UDC 517.9469

ON THE DIRICHLET PROBLEM FOR AN EQUATION WITH DEGENERATION AT INFINITY

Yu. S. Nikol’skii

The paper considers the differential equation

\[ L(u)=\sum_{|k|,\,|l|\leq r}(-1)^{|l|}\frac{\partial^l}{\partial x^l}\left(a_{kl}(x)u^{(k)}(x)\right)=F(x) \tag{1} \]

in the domain \(R_n^+\), which is the upper half-space of the \(n\)-dimensional space \(R_n\) of points \(x=(x_1,\ldots,x_n)\) with real coordinates and with coordinate \(x_n>0\). Here \(k=(k_1,\ldots,k_n)\), \(l=(l_1,\ldots,l_n)\) are nonnegative integer vectors \((k_i,l_j\geq 0\) are integers, \(i,j=1,\ldots,n)\), \(|k|=k_1+\cdots+k_n\), \(|l|=l_1+\cdots+l_n\), \(r>0\) is a natural number, and

\[ f^{(l)}=\frac{\partial^l f}{\partial x^l} = \frac{\partial^{|l|}f}{\partial x_1^{l_1}\cdots \partial x_n^{l_n}} . \]

The coefficients of equation (1) satisfy the symmetry condition

\[ a_{kl}(x)=a_{lk}(x) \tag{2} \]

and ellipticity with weight \(\varphi=\varphi(\rho)\), \(\rho=\sqrt{x_1^2+\cdots+x_n^2}\). Namely, it is assumed that the inequality

\[ \sum_{|k|,\,|l|\leq r} a_{kl}\xi_k\xi_l \geq \frac{\lambda}{[\varphi(\rho)]^2} \sum_{|k|=r}\xi_k^2 \tag{3} \]

holds for all \(x\in R_n^+\) and all \(\xi_k,\xi_l\) corresponding to the vectors \(k,l\), where \(\lambda>0\) is a constant independent of \(x,\xi_k,\xi_l\).

It is assumed that the weight function \(\varphi=\varphi(\rho)\) is positive, continuous, and bounded below by a positive number on every compact domain \(G\subset R_n^+\), and satisfies the following two conditions:

A. \(\varphi=\varphi(t)\) is nondecreasing on \([0,+\infty)\) and satisfies the inequality

\[ \varphi(2t)\leq c\,\varphi(t) \]

for sufficiently large \(t\).

B. If one sets \(\varphi=t^{a_0}\lambda(t)\), \(\left(a_0=\frac{n-2}{2}\right)\), then

\[ \int_1^\infty \frac{dz}{z\chi(z)}<\infty, \qquad \chi(z)=\left\{\min_{0\leq t<\infty}\left(\frac{\lambda(zt)}{\lambda(t)}\right)^2\right\}^{1/2}, \]

with \(\chi(z)\geq 1\) for sufficiently large \(z\).

The conditions A., B. are satisfied, for example, by the following functions:

\[ \varphi(t)=t^\alpha,\quad \varphi(t)=t^\alpha\ln t\left(\alpha>\frac{n-2}{2}\right). \]

To justify the existence of a generalized solution it is sufficient further to assume that the coefficients \(a_{kl}(x)\) are measurable on \(R_n^+\) and satisfy on \(R_n^+\) the inequalities

\[ |a_{kl}(x)|\leq \frac{M^2}{\left[(1+\rho)^{-\min(|k|,|l|)}\varphi(\rho)\right]^2}. \tag{4} \]

It will also be necessary to impose certain conditions on the function \(F=F(x)\), which will be given below.

For the existence of a classical solution of equation (1), as usual, one has to require the corresponding smoothness conditions on the coefficients \(a_{kl}\) and on the function \(F\).

With equation (1) there are associated two weighted classes \(L_{2,\varphi}^{(r)}(R_n^+)\) and \(W_{2,\varphi}^{(r)}(R_n^+)\), which are certain generalizations of the classes \(L_{2,\alpha}^{(r)}(R_n^+)\), \(W_{2,\alpha}^{(r)}(R_n^+)\) of L. D. Kudryavtsev [1].

By definition, a function \(f=f(x_1,\ldots,x_n)\), defined on \(R_n^+\) together with its generalized derivatives up to order \(r\) inclusive, belongs to the class \(L_{2,\varphi}^{(r)}(R_n^+)\) or \(W_{2,\varphi}^{(r)}(R_n^+)\), if for it the finite norm

\[ \|f\|_{L_{2,\varphi}^{(r)}(R_n^+)} = \sum_{|k|=r} \left\| \frac{f^{(k)}(x)}{\varphi(\rho)} \right\|_{L_2(R_n^+)} \tag{5} \]

or

\[ \|f\|_{W_{2,\varphi}^{(r)}(R_n^+)} = \|f\|_{L_2(\omega)} + \|f\|_{L_{2,\varphi}^{(r)}(R_n^+)}, \tag{6} \]

respectively, has meaning, where
\[ \|f\|_{L_2(g)}=\left\{\int_g |f|^2\,dg\right\}^{1/2} \]
(\(g\subset R_n\) is some domain); \(\omega\) is the intersection of \(R_n^+\) with the unit ball in \(R_n\) centered at the origin.

If the function \(f\) belongs to the class \(L_{2,\varphi}^{(r)}(R_n^+)\), then it is known [2] that on the \(n-1\)-dimensional subspace \(R_{n-1}=\{x:x_n=0\}\) the traces have meaning (in the sense of convergence in the mean)

\[ \left. \frac{\partial^i f}{\partial x_n^i} \right|_{R_{n-1}} = \psi_i,\quad i=0,1,\ldots,r-1, \tag{7} \]

belonging respectively to the fractional weighted classes \(L_{2,\varphi}^{(r-i-\frac12)}(R_{n-1})\). This circumstance makes it possible correctly to pose the first boundary-value problem for equation (1) in the class \(L_{2,\varphi}^{(r)}(R_n^+)\).

The fractional weighted classes \(L_{2,\varphi}^{(l)}(R_{n-1})\), \(W_{2,\varphi}^{(l)}(R_{n-1})\), where \(l\) is a positive noninteger and \(l=\bar l+\beta\), \(\bar l>0\) is an integer, \(0<\beta<1\), are defined as follows. By definition, a function \(\psi=\psi(x_1,\ldots,x_{n-1})\) belongs to the class \(L_{2,\varphi}^{(l)}(R_{n-1})\) or \(W_{2,\varphi}^{(l)}(R_{n-1})\), if it is given on \(R_{n-1}\) together with its generalized derivatives up to order \(\bar l\) inclusive and has the finite norm

\[ \|\psi\|_{L_{2,\varphi}^{(l)}(R_{n-1})} = \sum_{|k|=\bar l}\sum_{i=1}^{n-1} \left\{ \int_0^\infty \frac{dh}{h^{1+2\beta}} \left\| \frac{\Delta_i\bigl(\psi^{(k)},h\bigr)} {\varphi(\bar\rho+h)} \right\|_{L_2(R_{n-1})}^2 \right\}^{1/2} \]

or

\[ \|\psi\|_{W_{2,\varphi}^{(l)}(R_{n-1})} = \|\psi\|_{L_2(\omega^*)} + \|\psi\|_{L_{2,\varphi}^{(l)}(R_{n-1})}, \]

where \(\Delta_i(\psi,h)\) \((i=1,\ldots,n-1)\) denotes the first difference of the function \(\psi=\psi(x_1,\ldots,x_{n-1})\) with step \(h\) in the variable \(x_i\); \(\rho=|x_1|+\cdots+|x_{n-1}|\); \(\omega^*=\omega\cap R_{n-1}\).

For equation (1), posed on \(R_n^+\), we set the following boundary-value problem. Let functions be given on \(R_{n-1}\):

\[ \psi_i \in L_{2,\varphi}^{\left(r-i-\frac12\right)}(R_{n-1}) \qquad (i=0,1,\ldots,r-1). \]

It is required to find a function \(u\in L_{2,\varphi}^{(r)}(R_n^+)\) for which the following equalities hold:

\[ L(u)=F,\qquad x\in R_n^+, \tag{8} \]

\[ \left.\frac{\partial^i u}{\partial x_n^i}\right|_{R_{n-1}} = \psi_i \qquad (i=0,1,\ldots,r-1). \tag{9} \]

We prove that the boundary-value problem so posed has, and moreover uniquely, a generalized solution \(u\).

For the generalized solution \(u\) of problem (8), (9) the inequality

\[ \|u\|_{W_{2,\varphi}^{(r)}(R_n^+)} \le c\left\{ \sum_{i=0}^{r-1} \|\psi_i\|_{W_{2,\varphi}^{\left(r-i-\frac12\right)}(R_{n-1})} +\sigma_F \right\}, \]

holds, where the constant \(c>0\) does not depend on \(\psi_i\) or \(\sigma_F\), and where \(\sigma_F\) is the norm of \(F\) defined below.

From what has been said it follows, in particular, that in order for the generalized solution \(u\) of the boundary-value problem (8), (9) to belong to the class \(L_{2,\varphi}^{(r)}(R_n^+)\), it is necessary and sufficient that it have traces (7) belonging respectively to the classes

\[ L_{2,\varphi}^{\left(r-i-\frac12\right)}(R_{n-1}) \qquad (i=0,1,\ldots,r-1). \]

It is further proved that in the class \(L_{2,\varphi}^{(r)}(R_n^+)\) there exists, and moreover uniquely, a classical solution \(u\) of the indicated boundary-value problem. The proof of this fact is essentially based on a theorem asserting that every function \(f\in L_{2,\varphi}^{(r)}(R_n^+)\) having zero boundary functions (7) can be approximated in the metric \(L_{2,\varphi}^{(r)}(R_n^+)\) by functions with finite support.

The indicated results are proved by the variational method. In its justification an essential role is played by the embedding theorems obtained by the author in [2].

We note that these investigations develop the works of L. D. Kudryavtsev [3,4], 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 with coefficients degenerating at infinity.

Taking the opportunity, I consider it my pleasant duty to express my deep gratitude to Professor L. D. Kudryavtsev for posing the problem and for his constant attention during the execution of this work.

§ 1. EXISTENCE AND UNIQUENESS OF A GENERALIZED SOLUTION OF THE BOUNDARY-VALUE PROBLEM

Throughout the entire article we shall use embedding theorems*) obtained by the author in [2], and therefore we give their formulations for \(p=2\).

Let the weight function \(\varphi=\varphi(\rho)\) satisfy the conditions listed above (where condition \(B\) may be dropped). Then we have the following embedding theorems.

Theorem 1°. If the function \(f \in L_{2,\varphi}^{(r)}(R_n^+)\) or \(W_{2,\varphi}^{(r)}(R_n^+)\), then for it the boundary function has meaning

\[ \psi=\psi(x_1,\ldots,x_{n-1})=f\big|_{R_{n-1}} \in L_{2,\varphi}^{\left(r-\frac12\right)}(R_{n-1}) \quad \text{or} \quad W_{2,\varphi}^{\left(r-\frac12\right)}(R_{n-1}) \]

and, respectively, the embeddings

\[ L_{2,\varphi}^{(r)}(R_n^+)\to L_{2,\varphi}^{\left(r-\frac12\right)}(R_{n-1}),\qquad W_{2,\varphi}^{(r)}(R_n^+)\to W_{2,\psi}^{\left(r-\frac12\right)}(R_{n-1}) \]

hold.

Theorem 2°. (extension theorem). Let a system of functions be given on \(R_{n-1}\)

\[ \psi_i \in L_{2,\varphi}^{\left(r-i-\frac12\right)}(R_{n-1}),\, W_{2,\varphi}^{\left(r-i-\frac12\right)}(R_{n-1}) \quad (i=0,1,\ldots,r-1). \]

Then there exists a function \(f \in L_{2,\varphi}^{(r)}(R_n^+),\, W_{2,\varphi}^{(r)}(R_n^+)\) defined on \(R_n^+\), such that

\[ \left. \frac{\partial^i f}{\partial x_n^i} \right|_{R_{n-1}} =\psi_i \quad (i=0,1,\ldots,r-1), \]

and the inequality

\[ \|f\|_{L_{2,\varphi}^{(r)}(R_n^+)} \le c\sum_{i=0}^{r-1} \|\psi_i\|_{L_{2,\varphi}^{\left(r-i-\frac12\right)}(R_{n-1})} \]

is satisfied, and, respectively, a similar inequality holds in which \(L\) is to be replaced everywhere by \(W\).

Under our assumptions on the weight function \(\varphi\) (where condition \(A\) may now be dropped) the following embedding is valid.

Theorem 3°.

\[ W_{2,\varphi}^{(r)}(R_n^+)\to W_{2,(1+\rho)^k\varphi}^{(r-k)}(R_n^+) \quad (0\le k\le r). \]

We note that the obvious embedding holds

\[ W_{2,\varphi}^{(r)}(R_n^+)\to L_{2,\varphi}^{(r)}(R_n^+). \tag{1.1} \]

Conversely, if \(f\in L_{2,\varphi}^{(r)}(R_n^+)\), then the function \(f\) belongs to the class \(L_2^{(r)}(\omega)\). But then it is known [5] that for the half-sphere \(\omega\) the function \(f\in L_2(\omega)\), i.e. \(f\in W_{2,\varphi}^{(r)}(R_n^+)\). However, the embedding inverse to (1.1) does not hold, since norm (6) is not estimated by norm (5).

Let the function \(f\in L_{2,\varphi}^{(r)}(R_n^+)\) and

\[ D(f)= \int \sum_{|k|=r} \left| \frac{f^{(k)}(x)}{\varphi(\rho)} \right|^2 \,dR_n^+, \tag{1.2} \]

*) Let \(B_1,B_2\) be normed spaces. Then one says that \(B_1\) is embedded in \(B_2\), and writes \(B_1\to B_2\), if \(B_1\subset B_2\) and there exists a positive constant \(c\), independent of \(x\in B_1\), such that \(\|x\|_{B_2}\le c\|x\|_{B_1}\).

if the integral standing on the right-hand side of (1.2) is taken over the domain \(G \in R_n^+\), we shall denote it by \(D_G(f)\). It is clear that there exist two positive constants \(c_1, c_2\), independent of \(f\), such that

\[ c_1\sqrt{D(f)} \leq \|f\|_{L_{2,\varphi}^{(r)}(R_n^+)} \leq c_2\sqrt{D(f)}. \]

In view of this, in what follows we shall assume that

\[ \|f\|_{L_{2,\varphi}^{(r)}(R_n^+)}=\sqrt{D(f)}. \]

Positive constants independent of a number of sets under consideration will be denoted by the letter \(c\) with some subscript \((c_0=c)\); moreover, for simplicity of notation, in each estimate we shall begin the numbering of the subscripts anew.

Let \(m_0\) denote the class of functions \(v \in L_{2,\varphi}^{(r)}(R_n^+)\) having zero boundary functions \(\psi_i=0\) \((i=0,1,\ldots,r-1)\).

In solving the boundary-value problem (8), (9) posed above, we shall assume that the function \(F\) appearing on the right-hand side of equation (1) has the property that its norm

\[ \sigma_F=\|F\|_{\widetilde L_2^{(r)}(R_n^+)} =\sup_{\substack{\|v\|_{L_{2,\varphi}^{(r)}(R_n^+)}\leq 1\\ v\in m_0}} |(F,v)|, \tag{1.3} \]

is finite, where

\[ (F,v)=\int F\cdot v\,dR_n^+. \]

In particular, if

\[ A_F^2=\int (1+\rho)^{2r}\varphi^2 F^2\,dR_n^+<\infty, \tag{1.4} \]

then, by Bunyakovsky’s inequality, Theorem \(3^\circ\), and the Poincaré inequality for the Sobolev classes \(W_2^{(r)}(\omega)\) [5], we obtain

\[ |(F,v)|\leq A_F \sqrt{\int\left|\frac{v}{(1+\rho)^r\varphi}\right|^2 dR_n^+} \leq \]

\[ \leq cA_F\left[\|v\|_{L_2(\omega)}+\sqrt{D(f)}\right] \leq c_1 A_F\sqrt{D(f)}. \tag{1.5} \]

From (1.4) and (1.5) it follows that

\[ \sigma_F=\|F\|_{\widetilde L_2^{(r)}(R_n^+)}\leq c_1A_F<\infty. \]

Consider the bilinear functional

\[ E(f,g)=\int \sum_{|k|,\,|l|\leq r} a_{kl}(x) f^{(k)}(x) g^{(l)}(x)\,dR_n^+, \]

for the coefficients of which conditions (2)—(4) are fulfilled.

Put \(E(f)=E(f,f)\). If \(E(f)<\infty\), then, by (3),

\[ D(f)\leq \frac{1}{\lambda}E(f), \tag{1.6} \]

i.e.

\[ f\in L_{2,\varphi}^{(r)}(R_n^+),\quad W_{2,\varphi}^{(r)}(R_n^+). \]

Applying Bunyakovsky’s inequality and (4), on the basis of Theorem \(3^\circ\) we obtain

\[ E(f)\leq \sum_{|k|,\,|l|<r} \sqrt{\int \left|\frac{M\cdot f^{(k)}(x)} {(1+\rho)^{r-|k|}\varphi(\rho)}\right|^2\,dR_n^+}\times \]

\[ \times \sqrt{\int \left|\frac{M\cdot f^{(l)}(x)} {(1+\rho)^{r-|l|}\varphi(\rho)}\right|^2\,dR_n^+} \leq c\left(M\cdot \|f\|_{W_{2,\varphi}^{(r)}(R_n^+)}\right)^2 . \tag{1.7} \]

Thus, if \(f\in L_{2,\varphi}^{(r)}(R_n^+),\, W_{2,\varphi}^{(r)}(R_n^+)\), then \(E(f)<\infty\), and conversely. Consequently, \(L_{2,\varphi}^{(r)}(R_n^+)\) is the natural class of functions on which the functional \(E(f)\) is defined.

We now prescribe on \(R_{n-1}\) a fixed system of functions

\[ \psi_i\in L_{2,\varphi}^{\left(r-i-\frac12\right)}(R_{n-1}) \qquad (i=0,\,1,\ldots,\,r-1). \]

According to the converse embedding theorem \(2^\circ\), on \(R_n^+\) there exists a function \(\Phi\in L_{2,\varphi}^{(r)}(R_n^+)\) such that

\[ \left.\frac{\partial^i\Phi}{\partial x_n^i}\right|_{R_{n-1}} =\psi_i \qquad (i=0,\,1,\ldots,\,r-1), \]

and

\[ \|\Phi\|_{L_{2,\varphi}^{(r)}(R_n^+)} \leq c\sum_{i=0}^{r-1} \|\psi_i\|_{L_{2,\varphi}^{\left(r-i-\frac12\right)}(R_{n-1})}. \]

Define the class \(m\) (obviously nonempty) of all possible functions \(f\in L_{2,\varphi}^{(r)}(R_n^+)\) such that their boundary functions coincide respectively with the boundary functions of \(\Phi\), i.e.

\[ \left.\frac{\partial^i f}{\partial x_n^i}\right|_{R_{n-1}} = \left.\frac{\partial^i\Phi}{\partial x_n^i}\right|_{R_{n-1}} = \psi_i \qquad (i=0,\,1,\ldots,\,r-1). \]

We pose the following variational problem. It is required to determine in the class \(m\) of functions \(f\) a function \(u\) for which the minimum of the functional

\[ E(f)-2(F,f), \tag{1.8} \]

is attained, where the function \(F\) satisfies condition (1.3). Equation (1) for the functional (1.8) is the Euler equation.

Let us first prove that this functional is bounded from below on the class \(m\). Since

\[ E\left(\frac{f-\Phi}{2}\right)+ E\left(\frac{f+\Phi}{2}\right) = \frac12 E(f)+\frac12 E(\Phi), \tag{1.9} \]

we have

\[ E(f)-2(F,f) = \frac12 E(f-\Phi)-2(F,f-\Phi) +\frac12 E(f+\Phi)-E(\Phi)- \]

\[ -2(F,\Phi)\geq \frac12 E(f-\Phi)-2(F,f-\Phi)+A, \]

where

\[ A=-E(\Phi)-2(F,\Phi). \]

Further, because \(f-\Phi\in m_0\), on the basis of condition (1.3) and inequality (1.6) we shall have

\[ \left|(F,\, f-\Phi)\right|\leq \sigma_F \left[D(f-\Phi)\right]^{1/2} \leq \frac{\sigma_F}{\sqrt{\lambda}}\left[E(f-\Phi)\right]^{1/2}. \]

Therefore, taking the last inequality into account, we obtain

\[ \begin{aligned} E(f)-2(F,\, f) &\geq \frac{1}{2}E(f-\Phi) -\frac{2\sigma_F}{\sqrt{\lambda}}\left[E(f-\Phi)\right]^{1/2}+A \\ &= \left[ \frac{\sqrt{E(f-\Phi)}}{\sqrt{2}} -\frac{\sqrt{2}\,\sigma_F}{\sqrt{\lambda}} \right]^2 -\frac{2\sigma_F^2}{\lambda}+A \geq -\frac{2\sigma_F^2}{\lambda}+A =B>-\infty , \end{aligned} \tag{1.10} \]

where \(B\) depends only on \(F\) and \(\Phi\), but does not depend on \(f\in m\). This proves that the functional \(E(f)-2(F,f)\) is bounded below on the class \(m\).

Let

\[ d=\inf_{f\in m}\{E(f)-2(F,\, f)\} \]

and, for a sequence \(\{f_\nu\}\) of functions \(f_\nu\in m\),

\[ E(f_\nu)-2(F,\, f_\nu)=d+\varepsilon_\nu,\qquad \varepsilon_\nu\to 0,\quad \nu\to\infty . \]

Then

\[ \begin{aligned} \lambda D\left(\frac{f_\nu-f_\mu}{2}\right) &\leq E\left(\frac{f_\nu-f_\mu}{2}\right) \\ &= \frac{1}{2}\{E(f_\nu)-2(F,\, f_\nu)\} +\frac{1}{2}\{E(f_\mu)-2(F,\, f_\mu)\} \\ &\quad - \left\{ E\left(\frac{f_\nu-f_\mu}{2}\right) -2\left(F,\, \frac{f_\nu+f_\mu}{2}\right) \right\} \\ &\leq \frac{1}{2}\{d+\varepsilon_\nu\} +\frac{1}{2}\{d+\varepsilon_\mu\} -d = \frac{\varepsilon_\nu}{2}+\frac{\varepsilon_\mu}{2}, \end{aligned} \]

\[ \varepsilon_\nu,\ \varepsilon_\mu\to 0,\qquad \nu,\mu\to\infty , \tag{1.11} \]

since \(\dfrac{f_\nu+f_\mu}{2}\in m\). But \(f_\nu-f_\mu\in m_0\); therefore, taking into account Poincaré’s inequality (for \(\omega\)), we shall have

\[ \|f_\nu-f_\mu\|_{L_2(\omega)} \leq c\left[D(f_\nu-f_\mu)\right]^{1/2} \to 0,\qquad \nu,\mu\to\infty . \tag{1.12} \]

It is not difficult to prove that the space \(W^{(r)}_{2,\varphi}(R_n^+)\) is complete. Therefore, from (1.11) and (1.12) there follows the existence of a function \(u\in W^{(r)}_{2,\varphi}(R_n^+)\) such that

\[ \|f_\nu-u\|_{W^{(r)}_{2,\varphi}(R_n^+)}\to 0,\qquad \nu\to\infty . \]

By Theorem \(1^\circ\), the function \(u\) has boundary functions

\[ \left.\frac{\partial^i u}{\partial x_n^i}\right|_{R_{n-1}} = \psi_i^0 \qquad (i=0,\,1,\ldots, r-1). \]

Let us prove that \(\psi_i^0=\psi_i\) \((i=0,\,1,\ldots,r-1)\). By virtue of the known properties of the Sobolev classes \(W_2^{(r)}(\omega)\) [5], we have

\[ \sum_{i=0}^{r-1} \|\psi_i^0-\psi_i\|_{L_2(\omega^*)} \leq c\|u-f_\nu\|_{W_2^{(r)}(\omega)} \leq c_1\|u-f_\nu\|_{W^{(r)}_{2,\varphi}(R_n^+)} \to 0,\qquad \nu\to\infty . \tag{1.13} \]

Next, on the basis of Theorem \(1^\circ\), which must be applied to each of the functions
\[ \frac{\partial^i u}{\partial x_n^i}-\frac{\partial^i f_\nu}{\partial x_n^i} \qquad (i=0,1,\ldots,r-1), \]
replacing \(r\) by \(r-i\), we obtain
\[ \sum_{i=0}^{r-1}\|\psi_i^0-\psi_i\|_{L_{2,\varphi}^{\left(r-i-\frac12\right)}(R_{n-1})} \leq c\sqrt{D(u-f_\nu)}\to 0,\qquad \nu\to\infty . \tag{1.14} \]

From (1.13) and (1.14) it follows that
\[ \|\psi_i^0-\psi_i\|_{L_2(\omega^*)}=0,\qquad \|\psi_i^0-\psi_i\|_{L_{2,\varphi}^{\left(r-i-\frac12\right)}(R_{n-1})}=0, \]
whence it follows that
\[ \psi_i^0=\psi_i \qquad (i=0,1,\ldots,r-1), \]
on \(R_{n-1}\). Consequently, \(u\in m\).

It is proved in the usual way that the minimum of the variational problem is attained for \(u\), and that this function is unique (see, for example, [6]).

The function \(u\) for which the minimum of our variational problem is attained may also be defined as a function \(u\in m\) such that for it the equality holds (see, for example, [6])
\[ E(u,v)-(F,v)=0,\quad \text{for every } v\in m_0 . \tag{1.15} \]

A function \(u\) satisfying condition (1.15) is called a generalized solution of the boundary-value problem (8), (9).

Let us introduce a new class of functions \(m_{00}\). By definition, a function \(v\) belongs to the class \(m_{00}\) if it belongs to the class \(L_{2,\varphi}^{(r)}(R_n^+)\) and is identically equal to zero outside some closed bounded set \(G\subset R_n^+\), depending on it.

From (1.15) it follows that
\[ E(u,v)-(F,v)=0,\quad \text{for every } v\in m_{00}, \tag{1.16} \]
and since in § 3 it will be proved that the class \(m_{00}\) is everywhere dense in the class \(m_0\) in the sense of the metric of \(L_{2,\varphi}^{(r)}(R_n^+)\), it is easy to show that (1.15) follows from (1.16).

Consequently, the generalized solution of the boundary-value problem (8), (9) may be defined as a function \(u\in m\) for which (1.16) holds.

Since the generalized solution \(u\) of the boundary-value problem (8), (9) gives a minimum to the functional (1.8), it is unique.

Thus we obtain the following theorem.

Theorem \(4^\circ\). In the class \(m\) there exists, and moreover a unique, generalized solution of the boundary-value problem (8), (9).

§ 2. ESTIMATE OF THE GENERALIZED SOLUTION OF THE BOUNDARY-VALUE PROBLEM

In solving our boundary-value problem we proceeded from the given functions
\[ \psi_i\in L_{2,\varphi}^{\left(r-i-\frac12\right)}(R_{n-1}) \qquad (i=0,1,\ldots,r-1). \]

From these functions we defined a function \(\Phi\in L_{2,\varphi}^{(r)}(R_n^+)\) such that
\[ \left.\frac{\partial^i\Phi}{\partial x_n^i}\right|_{R_{n-1}} =\psi_i \qquad (i=0,1,\ldots,r-1), \]

\[ \sqrt{D(\Phi)}=\|\Phi\|_{L_{2,\varphi}^{(r)}(R_n^+)} \leq c\sum_{i=0}^{r-1}\|\psi_i\|_{L_{2,\varphi}^{\left(r-i-\frac12\right)}(R_{n-1})}. \tag{2.1} \]

The function \(\Phi\) defined the class \(m\), and at the same time it belonged to \(m\). Since the generalized solution \(u\) of the boundary-value problem (8), (9), \(\Phi\) and \(\frac12(u+\Phi)\) belong to \(m\), and \(u\) satisfies the minimal property, using the equalities (1.9) and (1.10), we obtain

\[ \begin{aligned} E(u-\Phi) &=2\bigl[E(u)-2(F,u)\bigr]+2\bigl[E(\Phi)-2(F,\Phi)\bigr]-\\ &\quad -4\left[E\left(\frac{\Phi+u}{2}\right)-2\left(F,\frac{\Phi+u}{2}\right)\right]\\ &\leq 2\bigl[E(\Phi)-2(F,\Phi)\bigr]-2\bigl[E(u)-2(F,u)\bigr]\\ &\leq 2\bigl[E(\Phi)-2(F,\Phi)\bigr]+\frac{4\sigma_F^2}{\lambda}+2E(\Phi)+4(F,\Phi)\\ &=4E(\Phi)+\frac{4\sigma_F^2}{\lambda}, \end{aligned} \tag{2.2} \]

where the constants \(\sigma_F\) and \(\lambda\) are defined above.

Taking (1.6) into account, we shall have

\[ \|u\|_{W_{2,\varphi}^{(r)}(R_n^+)}^2 \leq 2\{\|u\|_{L_2(\omega)}^2+D(u)\}\leq \]

\[ \leq 2\left\{\|u\|_{L_2(\omega)}^2+\frac1\lambda E(u)\right\}\leq \]

\[ \leq 4\left\{ \left[\|\Phi\|_{L_2(\omega)}^2+\frac1\lambda E(\Phi)\right] + \left[\|u-\Phi\|_{L_2(\omega)}^2+\frac1\lambda E(u-\Phi)\right] \right\}. \tag{2.3} \]

Since \(u-\Phi\in m_0\), applying the Poincaré inequality (for \(\omega\)) and (1.6), (2.2), we obtain

\[ \|u-\Phi\|_{L_2(\omega)}^2 \leq cD(u-\Phi)\leq \frac{c}{\lambda}E(u-\Phi) \leq \frac{4c}{\lambda}\left[E(\Phi)+\frac{\sigma_F^2}{\lambda}\right]. \tag{2.4} \]

Further, by virtue of (1.7),

\[ E(\Phi)\leq c\{\|\Phi\|_{L_2(\omega)}^2+D(\Phi)\}. \tag{2.5} \]

By virtue of the known properties of \(W_2^{(r)}(\omega)\), taking into account that \(\omega^*=\overline{\omega}\cap R_{n-1}\), and having in mind (2.1), we obtain

\[ \|\Phi\|_{L_2(\omega)} \leq c_1\left\{\sum_{i=0}^{r-1}\|\psi_i\|_{L_2(\omega^*)}+\sqrt{D_\omega(\Phi)}\right\}\leq \]

\[ \leq c_2\sum_{i=0}^{r-1}\left[ \|\psi_i\|_{L_2(\omega^*)} +\|\psi_i\|_{L_{2,\varphi}^{\left(r-i-\frac12\right)}(R_{n-1})} \right] = c_2\sum_{i=0}^{r-1}\|\psi_i\|_{W_{2,\varphi}^{\left(r-i-\frac12\right)}(R_{n-1})}. \tag{2.6} \]

From (2.5), (2.6) we have

\[ \sqrt{E(\Phi)} \leq c\sum_{i=0}^{r-1} \|\psi_i\|_{W_{2,\varphi}^{\left(r-i-\frac12\right)}(R_{n-1})}. \tag{2.7} \]

From (2.3), (2.4), (2.5), (2.6), and (2.7) we finally obtain

\[ \|u\|_{W^{(r)}_{2,\varphi}(R_n^+)} \leq c\left\{ \sum_{i=0}^{r-1} \|\psi_i\|_{W^{(r-i-\frac12)}_{2,\varphi}(R_{n-1})} +\sigma_F \right\}, \]

where \(c\) is a constant independent of the boundary functions \(\psi_i\) and of the quantity \(\sigma_F\).

Thus, we have obtained an estimate of the generalized solution \(u\), belonging to the class \(m\), of equation (1) in terms of the boundary functions \(\psi_i \in L^{(r-i-\frac12)}_{2,\varphi}(R_{n-1})\) \((i=0,1,\ldots,r-1)\) and \(F\).

§ 3. EXISTENCE AND UNIQUENESS OF THE CLASSICAL SOLUTION OF THE BOUNDARY-VALUE PROBLEM

A function \(u\) is called a classical solution of equation (1), given on \(R_n^+\), if it has continuous derivatives on \(R_n^+\) up to order \(2r\) inclusive and satisfies this equation. It is known that if \(s>n/2\), \(a_{kl}\in C^{(r+s)}(R_n^+)\), and \(F\in W_2^{(s)}\) locally on \(R_n^+\), then the generalized solution \(u\) of the boundary-value problem (8), (9) has continuous partial derivatives up to order \(2r\) inclusive on \(R_n^+\) (see, for example, [7—8]), i.e. the generalized solution \(u\) becomes a classical solution of equation (1), belonging to the class \(m\).

We shall show that in the class \(m\) there exists only one classical solution of equation (1). To prove this assertion we shall need the following theorem.

Theorem 5. The class \(m_{00}\) is everywhere dense in the class \(m_0\) in the sense of the metric

\[ D(f)=\int \sum_{|k|=r}\left|\frac{f^{(k)}(x)}{\varphi(\rho)}\right|^2\,dR_n^+ . \]

Proof. Let \(\delta>0\), and let \(R_\delta^+\) be the set of points \(x=(x_1,\ldots,x_n)\) with coordinate \(x_n>\delta\). Put

\[ \bar{\eta}(x)= \begin{cases} 0, & 0<x_n\leq 2\delta,\\ 1, & 2\delta<x_n . \end{cases} \]

Perform the \(\delta\)-averaging of this function in the sense of Sobolev; then we obtain an infinitely differentiable function \(\eta(x)\) on \(R_n^+\), not exceeding one, and such that

\[ \eta(x)=\eta_\delta(x)= \begin{cases} 0, & 0<x_n\leq \delta,\\ 1, & 3\delta<x_n, \end{cases} \]

having the property

\[ |\eta_\delta^{(k)}(x)|\leq \frac{c}{\delta^{|k|}}, \tag{3.1} \]

where \(c>0\) is a constant independent of \(x\) and \(\delta\), but depending on \(k\) (see [5]).

Let \(v\) be an arbitrary function from \(m_0\); then \(\eta(x)\cdot v(x)\) again belongs to \(m_0\) and has support in \(R_\delta^+\). We shall show that

\[ D(v-\eta\cdot v)\to 0,\qquad \delta\to 0. \tag{3.2} \]

Indeed, putting \(R_n^+ - R_\delta^+=\Omega_\delta\), we shall have

\[ D(v-\eta v)=D_{\Omega_{3\delta}}(v-\eta\cdot v)\leq 2D_{\Omega_{3\delta}}(v)+2D_{\Omega_{3\delta}}(\eta v). \tag{3.3} \]

Obviously,

\[ D_{\Omega_{3\delta}}(v)\to 0,\qquad \delta\to 0. \tag{3.4} \]

It remains to prove that

\[ D_{\Omega_{3\delta}}(\eta v)\to 0,\qquad \delta\to 0. \tag{3.5} \]

We have

\[ D_{\Omega_{3\delta}}(\eta v) = \sum_{|k|=r} \int \left| \frac{(\eta v)^{(k)}}{\varphi(\rho)} \right|^{2} \,d\Omega_{3\delta}. \]

But

\[ (\eta v)^{(k)} = \sum_{l\leq k} C_k^l \eta^{(l)}v^{(k-l)}, \]

where the sum is extended over all integral nonnegative vectors
\(l=(l_1,\ldots,l_n)\leq (k_1,\ldots,k_n)=k\), in other words, such that
\(l_i\leq k_i\) \((i=1,\ldots,n)\), and
\(C_k^l=C_{k_1}^{l_1}\cdots C_{k_n}^{l_n}\).
Using inequality (3.1), we obtain

\[ I= \int \left| \frac{\eta^{(l)}\cdot v^{(k-l)}}{\varphi(\rho)} \right|^{2} d\Omega_{3\delta} \leq \frac{c}{\delta^{2|l|}} \int \left| \frac{v^{(k-l)}}{\varphi(\rho)} \right|^{2} d\Omega_{3\delta}. \tag{3.6} \]

Expand the function \(v^{(k-l)}(x)\) by Taylor’s formula in a neighborhood of the point
\((x_1,\ldots,x_{n-1},0)\) in powers of \(x_n\); then, taking into account that
\(v\in m_0\), we shall have

\[ v^{(k-l)}(x) = \frac{1}{(|l|-1)!} \int_{0}^{x_n} (x_n-t)^{|l|-1} g(x_1,\ldots,x_{n-1},t)\,dt, \tag{3.7} \]

where

\[ g(x_1,\ldots,x_{n-1},t) = \frac{\partial^{|l|}v^{(k-l)}(x_1,\ldots,x_{n-1},t)} {\partial x_n^{|l|}}. \]

Thus, taking (3.6) and (3.7) into account, we obtain

\[ I^{1/2} \leq \frac{c}{\delta} \left\| \frac{1}{\varphi(\rho)} \int_{0}^{x_n} |g(x_1,\ldots,x_{n-1},t)|\,dt \right\|_{L_2(\Omega_{3\delta})}. \]

Next, making the substitution \(t=x_nu\) and applying the generalized Minkowski inequality, then again making the substitution \(x_nu=z\) and using the monotonicity of the function \(\varphi(t)\), we shall have

\[ I^{1/2} \leq c\int_{0}^{1} \left\| \frac{g(x_1,\ldots,x_{n-1},x_nu)} {\varphi(\rho)} \right\|_{L_2(\Omega_{3\delta})} \,du = \]

\[ = c\int_{0}^{1} \frac{du}{u^{1/2}} \left\{ \int\!\!\int_{0}^{3\delta u} \left| \frac{ g(x_1,\ldots,x_{n-1},z) }{ \varphi\!\left(\sqrt{x_1^2+\cdots+x_{n-1}^2+\left(\frac{z}{u}\right)^2}\right) } \right|^{2} \,dz\,dR_{n-1} \right\}^{1/2} \leq \]

\[ \leq c_1 \left\{ \int\!\!\int_{0}^{3\delta} \left| \frac{g(x_1,\ldots,x_{n-1},z)} {\varphi(\rho)} \right|^{2} \,dz\,dR_{n-1} \right\}^{1/2} \to 0,\qquad \delta\to 0. \tag{3.8} \]

It follows from (3.8) that (3.5) is valid, and consequently, taking into account (3.4) and (3.3), (3.2) is also valid.

Let

\[ \bar\mu(x)=\bar\mu_N(x)= \begin{cases} 1, & \rho \leqslant 2N,\\ 0, & \rho>2N \end{cases} \qquad \left(\rho=\sqrt{x_1^2+\cdots+x_n^2}\right) \]

and let the function \(\mu(x)\) be the \(N\)-averaging, in Sobolev’s sense, of \(\bar\mu(x)\); hence \(\mu(x)\) is an infinitely differentiable function satisfying the conditions

\[ \mu(x)=\mu_N(x)= \begin{cases} 1, & \rho \leqslant N,\\ 0, & \rho>3N \end{cases} \]

and

\[ \left|\mu^{(k)}(x)\right|\leqslant \frac{c}{N^{|k|}}, \]

where \(c>0\) depends only on \(k\).

Let now \(v\in m_0\). We shall prove that

\[ D(v-\mu v)\to 0,\qquad N\to\infty . \tag{3.9} \]

Indeed, denoting by \(\omega_N\) the intersection of \(R_n^+\) with the ball centered at the origin of radius \(N\), as \(N\to\infty\) we obtain

\[ D(v-\mu v)=D_{R_n^+-\omega_N}(v-\mu v)\leqslant \]

\[ \leqslant 2D_{R_n^+-\omega_N}(v)+2D_{R_n^+-\omega_N}(\mu v) =2D_{R_n^+-\omega_N}(\mu v)+o(1)= \]

\[ =2\sum_{|k|=r}\int_{\omega_{3N}-\omega_N} \left|(\mu v)^{(k)}\cdot \frac{1}{\varphi(\rho)}\right|^2 dx+o(1). \tag{3.10} \]

Further, putting \(\omega_{3N}-\omega_N=\sigma_N\), we shall have

\[ \sum_{|k|=r}\int \left|(\mu v)^{(k)}\cdot \frac{1}{\varphi(\rho)}\right|^2\,d\sigma_N \leqslant c\sum_{|k|=r}\sum_{l\leq k}\int \left|\mu^{(k-l)}\right|^2\cdot \left|\frac{v^{(l)}}{\varphi(\rho)}\right|^2\,d\sigma_N \leqslant \]

\[ \leqslant c_1\sum_{|k|=r}\sum_{l\leq k} \frac{1}{N^{2|k-l|}} \int \left|v^{(l)}\cdot \frac{1}{\varphi(\rho)}\right|^2\,d\sigma_N \leqslant \]

\[ \leqslant c_1\sum_{|k|=r}\sum_{l\leq k} \frac{(3N)^{2|k-l|}}{N^{2|k-l|}} \int \left|\frac{v^{(l)}}{\varphi(\rho)}\cdot \frac{1}{\rho^{r-|l|}}\right|^2 \,d\sigma_N \leqslant \]

\[ \leqslant c_2\sum_{|k|=r}\sum_{l\leq k} \int_{R_n^+-\omega_N} \left|\frac{v^{(l)}}{(1+\rho)^{r-|l|}\varphi(\rho)}\right|^2 dx \to 0,\qquad N\to\infty, \tag{3.11} \]

since, according to Theorem \(3^\circ\), we have

\[ \int \left|\frac{v^{(l)}}{(1+\rho)^{r-|l|}\varphi(\rho)}\right|^2 dR_n^+ \leqslant c\|v\|^2_{W^{(r)}_{2,\varphi}(R_n^+)}<\infty . \]

From (3.10) and (3.11) we obtain (3.9).

Again let an arbitrary function \(v\in m_0\) and \(\varepsilon>0\) be given. It follows from (3.2) that, for sufficiently small \(\delta>0\),

\[ \sqrt{D(v-\eta_\delta v)}<\varepsilon, \tag{3.12} \]

and since \(\eta_\delta\cdot v \in m_0\), by virtue of (3.9) one can choose a sufficiently large \(N\), depending on \(\delta\), such that

\[ \sqrt{D(\eta_\delta v-\mu_N\eta_\delta v)}<\varepsilon. \tag{3.13} \]

From (3.12) and (3.13) it follows that

\[ \sqrt{D(v-\mu_N\eta_\delta v)}<2\varepsilon, \]

i.e., we have proved our theorem, since it is obvious that \(\mu_N\eta_\delta v \in m_{00}\).

Suppose that the coefficients \(a_{kl}(x)\) of equation (1) are \(r\) times continuously differentiable on \(R_n^+\), and that in the class \(m\) there exist two classical solutions \(u\) and \(u_1\) of equation (1). Then \(z=u-u_1 \in m_0\) and \(L(z)=0\), \(x_n>0\). Let, further, \(v\) be an arbitrary function of class \(m_0\). Choose a sequence of functions \(v_n \in m_{00}\) such that

\[ D(v-v_n)\to 0,\quad n\to\infty, \]

which is possible according to Theorem \(4^\circ\). By integration by parts we obtain the equality

\[ E(z,v_n)=\int v_n L(z)\,dR_n^+=0, \]

which is possible since \(z\) is continuously differentiable \(2r\) times, while \(v_n\) has compact support strictly inside \(R_n^+\). Therefore

\[ |E(z,v)|=|E(z,v-v_n)|\leq \sqrt{E(z)}\sqrt{E(v-v_n)}. \tag{3.14} \]

Since \(u,u_1\in m\subset L_{2,\varphi}^{(r)}(R_n^+)\), we also have \(z\in L_{2,\varphi}^{(r)}(R_n^+)\) and \(E(z)<\infty\). Further, since \(v-v_n\in m_0\), using (1.7) and the Poincaré inequality for \(W_2^{(r)}(\omega)\), we obtain

\[ E(v-v_n)\leq c\{\|v-v_n\|_{L_2(\omega)}^2+D(v-v_n)\}\leq \]

\[ \leq c_1D(v-v_n)\to 0,\quad n\to\infty. \]

Thus, the right-hand side of inequality (3.14) tends to zero as \(n\to\infty\), i.e.

\[ E(z,v)=0. \]

The last equality is true for all \(v\in m_0\); therefore \(z\) gives a minimum of the variational problem

\[ \min_{f\in m_0}E(f). \]

But this problem evidently attains its minimum only for \(f=0\); consequently,

\[ z=u-u_1=0, \]

i.e.

\[ u=u_1. \]

Thus, if the corresponding smoothness conditions indicated above are imposed on the coefficients \(a_{kl}(x)\) and \(F(x)\) of equation (1), then we obtain the following theorem.

Theorem \(5^\circ\). In the class \(m\) there exists, and moreover uniquely, a classical solution of equation (1).

References

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5. Sobolev S. L. Some applications of functional analysis in mathematical physics. L., 1950.
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Received by the editors
October 18, 1966

Moscow Institute of Physics and Technology