots D_{\gamma_{2m-2}} \times \]

\[ \times \left[g(r)\psi\left(\frac rh\right)\right]\,d\Omega. \tag{50} \]

Hence, and from the properties of the volume potential, we conclude that the function \(\bar u\in C^{(2m)}(g)\), \(\bar g\subset\Omega\), and, consequently, the function \(u=\bar u\Xi^{-1/2}\) (see (36)) also belongs to \(C^{(2m)}(g)\). Taking into account the smoothness properties of the eigenfunction \(u\) of the operator \(G_{\psi,\varphi}\) and the finiteness of the function \(v\), from equation (33) we obtain equation (32) in the usual way.

Theorem 7 is proved.

Remark. If the coefficients \(C_{k,l}\) and \(\Psi_{k,l}\) are sufficiently smooth functions, then, obviously, by this method one can obtain correspondingly better smoothness of the eigenfunctions of the corresponding operators.

If the operator \(\mathscr L_m u\) does not admit a fundamental solution, then, using the scheme of arguments presented in the justification of differentiability of solutions in the manuscript of S. M. Nikol’skii, one can, just as in [27], prove the following result.

Theorem 8. If the coefficients

\[ A_{k,l}(x)\in C^{(\max(|k|,|l|))}(g),\qquad 0\le |k|,\ |l|\le m; \]

\[ \psi_{k,l}(x)\in C^{(\max(|k|,|l|))}(g),\qquad 0\le |k|,\ |l|\le m-1,\quad \bar g\subset\Omega, \]

then every eigenfunction of the operator \(G_{\psi,\varphi}\) corresponding to a nonzero eigenvalue \(\lambda\) has in \(\Omega\) all generalized (in the sense of S. L. Sobolev) partial derivatives up to order \(2m\) inclusive, belonging to \(L_2(g)\), and almost everywhere in the domain \(\Omega\) satisfies equation (27).

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Received by the editors
January 3, 1967

Moscow Engineering Physics
Institute