UDC 517.946.9 : 513.881

ON AN INEQUALITY FOR FUNCTIONS FROM WEIGHT CLASSES AND BOUNDARY VALUE PROBLEMS WITH STRONG DEGENERATION ON A BOUNDARY CONSISTING OF A CURVE AND A POINT

Yu. D. Salmanov

INTRODUCTION

Let \(g\) be a bounded domain with boundary \(\Gamma\) of class \(C^l\), \(l \geq 1\), a natural number, and let \(\sigma=\sigma(x)\) be the distance from the point \(x=(x_1,\ldots,x_m)\) to the boundary \(\Gamma\) of the domain \(g\). By definition, a function \(f\) belongs to the weight class \(W^{(l)}_{p,\alpha}(g)\) if \(f\) has all generalized derivatives up to order \(l\) inclusive and if the norm

\[ \|f\|_{W^{(l)}_{p,\alpha}(g)} = \|f\|_{L_p(g)} + \left\| \frac{D^l f}{\sigma^\alpha} \right\|_{L_p(g)} <\infty, \tag{1} \]

is finite, where \(\alpha\) is a real number; \(1<p<\infty\); \(D^l f\) denotes the sum of the moduli of all derivatives of order \(l\).

S. M. Nikol’skii and P. I. Lizorkin [1] proved the following theorem: if
\[ s_0-l+\frac1p-1<\alpha<s_0-l+\frac1p \]
(\(s_0\geq 1\) is an integer) and \(1/2\leq s_0\leq l\), then there exists a constant \(C>0\), independent of \(f\in W^{(l)}_{p,\alpha}(g)\), such that the inequality

\[ \|f\|_{L_p(g)} \leq C \left( \sum_{k=0}^{s_0-1} \left\| \frac{\partial^k f}{\partial n^k} \right\|_{L_p(\Gamma)} + \left\| \frac{D^l f}{\sigma^\alpha} \right\|_{L_p(g)} \right), \tag{2} \]

holds, where
\[ \left\| \frac{\partial^k f}{\partial n^k} \right\|_{L_p(\Gamma)} \]
denotes the norm of the boundary values of the normal derivatives
\[ \left. \frac{\partial^k f}{\partial n^k} \right|_{\Gamma} \]
on \(\Gamma\).

Let us note a characteristic distinction between inequality (2) and other similar inequalities for functions from weight classes known in the literature. In the usual inequalities of this kind, if degeneration of the weight \(\sigma\) along the entire boundary \(\Gamma\) was allowed, then it was assumed to be so weak that it entailed the existence, for functions \(f\) with finite norm (1), of \(l\) boundary functions
\[ \left. \frac{\partial^k f}{\partial n^k} \right|_{\Gamma}, \qquad k=0,1,\ldots,l-1. \]

In the case of strong degeneration, however, it was always assumed that it (the strong degeneration) occurs only on part of the boundary, while on the other part there is no degeneration at all.

The inequality (2) holds, generally speaking, in the case of strong degeneracy along the entire boundary, because in it the number \(s_0\) may be smaller than \(l\).

The authors of the article [1] considered strong degeneracy along an \((m-1)\)-dimensional boundary of a domain. But a similar formulation of the question is possible for boundaries having a smaller number of dimensions, \(m-2,\ m-3,\ldots\)

In the present work such a formulation of the question is carried out in one comparatively simple case.

We consider a bounded domain \(g\), lying inside a closed curve \(\Gamma\) of class \(C^l\) with the point \(O\) removed. Strong degeneracy takes place on \(\Gamma\) and at \(O\). In this situation we prove inequality (26) for the class \(W^{(l)}_{p,\alpha,\beta}(g)\), whose definition is given at the beginning of § 1.

The obtained inequality (26) is applied (see § 3) to the solution of a boundary value problem for a differential equation strongly degenerating on \(\Gamma\) and at \(O\).

In § 4 it is shown that the generalized solution of equation (40) is classical if condition (42) is satisfied.

§ 1. DEFINITIONS AND AUXILIARY PROPOSITIONS

1. Let \(g\) be a bounded domain of points \(P=(x,y)\) in the plane, \(R\) the real two-dimensional space with boundary \(\Gamma\) differentiable \(l\) times continuously (\(l \geq 1\) an integer); \(r=r(P)=\sqrt{x^2+y^2}\) \((O=(0,0)\in g)\); \(\rho(P,P')\) the distance between points \(P\) and \(P'\) of the plane, and \(\sigma=\sigma(P)=\inf \rho(P,P')\) \((P'\in\Gamma)\) the distance from the point \(P\) to the boundary \(\Gamma\) of the domain \(g\).

By definition, a function \(f(x,y)\) belongs to the weighted space \(W^{(l)}_{p,\alpha,\beta}(g)\), \(1<p<\infty,\ l=1,2,\ldots;\ \alpha\) and \(\beta\) are real numbers, if \(f\) has generalized derivatives on \(g\) up to order \(l\) inclusive and if the norm is finite

\[ \|f\|_{W^{(l)}_{p,\alpha,\beta}(g)} = \|f\|_{L_p(g)} + \left\| \frac{D^l f}{r^\alpha \sigma^\beta} \right\|_{L_p(g)} <\infty, \tag{3} \]

where

\[ \|f\|_{L_p(g)} = \left( \int_g |f|^p\,dx\,dy \right)^{1/p}, \]

\[ D^l f = \sum_{k_1+k_2=l} \left| \frac{\partial^{k_1+k_2} f(x,y)} {\partial x^{k_1}\partial y^{k_2}} \right|. \tag{4} \]

Let \(Q_1=g-Q\), where \(Q\) is a disk of radius \(d>0\) with center at the zero point \(O=(0,0)\) of the plane \(R\), lying strictly inside \(g\).

Obviously,

\[ W^{(l)}_{p,\alpha,\beta}(g)\to W^{(l)}_{p,\alpha,0}(Q), \tag{5} \]

\[ W^{(l)}_{p,\alpha,\beta}(g)\to W^{(l)}_{p,0,\beta}(Q_1), \tag{6} \]

\[ W^{(l)}_{p,\alpha,0}(Q)\to W^{(l-2/p)}_{\infty}(Q) \tag{7} \]

\[ (\alpha \geq 0,\quad l-2/p>0). \]

The following embedding holds:

\[ W^{(l)}_{p,0,\beta}(Q_1)\to B^{(l+\beta-1/p)}_{p}(\Gamma) \tag{8} \]

\[ (\beta \leq 0,\quad l+\beta-1/p>0). \]

The proof of (8) was given by A. A. Vasharin [8] for \(p=2\), \(l=1\); the general case by P. I. Lizorkin [6] and S. V. Uspenskii [7] (see the survey article of S. M. Nikol’skii [3]). We note that the first systematic study of weight classes for \(\beta \leqslant 0\) was carried out by L. D. Kudryavtsev [2] in terms of \(H\)-classes.

Apparently, the embedding (8) also holds in the case \(\beta>0\), \(l+\beta-1/p>0\). But we find it difficult to give an exact reference. For our purposes the following assertion is sufficient: if \(m<l+\beta-1/p\) (\(m=m_1+m_2\) is an integer nonnegative), then

\[ \left(f^{(m)} \equiv \frac{\partial^{m_1+m_2} f(x,y)} {\partial x^{m_1}\partial y^{m_2}}\right)_{\Gamma} \in L_p(\Gamma). \]

But this is already not hard to prove (see, for example, [5]).

There is an embedding

\[ W_{p,\alpha,0}^{(l)}(Q)\to C^m(Q) \tag{9} \]

\[ \left( -\frac{2}{p'}<\alpha\leqslant 0,\quad \frac{1}{p}+\frac{1}{p'}=1,\quad m<l+\alpha-\frac{2}{p} \right). \]

This is easily proved with the aid of S. L. Sobolev’s integral identity [4].

II. Let \(A\in\Gamma\). If the ray \(L\), issuing from the point \(O\) and passing through the point \(A(O\ne g)\), does not touch \(\Gamma\) at \(A\), then the point \(A\) will be called regular, and otherwise irregular (with respect to \(O\)).

From \(O\) draw the ray \(L\). We denote by \(A_L\) the nearest point of intersection of the ray \(L\) with the boundary \(\Gamma\). To each ray \(L\) there corresponds its segment \([O,A_L]\). The set-theoretic sum of the segments \([O,A_L]\) corresponding to all rays \(L\) will be denoted by \(B\).

Put

\[ \gamma=B\cap\Gamma \tag{10} \]

and

\[ \Gamma_2=\Gamma-\Gamma_1, \tag{11} \]

where \(\Gamma_1\) consists of the regular and \(\Gamma_2\) of the irregular points of \(\Gamma\). We shall also say that \(\Gamma_1\) is the regular, and \(\Gamma_2\) the irregular part of \(\Gamma\).

Let \(\Omega\) be an arbitrary bounded set. Introduce the set

\[ E(\Omega;\xi)=\bigcup_{A\in\Omega} Q(A;\xi), \tag{12} \]

where \(Q(A;\xi)\) is the disk with center at \(A\) and radius \(\xi>0\), which we shall call the \(\xi\)-neighborhood of \(\Omega\).

We use the notion of a regular bridge. It was introduced in [1]. For convenience we give the definition of this notion.

Let \(\Omega\) be a domain of the \(m\)-dimensional space \(R_m\) of points \(y=(y_1,\ldots,y_m)=(\bar y,y_m)\), \(\bar y=(y_1,\ldots,y_{m-1})\), defined by the inequalities

\[ \psi_1(\bar y)<y_m<\psi_2(\bar y), \]

where \(\psi_1,\psi_2\) are continuous functions on some \((m-1)\)-dimensional sphere—the projection of the domain \(\Omega\) onto the hyperplane \(y_m=0\). The domain \(\Omega\) is called elementary. The (closed) surfaces \(y_m=\psi_1(\bar y)\), \(y_m=\psi_2(\bar y)\) will be called the bases of \(\Omega\); we shall also denote them by \(\partial\Omega\); the remaining part of the boundary \(\Omega\) is the lateral surface of \(\Omega\). The elementary domain \(\Omega\) is called

a bridge of the domain \(g\), if \(\overline{\Omega}\subset \overline{g}\), the bases of \(\Omega\) belong to the boundary of \(g\), and the lateral surface of \(\Omega\) does not belong to the boundary of \(g\). If the straight lines parallel to the axis of the bridge do not meet its bases, then it is called a regular bridge of the domain \(g\).

Lemma 1 (see [1]). A bounded domain \(g\) with continuously differentiable boundary can be covered by a finite number of regular bridges.

Let \(l, s_1\), and \(s_2\) be natural numbers; \(\alpha_1, \alpha_2\) real numbers for which
\[ 1 \leq s_1 \leq l,\qquad 1 \leq s_2 \leq l,\qquad s_1+s_2 \geq l, \]
\[ s_1-l+\frac{1}{p}-1<\alpha_1<s_1-l+\frac{1}{p}, \tag{z} \]
\[ s_2-l+\frac{1}{p}-1<\alpha_2<s_2-l+\frac{1}{p}. \tag{\(\tau\)} \]

Lemma 2 (see [1]). For a function \(f\) with finite norm
\[ \left\| \frac{f^{(l)}}{(x-a)^{\alpha_1}(b-x)^{\alpha_2}} \right\|_{L_p(a,b)} <\infty \]
the inequality
\[ \|f\|_{L_p(a,b)} \leq C_1\left( \sum_{j=0}^{s_1-1}|f^{(j)}(a)| + \sum_{j=0}^{s_2-1}|f^{(j)}(b)| + \left\| \frac{f^{(l)}}{(x-a)^{\alpha_1}(b-x)^{\alpha_2}} \right\|_{L_p(a,b)} \right), \tag{13} \]
holds, where \(C_1\) does not depend on \(f\).

We note that this lemma remains valid if \((z)\) and \((\tau)\) are replaced respectively by the inequalities
\[ s_1-l+\frac{1}{p}-1<\alpha_1<\frac{1}{p}, \tag{\(z_1\)} \]
\[ s_2-l+\frac{1}{p}-1<\alpha_2<\frac{1}{p}. \tag{\(\tau_1\)} \]

The inequalities \(\alpha_1<s_1-l+1/p\) and \(\alpha_2<s_2-l+1/p\) were used in the proof of the lemma only in establishing the fact that the derivatives \(f^{(j)}(x)\) are continuous on \((a,b)\), including the point \(a\) (for \(j=0,1,\ldots,s_1-1\)) and the point \(b\) (for \(j=0,1,\ldots,s_2-1\)). But this is established without using the inequality \(\alpha_1<s_1-l+1/p\) and \(\alpha_2<s_2-l+1/p\).

Choose \(\varepsilon_0\) for which the intersection
\[ \overline{Q}(O;\varepsilon_0)\cap E(\Gamma_2;3\varepsilon_0) \tag{*} \]
is the empty set, and consider the function
\[ \Psi_{\varepsilon_0}(x,y)= \begin{cases} 0 \to \text{ on } E(\Gamma_2;2\varepsilon_0),\\ 1 \to \text{ at the remaining points } (x,y)\in R. \end{cases} \tag{14} \]

By \(\psi\) we denote the Sobolev-averaged function for the function \(\Psi_{\varepsilon_0}\) with averaging radius \(\varepsilon_0\).

If \(f \in W_{p,\alpha,\beta}^{(l)}(g)\), then \(F \in W_{p,\alpha,\beta}^{(l)}(g)\), where

\[ F=f\psi . \tag{15} \]

Indeed, denoting \(E(\Gamma_2;\delta)\cap \bar g\) by \(E_\delta\) \((\delta>0)\) for brevity, we have \(F \in W_{p,\alpha,\beta}^{(l)}(g-E_{3\varepsilon_0})\), since \(F\equiv f\) on \(g-E_{3\varepsilon_0}\). Therefore, by virtue of \((*)\) it is enough to show that

\[ F \in W_{p,\alpha,\beta}^{(l)}(Q_1)=W_{p,0,\beta}^{(l)}(Q_1). \]

We have

\[ \left\|\frac{D^{(l)}F}{\sigma^\beta}\right\|_{L_p(Q_1)} \leq C_2\sum_{k=0}^{l}\left\|\frac{D^{l-k}fD^k\psi}{\sigma^\beta}\right\|_{L_p(Q_1)} \leq C_3\sum_{k=0}^{l}\left\|\frac{D^kf}{\sigma^\beta}\right\|_{L_p(Q_1)} . \]

Hence, by the theorem of S. V. Uspenskii [7], it follows that

\[ \left\|\frac{D^kF}{\sigma^\beta}\right\|_{L_p(Q_1)} \leq C_4\left(\|f\|_{L_p(Q_1)}+ \left\|\frac{D^lf}{\sigma^\beta}\right\|_{L_p(Q_1)}\right). \tag{16} \]

Consequently,

\[ F \in W_{p,\alpha,\beta}^{(l)}(g). \]

Let

\[ x=\rho\cos\Theta, \]

\[ y=\rho\sin\Theta \tag{17} \]

and we shall assume that \(\bar B\) and \(\bar E_\delta\) are the images in polar coordinates of \(B\) and \(E_\delta\), respectively,

\[ \rho=\varphi(\Theta)\qquad (0\leq \Theta\leq 2\pi) \tag{18} \]

is the equation of \(\bar\gamma\), and \(\varphi(\Theta)\) is a bounded single-valued function defined on \([0,2\pi]\).

We shall prove that there exist constants \(C_1(\varepsilon_0)\) and \(C_2(\varepsilon_0)\) such that on the set \((\rho,\Theta)\in \bar B-\bar E_{\varepsilon_0}\) the inequalities

\[ \frac{C_1(\varepsilon_0)}{\bar\sigma(\rho,\Theta)} \leq \frac{1}{|\varphi(\Theta)-\rho|} \leq \frac{C_2(\varepsilon_0)}{\bar\sigma(\rho,\Theta)} \tag{19} \]

hold.

Indeed, put

\[ \bar B-\bar E_{\varepsilon_0}=A_{\varepsilon_0/2}+N_{\varepsilon_0/2}, \]

where \(A_{\varepsilon_0/2}\) is the set of all points \((\rho,\Theta)\) belonging to \(\bar B-\bar E_{\varepsilon_0}\) and lying at a distance not less than \(\varepsilon_0/2\) from \(\bar\gamma\); \(N_{\varepsilon_0/2}\) is the remaining part of \(\bar B-\bar E_{\varepsilon_0}\).

The validity of inequality (19) for \(A_{\varepsilon_0/2}\) with some constants \(C_1'(\varepsilon_0)\) and \(C_2'(\varepsilon_0)\) is obvious, because for all points \((\rho,\Theta)\in A_{\varepsilon_0/2}\) the quantities \(\bar\sigma(\rho,\Theta)\) and \(|\varphi(\Theta)-\rho|\) are bounded from below and from above by positive constants.

From the definition of the sets \(\bar B-\bar E_{\varepsilon_0}\) and \(N_{\varepsilon_0/2}\) it follows that for any point of \(N_{\varepsilon_0/2}\) the nearest boundary point is separated from the singular points by a distance not less than \(\varepsilon_0/2\).

Let \(S_{\varepsilon_0/2}\) be the closure of \(\bar\gamma-\Lambda_{\varepsilon_0/2}\); it lies at a distance not less than \(\varepsilon_0/2\) from \(\bar\Gamma_2\), where \(\Lambda_\delta=\bar E(\Gamma_2;\delta)\cap\bar\Gamma\); \(\Lambda_\delta\) is the image of \(\Lambda_\delta\) in polar coordinates.

Let \(\omega(\Theta)\) be the angle formed by the ray emanating from \(O\) and having polar angle \(\Theta\) with the tangent to \(\bar\gamma\); \(\omega(\Theta)\) is a continuous single-valued function of \(\Theta\) on the set \(S_{\varepsilon_0/2}\), and there exists \(\Theta_0>0\) such that \(\omega(\Theta)\) belongs to none of the intervals \([0,\Theta_0]\), \([\pi-\Theta_0,\pi+\Theta_0]\), \([2\pi-\Theta_0,2\pi]\) for \((\rho,\Theta)\in S_{\varepsilon_0/2}\).

Therefore, from the equality

\[ \frac{d\varphi(\Theta)}{d\Theta}=\varphi(\Theta)\operatorname{ctg}\omega(\Theta) \]

in view of the boundedness of \(\varphi(\Theta)\) on \([0,2\pi]\), we obtain

\[ \left|\frac{d\varphi(\Theta)}{d\Theta}\right|\leq C_5\bigl((\rho,\Theta)\in S_{\varepsilon_0/2}\bigr). \tag{20} \]

Divide the segment \([0,2\pi]\) into four parts:
\[ \left[\frac{(k-1)\pi}{2},\,\frac{\pi k}{2}\right],\qquad k=1,2,3,4. \]
The inequalities (19) are proved for each of these segments in the same way; hence it suffices to prove them for \([0,\pi/2]\).

Denoting by \((\rho_*,\Theta_*)\) the nearest boundary point to the point \((\rho,\Theta)\in N_{\varepsilon_0/2}\), we shall have \((0\leq \Theta\leq \pi/2)\)

\[ \begin{aligned} 1 &\leq \frac{|\varphi(\Theta)-\rho|}{\delta(\rho,\Theta)} = \frac{|\varphi(\Theta)-\rho|} {\sqrt{\rho^2+\rho_*^2-2\rho\rho_*\cos(\Theta-\Theta_*)}} \\ &= \frac{|\varphi(\Theta)-\rho|} {\sqrt{(\rho-\rho_*)^2+2\rho\rho_*\bigl(1-\cos(\Theta-\Theta_*)\bigr)}} \\ &= \frac{|\varphi(\Theta)-\rho|} {\sqrt{(\rho-\rho_*)^2+4\rho\rho_*\sin^2\frac{\Theta-\Theta_*}{2}}} \leq C_6\frac{|\varphi(\Theta)-\rho|} {\bigl[|\rho-\rho_*|+|\Theta-\Theta_*|\bigr]} \\ &\leq \frac{|\varphi(\Theta)-\varphi(\Theta_*)|+|\varphi(\Theta_*)-\rho|} {\bigl[|\rho-\rho_*|+|\Theta-\Theta_*|\bigr]} \leq C_7 . \end{aligned} \]

The inequalities (19) are proved.

III. Let the curve \(\Gamma\) be sufficiently many times differentiable and have a direction \(\rho\) continuously differentiable, changing the required number of times together with \(s\), the arc length of \(\Gamma\).

Given

\[ \left.\frac{\partial^k u}{\partial n^k}\right|_\Gamma =\lambda_k(s)\in L_p(\Gamma)\qquad (k=0,1,\ldots,s_0-1). \]

Suppose that \(\lambda_k(s)\) has derivatives up to order \(s_0-1-k\) inclusive (with respect to \(s\)). It is necessary to estimate
\(\left.\partial^k u/\partial \rho^k\right|_\Gamma\) in the sense of the metric \(L_p(\Gamma)\). Fix a point \(A=(x,y)\in\Gamma\). With the point \(A\) associate a new coordinate system \((n,s)\). Then our function \(u(x,y)\) becomes the function \(\bar u(n,s)\). Obviously,

\[ \left.\frac{\partial^k\bar u}{\partial n^{k_1}\partial s^{k_2}}\right|_A = \left.\frac{\partial^k u}{\partial n^{k_1}\partial s^{k_2}}\right|_A \qquad (k_1+k_2=k). \]

Therefore

\[ \left.\frac{\partial^k u}{\partial \rho^k}\right|_A = \sum_{j=0}^{k} C_k^j \left.\frac{\partial^k u}{\partial n^j\partial s^{k-j}}\right|_A \cos^j(n,\rho)\cos^{k-j}(s,\rho). \]

On the basis of the last equality we obtain the estimate

\[ \left\|\frac{\partial^k u}{\partial \rho^k}\right\|_{L_p(\Gamma)} \leq C_8 \sum_{j=0}^k \left\|\frac{\partial^k u}{\partial n^j\,\partial s^{k-j}}\right\|_{L_p(\Gamma)} \tag{21} \]

\[ (k=0,1,\ldots,s_0-1). \]

§ 2. On an Inequality for Functions of the Class \(W^{(l)}_{p,\alpha,\beta}(g)\)

Let \(l, s_0, m \geq 1\) be natural numbers; \(1<p<\infty\); \(\alpha\) and \(\beta\) real numbers, and suppose that the inequalities

\[ \frac{l}{2} \leq s_0 \leq l,\qquad m\leq l,\qquad s_0+m>l, \tag{22} \]

\[ s_0-l+\frac{1}{p}-1<\beta<s_0-l+\frac{1}{p}, \tag{23} \]

\[ \max\left(-\frac{2}{p'},\, m-l-1+\frac{2}{p}\right)<\alpha<\frac{2}{p}, \tag{24} \]

\[ \left(\frac{1}{p}+\frac{1}{p'}=1\right), \]

are satisfied; for \(p\leq 2\) we shall also assume that

\[ m-l-1+\frac{2}{p}<0. \tag{25} \]

Theorem. Let \(g\) be a bounded (open) domain with a boundary that is \(l\) times continuously differentiable, and suppose that inequalities (22)—(25) are satisfied. Then there exists a constant \(C_1>0\), independent of \(f\in W^{(l)}_{p,\alpha,\beta}(g)\), such that the inequality

\[ \left\|\frac{f}{r^{1/p}}\right\|_{L_p(g)} \leq C_1\left( \sum_{|k|=0}^{m-1} |f^{(k)}(0,0)| + \sum_{k=0}^{s_0-1}\sum_{j=0}^{k} \left\|\frac{\partial^k f}{\partial n^j\,\partial s^{k-j}}\right\|_{L_p(\Gamma)} + \left\|\frac{D^l f}{r^\alpha \sigma^\beta}\right\|_{L_p(g)} \right), \tag{26} \]

where \(\left\|\dfrac{\partial^k f}{\partial n^k}\right\|_{L_p(\Gamma)}\) denote the norms of the boundary values of the normal derivatives on \(\Gamma\), and \(s\) is the arc length of \(\Gamma\).

Before proving this theorem, we shall prove one lemma. Let, as before, \(Q_1=g-Q\), where \(Q=Q(0;d)\) is the (open) disk of radius \(d\) with center at the point \(O=(0,0)\), and let \(x^0\in \overline{Q_1}\). From the point \(x^0\) draw a straight line \(X_0\) not intersecting \(Q\). The nearest points of intersection of the line \(X_0\) with \(\Gamma\) (there are two of them) will be denoted by \(x_+^0\) and \(x_-^0\) (if \(x^0\in\Gamma\), then one of the points \(x_+^0\) or \(x_-^0\) coincides with \(x^0\)). The points \(x_+^0\) and \(x_-^0\) will be called the “points of impingement” of the line \(X_0\) on \(\Gamma\).

If the points \(x_+^0\) and \(x_-^0\) are not regular (with respect to \(x^0\)), then, owing to the differentiability of the boundary, we can rotate the line \(X_0\)

at a point \(x^0\) so that, for the obtained line \(X'_0\), the nearest points of intersection with the boundary \(\Gamma\) will already be regular.

Denote by \(\Pi(X)\) the circular cylinder with axis \(X\). All lines belonging to \(\Pi(X'_0)\) and parallel to \(X'_0\) will meet \(\Gamma\) regularly, provided only that the cylinder \(\Pi(X'_0)\) is sufficiently thin.

Let now \(X\) denote an arbitrary line belonging to \(\Pi(X'_0)\) (the line \(X\) is parallel to the axis \(X'_0\)), and let \(x_+\) and \(x_-\) be the points where the line \(X\) meets the boundary \(\Gamma\). It is clear that the set-theoretic sum of the segments \([x_+, x_-]\) corresponding to all lines \(X \in \Pi(X'_0)\) constitutes a regular bridge for the domain \(g\). We denote this bridge by \(\Omega(x^0)\) and shall call it the regular bridge covering the point \(x^0\).

When the line \(X_0\) is rotated at the point \(x^0\) through an angle \(\Theta\), the points of intersection \(g \cap X_0\) will be displaced by a distance not exceeding \(\Theta D\), where \(D\) denotes the diameter of the domain \(g\). Consequently, the intersection \(g \cap X'_0\) will be at a distance from the point \(O\) not less than \(\sigma(O; X_0)-\Theta D\). The cylinder \(\Pi(X'_0)\), and therefore also the bridge \(\Omega(x^0)\), will be at a distance from the point \(O\) not less than \(\sigma(O; X_0)-\Theta D-\delta\), where \(\delta\) is the radius of the cylinder \(\Pi(X'_0)\).

Let \(\Theta\) and \(\delta\) be so small that the inequality

\[ \sigma(O; X_0)-\Theta D-\delta \geq \frac{d}{2} \]

holds.

Thus we have obtained that every point \(x^0 \in \overline{Q}_1\) can be covered by a regular bridge for the domain \(g\) in such a way that it will be at a distance from the point \(O\) not less than \(d/2\).

Hence, on the basis of the Heine–Borel lemma, we obtain the lemma.

Lemma 3. The domain \(\overline{Q}_1\) (here it suffices to require only continuous differentiability of the boundary \(\Gamma\)) can be covered by a finite number of regular bridges (for the domain \(g\)) in such a way that these bridges will be at a distance from the point \(O\) not less than \(d/2\).

Proof of the theorem. It suffices to prove inequality (26) separately for the domains \(Q_1\) and \(Q \subset B-E_{3\varepsilon_0}\). Let \(\Omega_1, \ldots, \Omega_t\) be bridges covering \(Q_1\).

In view of (22), (23) and Lemma 2 of [1], the following inequality holds:

\[ \|f\|_{L_p(\Omega_j)} \leq C_2\left(\sum_{k=0}^{s_0-1} \left\|\frac{\partial^k f}{\partial y_j^k}\right\|_{L_p(\partial\Omega_j)} + \left\|\frac{D^l f}{\sigma^\beta}\right\|_{L_p(\Omega_j)} \right), \]

where \(y_j\) is the axis of the regular bridge \(\Omega_j\) \((j=1,\ldots,t)\); the constant \(C_2\) does not depend on \(f \in W^{(l)}_{p,\alpha,\beta}(g)\). Hence, taking into account inequality (21), we obtain for the domains \(\Omega_j\) the inequalities we need

\[ \|f\|_{L_p(\Omega_j)} \leq C_3\left( \sum_{k=0}^{s_0-1}\sum_{j=0}^{k} \left\| \frac{\partial^k f}{\partial n^j\, \partial s^{k-j}} \right\|_{L_p(\partial\Omega_j)} + \left\|\frac{D^l f}{\sigma^\beta}\right\|_{L_p(\Omega_j)} \right) \leq \]

\[ \leq C_4\left( \sum_{k=0}^{s_0-1}\sum_{j=0}^{k} \left\| \frac{\partial^k f}{\partial n^j\, \partial s^{k-j}} \right\|_{L_p(\Gamma)} + \left\|\frac{D^l f}{r^\alpha\sigma^\beta}\right\|_{L_p(g)} \right). \tag{27} \]

We now proceed to obtain the required inequality for the domain \(B\). As we already know, for \(F=f\psi\) (see (15)) the inequality

\[ \left\|\frac{D^lF}{r^\alpha\sigma^\beta}\right\|_{L_p(B)}<\infty . \tag{28} \]

is meaningful.

Passing to the polar coordinate system transforms inequality (28) into the following inequality:

\[ \left\|\frac{\overline{D^lF}}{\rho^{\alpha-1/p}\sigma^\beta}\right\|_{L_p(\overline{B})}<\infty, \]

which is equivalent, by virtue of (19), to the inequality

\[ \left\|\frac{\overline{D^lF}}{\rho^{\alpha-1/p}\,|\varphi(\Theta)-\rho|^\beta}\right\|_{L_p(\overline{B})}<\infty, \]

since \(\overline{F}=0\) on \(\overline{E}_{\varepsilon_0}\).

From the last inequality it follows that

\[ \left\|\frac{\partial^l\overline{F}/\partial\rho^l}{\rho^{\alpha-1/p}\,|\varphi(\Theta)-\rho|^\beta}\right\|_{L_p(\overline{B})}<\infty, \tag{29} \]

since

\[ \frac{\partial^q\overline{F}(\rho,\Theta)}{\partial\rho^q} = \sum_{i=0}^{q} C_q^i \frac{\partial^q F(x,y)}{\partial x^i\,\partial y^{q-i}} \cos^i\Theta\,\sin^{q-i}\Theta . \tag{30} \]

By Fubini’s theorem it follows from this that for almost all \(\Theta\in[0,2\pi]\) the finite integral

\[ \int_{0}^{\varphi(\Theta)} \left| \frac{\partial^l\overline{F}/\partial\rho^l} {\rho^{\alpha-1/p}\,|\varphi(\Theta)-\rho|^\beta} \right|^p \,d\rho <\infty \]

exists.

Hence, by Lemma 2, we obtain

\[ \int_{0}^{\varphi(\Theta)} |\overline{F}|^p\,d\rho \le C_5 \left( \sum_{j=0}^{m-1} \left| \frac{\partial^j\overline{F}(0,\Theta)}{\partial\rho^j} \right|^p + \sum_{j=0}^{s_0-1} \left| \frac{\partial^j\overline{F}(\varphi(\Theta),\Theta)}{\partial\rho^j} \right|^p + \int_{0}^{\varphi(\Theta)} \left| \frac{\partial^l\overline{F}/\partial\rho^l} {\rho^{\alpha-1/p}\,|\varphi(\Theta)-\rho|^\beta} \right|^p \,d\rho \right), \]

where the constant \(C_5\) depends on the function \(\varphi\), but not on \(\Theta\).

Taking into account that \(\overline{F}\equiv\overline{f}\) on \(\overline{g}-\overline{E}_{3\varepsilon_0}\), from the last inequality, by virtue of (30), we obtain

\[ \int_{0}^{\varphi(\Theta)} |\overline{F}|^p\,d\rho \le C_6 \left( \sum_{|k|=0}^{m-1} |f^{(k)}(0,0)|^p + \sum_{k=0}^{s_0-1} \left| \frac{\partial^k\overline{F}(\varphi(\Theta),\Theta)}{\partial\rho^k} \right|^p + \int_{0}^{\varphi(\Theta)} \left| \frac{\partial^l\overline{F}/\partial\rho^l} {\rho^{\alpha-1/p}\,|\varphi(\Theta)-\rho|^\beta} \right|^p \,d\rho \right). \]

Integrating both sides of the last inequality with respect to \(\Theta\) from \(0\) to \(2\pi\), we have

\[ \int_{0}^{2\pi} d\Theta \int_{0}^{\varphi(\Theta)} |F|^p\, d\rho \leq C_7 \left( \sum_{|k|=0}^{m-1} |f^{(k)}(0,0)|^p +\sum_{k=0}^{s_0-1}\int_{0}^{2\pi} \left|\frac{\partial^k \overline{F}(\varphi(\Theta),\Theta)}{\partial \rho^k}\right|^p d\Theta +\int_{0}^{2\pi} d\Theta \int_{0}^{\varphi(\Theta)} \left|\frac{\partial^l \overline{F}/\partial \rho^l} {\rho^{\alpha-1/p}|\varphi(\Theta)-\rho|^\beta}\right|^p d\rho \right). \]

Hence, taking again (30) and (19) into account, we obtain that

\[ \|\overline{F}\|_{L_p(\overline{B})} \leq C_8 \left( \sum_{|k|=0}^{m-1} |f^{(k)}(0,0)| +\sum_{k=0}^{s_0-1} \left(\int_{0}^{2\pi} \left|\frac{\partial^k \overline{F}(\varphi(\Theta),\Theta)} {\partial \rho^k}\right|^p d\Theta\right)^{1/p} +\left\| \frac{D^l F}{\rho^{\alpha-1/p}\sigma^\beta} \right\|_{L_p(\overline{B})} \right). \]

Hence, passing to the former rectangular Cartesian coordinate system, we have \((r \equiv \rho)\)

\[ \left\|\frac{F}{r^{1/p}}\right\|_{L_p(B)} \leq C_9 \left( \sum_{|k|=0}^{m-1} |f^{(k)}(0,0)| +\sum_{k=0}^{s_0-1} \left\|\frac{\partial^k F}{\partial \rho^k}\right\|_{L_p(\gamma)} +\left\|\frac{D^l F}{r^\alpha \sigma^\beta}\right\|_{L_p(B)} \right). \tag{31} \]

Taking here inequalities (16) and (21) into account, we obtain \((F \equiv f\) on \(B-E_{3\varepsilon_0})\) that

\[ \left\|\frac{f}{r^{1/p}}\right\|_{L_p(B-E_{3\varepsilon_0})} \leq C_{10} \left( \sum_{|k|=0}^{m-1} |f^{(k)}(0,0)| +\sum_{k=0}^{s_0-1}\sum_{j=0}^{k} \left\|\frac{\partial^k f}{\partial n^j\,\partial s^{k-j}}\right\|_{L_p(\Gamma)} +\left\|\frac{D^l f}{r^\alpha \sigma^\beta}\right\|_{L_p(g)} \right). \tag{32} \]

From (27) and (32) follows (26). For \(s_0+m<l\), an inequality of type (26), generally speaking, does not hold. We note that the generalization of inequality (26) to a domain having \(n\) dimensions presents no special difficulties.

§ 3. On boundary value problems for equations with strong degeneracy on the boundary

Let \(l, s_0, m, \alpha, \beta\) be the numbers specified in § 2 satisfying inequalities (22)—(25) for \(p=2\), and let \(\Phi \in W^{(l)}_{2,\alpha,\beta}(g)\) be a given function for which the boundary functions

\[ \left.\frac{\partial^k \Phi}{\partial n^k}\right|_{\Gamma} = \mu_k(s) \in L_2(\Gamma) \quad (k=0,1,\ldots,s_0-1), \tag{33} \]

as well as the values of the numbers

\[ \Phi^{(k)}(0,0) = \frac{\partial^{k_1+k_2}\Phi(0,0)}{\partial x^{k_1}\partial y^{k_2}} \quad (|k|=0,1,\ldots,m-1). \tag{34} \]

Introduce the class \(\mathbf M(\Phi)\) of all functions \(f \in W_{2,\alpha,\beta}^{(l)}(g)\) having the boundary values (33) and (34). Denote by \(\mathbf M_0\) the set of all functions \(f \in W_{2,\alpha,\beta}^{(l)}(g)\) such that

\[ \left. \frac{\partial^k f}{\partial n^k} \right|_{\Gamma} = 0 \qquad (k=0,1,\ldots,s_0-1) \]

and

\[ f^{(k)}(0,0)=0 \qquad (|k|=0,1,\ldots,m-1). \]

From the theorem proved in § 2 it follows that there exists a constant \(C_1>0\), independent of \(f \in \mathbf M_0\), such that the inequality

\[ \|f\|_{L_2(g)} \leq C_1 \left\| \frac{D^l f}{r^\alpha \sigma^\beta} \right\|_{L_2(g)} = C_1 D_g(f) \tag{35} \]

holds.

Let to each pair of vectors \((\mathbf k,\mathbf p)\) with \(|\mathbf k|\leq l,\ |\mathbf p|\leq l\) \((\mathbf k=(k_1,k_2),\ \mathbf p=(p_1,p_2))\) there be assigned a measurable function \(a_{\mathbf k\mathbf p}(x,y)\), symmetric with respect to \(\mathbf k\) and \(\mathbf p\):

\[ |a_{\mathbf k\mathbf p}| \leq \frac{M^2}{r^{2\alpha}\sigma^{2\beta_{\mathbf k\mathbf p}}}, \tag{36} \]

where \(M\) is a constant independent of \((x,y)\in g\), and

\[ \beta_{\mathbf k\mathbf p}= \begin{cases} l+\beta-\max\{|\mathbf k|,|\mathbf p|\}, & \text{if } \max\{|\mathbf k|,|\mathbf p|\}>s_0,\\ \beta+l-s_0, & \text{if } \max\{|\mathbf k|,|\mathbf p|\}<s_0. \end{cases} \]

Suppose that there exists a constant \(\varkappa>0\) such that the inequality

\[ \sum_{|\mathbf k|,|\mathbf p|\leq l} a_{\mathbf k\mathbf p}\,\xi_{\mathbf p}\xi_{\mathbf k} \geq \varkappa r^{-2\alpha}\sigma^{-2\beta} \sum_{|\mathbf k|=l}\xi_{\mathbf k}^2 \tag{37} \]

holds for any real numbers \(\xi_{\mathbf k},\xi_{\mathbf p}\) corresponding to the vectors \(\mathbf k\) and \(\mathbf p\); \(\varkappa\) is independent of \((x,y)\in g\).

Consider the bilinear functional

\[ E_g(f,v)= \sum_{|\mathbf k|,|\mathbf p|\leq l} \int_g a_{\mathbf k\mathbf p} f^{(\mathbf k)} v^{(\mathbf p)}\,dx\,dy. \]

From (37) it follows \(\bigl(E_g(f)\equiv E_g(f,f)\bigr)\) that

\[ E_g(f)\geq \varkappa D_g(f). \tag{38} \]

Let \(F\in L_2(g)\), and consider the new functional

\[ \mathrm H(f)=E_g(f)-2(F,f), \]

where \((F,f)\) is the scalar product of \(F\) and \(f\). Pose the following variational problem: find the minimum of the functional \(\mathrm H(f)\) among functions \(f\) of the class \(\mathbf M(\Phi)\):

\[ \min H(f)=b, \tag{39} \]

\[ f\in \mathbf M(\Phi). \]

Theorem. There exists a function \(u\in \mathbf M(\Phi)\), moreover a unique one, for which the minimum of the variational problem is attained. This function is a generalized solution of the equation

\[ Lu=\sum_{|\mathbf k|,\,|\mathbf p|\le l}(-1)^{|\mathbf p|}\bigl(a_{\mathbf k\mathbf p}u^{(\mathbf k)}\bigr)^{(\mathbf p)}=F \tag{40} \]

in the sense that

\[ E_g(u,v)-(F,v)=0 \tag{41} \]

for any \(v\in \mathbf M_0\).

The proof is carried out in the usual way (see, for example, [4, 9]). Therefore we shall confine ourselves to giving only the scheme of the proof.

Let \(f_q\) be a minimizing sequence of functions of the class [[unclear: beginning of line cut off]], i.e. \(H(f_q)=b+\varepsilon_q\) \((\varepsilon_q\ge 0,\ \varepsilon_q\to 0\) as \(q\to\infty)\). First it is shown (see [4], p. 108) that \(E_g(f_q-f_s)\to 0\) \((q,s\to\infty)\). Then, by virtue of (38), it follows that \(D_g(f_q-f_s)\to 0\) \((q,s\to\infty)\). Therefore it follows from [[unclear: beginning of line cut off]] that \(\|f_q-f_s\|_{W^{(l)}_{2,\alpha,\beta}(g)}\to 0\). By completeness of the space [[unclear: beginning of line cut off]] \((g)\) there exists a function \(u\in W^{(l)}_{2,\alpha,\beta}(g)\) such that

\[ \|f_q-u\|_{W^{(l)}_{2,\alpha,\beta}(g)}\to 0\quad (q\to\infty). \]

\[ \left\|\frac{\partial^k(u-\Phi)}{\partial n^k}\right\|_{L_2(\Gamma)} = \left\|\frac{\partial^k(u-f_q)}{\partial n^k}\right\|_{L_2(\Gamma)} \le \]

\[ \le C_2\|u-f_q\|_{W^{(l)}_{2,\alpha,\beta}(g)}\to 0 \quad (k=0,1,\ldots,s_0-1) \]

so that

\[ \left.\frac{\partial^k u}{\partial n^k}\right|_\Gamma = \left.\frac{\partial^k\Phi}{\partial n^k}\right|_\Gamma \quad (k=0,1,\ldots,s_0-1). \]

Similarly it is shown that

\[ u^{(\mathbf k)}(0,0)=\Phi^{(\mathbf k)}(0,0)\quad (|\mathbf k|=0,1,\ldots,m-1). \]

Consequently, \(u\in \mathbf M(\Phi)\). \(H(u)=b\), and equality (41) is proved in the usual way.

§ 4. ON THE DIFFERENTIABILITY OF GENERALIZED SOLUTIONS

Let \(g'\) be an arbitrary domain such that \(\overline{g'}\subset g\) and \(O\notin \overline{g'}\). Introduce the notation

\[ l_{\mathbf k\mathbf p}=\max\{|\mathbf k|,|\mathbf p|\}. \]

Theorem. Suppose that the coefficients \(a_{\mathbf k\mathbf p}\) of equation (40) have, on \(g\), except possibly at the point \(O\), continuous derivatives up to order \(l_{\mathbf k\mathbf p}-1\) inclusive, satisfying on every domain \(g'\) the Lipschitz condition

\[ \left| \Delta_i \frac{\partial^{l_{\mathbf k\mathbf p}-1}a_{\mathbf k\mathbf p}(X)} {\partial x_i^{\,l_{\mathbf k\mathbf p}-1}} \right| \le M_{g'}|h|, \tag{42} \]

where the constant \(M_{g'}\) does not depend on \(X\), \(X+he_i\in g'\); \(e_i\) is the unit vector having the positive direction of the \(x_i\)-axis; \(\Delta_i f(X)\) denotes the difference of the function \(f\) in the direction of the \(x_i\)-axis with step \(h\). Then the generalized-

We shall now formulate a lemma on which the proof of the theorem is based.

Lemma 4 (S. M. Nikol’skii). Let the function \(F\) belong to the class \(L_2(\Omega)\), and let \(\psi\) be a finite function on \(\Omega\) with support [[unclear: truncated at right margin]] having \(s\) times continuously differentiable derivatives. Then

\[ \left|\int_{\Omega_1}\frac{\Delta^s F}{h^s}\,\psi\,dx\right| \leq C_1\int_{\Omega_1}\left|F\, \frac{\partial^s\psi}{\partial x_1^{s_1}\ldots \partial x_n^{s_n}}\right|\,dx \leq \]

\[ \leq C_2 \left( \int_{\Omega_1} \left| \frac{\partial^s\psi}{\partial x_1^{s_1}\ldots \partial x_n^{s_n}} \right|^2 dx \right)^{1/2} \left( \int_{\Omega}|F|^2\,dx \right)^{1/2}, \]

where

\[ \Delta^s F=\Delta_1^{s_1}\ldots \Delta_n^s F. \]

We now pass to the proof of the theorem. We note that the notations which occur below are taken from [10]. Repetition of the proof of Theorem § 2 of [10] leads us to the following inequality, analogous to inequality (2.5) of the cited work:

\[ \varkappa D_{\Delta_2}\left(\eta\,\frac{\Delta_i^s u}{h^s}\right) \leq -J_1-J_2+J_3+J_4-J_5+J_6. \]

Here it is only necessary in the integrals \(J_1,\ldots,J_5\) to replace the sum

\[ \sum_{|k|,\ |p|\leq l} \]

by the sum

\[ \sum_{k,\ l\in E} \]

and

\[ J_6=\int_{\Delta_2} \frac{\Delta_i^s u}{h^s}\, \frac{\Delta_i^s F}{h^s}\, \eta_1\,dx. \]

The essence of the proof of the theorem consists in showing that the integrals \(J_1,\ldots,J_6\) are bounded by some constant independent of \(h\), or else do not exceed

\[ C_3\sqrt{D_{\Delta_2}\left(\eta\,\Delta_i^s u/h^s\right)}+C_4 \]

(\(C_3\) and \(C_4\) are constants independent of \(h\)).

We write out the integral \(J_1\), since in estimating \(J_1\) and \(J_6\) additional arguments are required:

\[ J_1= \sum_{\lambda=1}^{s} \sum_{|k|,\ |p|\leq l} C_s^{\lambda} \int_{\Delta_2} \frac{\Delta_i^{\lambda}a_{kp}}{h^{\lambda}}\, \frac{\Delta_i^{s-\lambda}u^{(k)}}{h^{s-\lambda}} \times \]

\[ \times \sum_{0\leq \tau\leq p} C_p^{\tau} \frac{\Delta_i^s u^{(p-\tau)}}{h^s}\, \eta_1^{(\tau)}\,dx. \]

Let us estimate \(J_1\). It is sufficient to estimate one integral entering into \(J_1\). \(s=1\). If \(l_{kp}\geq 1\), then

\[ [[unclear: displayed formula continues below the visible page]] \]

\[ \le \begin{cases} C_5, & \text{if } |\mathbf p|<l,\\[6pt] C_6\sqrt{D_{\Delta_2}\!\left(\eta\,\dfrac{\Delta_i u}{h}\right)}, & \text{if } |\mathbf p|=l. \end{cases} \]

If \(l_{\mathrm{cr}}=0\), i.e., \(\mathbf k=0\) and \(\mathbf p=0\), then on the basis of a lemma of S. M. Nikolskii we have

\[ |J_1^{\,j}|= \left| \int_{\Delta_2} \dfrac{\Delta_i a_{00}}{h}\,u\,\dfrac{\Delta_i u}{h}\,\eta_1\,dx \right| \le C_7 \int_{\Delta_2} \left| a_{00}\,\dfrac{\partial}{\partial x_i} \left( \eta_1 u\,\dfrac{\Delta_i u}{h} \right) \right|\,dx \le \]

\[ \le \begin{cases} C_8, & \text{if } l>1,\\[6pt] C_9+C_{10}\sqrt{D_{\Delta_2}\!\left(\eta\,\dfrac{\Delta_i u}{h}\right)}, & \text{if } l=1. \end{cases} \]

Now we estimate the integral \(J_6\):

\[ |J_6|= \left| \int_{\Delta_2} \dfrac{\Delta_i u}{h}\, \dfrac{\Delta_i F}{h}\,\eta_1\,dx \right| \le C_{11} \int_{\Delta_2} \left| F\,\dfrac{\partial}{\partial x_i} \left( \eta_1\,\dfrac{\Delta_i u}{h} \right) \right|\,dx \le \]

\[ \le C_{11}\|F\|_{L_2(g)} \left\| \dfrac{\partial}{\partial x_i} \left( \eta_1\,\dfrac{\Delta_i u}{h} \right) \right\|_{L_2(\Delta_2)} \le \]

\[ \le \begin{cases} C_{12}, & \text{if } l>1,\\[6pt] C_{13}+C_{14}\sqrt{D_{\Delta_2}\!\left(\eta\,\dfrac{\Delta_i u}{h}\right)}, & \text{if } l=1. \end{cases} \]

The theorem is then proved by induction.

In conclusion the author expresses his deep gratitude to Prof. S. M. Nikolskii for posing the problem and for valuable advice.

References

1. Nikolskii S. M., Lizorkin P. I. DAN SSSR, 159, No. 3, 1964.
2. Kudryavtsev L. D. Proceedings of the V. A. Steklov Mathematical Institute, 55, 1959.
3. Nikolskii S. M. UMN, vol. XVI, issue 5 (101), 1961.
4. Sobolev S. L. Some applications of functional analysis in mathematical physics. Novosibirsk, Publishing House of the Siberian Branch of the Academy of Sciences of the USSR, 1962.
5. Kondrashov V. I. DAN SSSR, 142, No. 6, 1962.
6. Lizorkin P. I. DAN SSSR, 126, No. 4, 1959.
7. Uspenskii S. V. Proceedings of the V. A. Steklov Mathematical Institute, vol. X, 1961, pp. 282–303.
8. Vasharin A. A. Izv. AN SSSR, ser. matem., 23, 421–454, 1959.
9. Nikolskii S. M. Matem. sb., 62 (104): 1, 1963.
10. Salmanov Yu. D. Differential Equations, 1, No. 10, 1321–1337, 1965.

Received by the editors
April 5, 1966

Institute of Mathematics
and Mechanics, Academy of Sciences of the Azerbaijan SSR