V. R. PORTNOV

TWO EMBEDDING THEOREMS FOR THE SPACE \(L_{p,b}^{(1)}(\Omega \times R_+)\) AND THEIR APPLICATIONS

(Presented by Academician S. L. Sobolev on 14 XII 1963)

Let us denote: \(E_n\) is the \(n\)-dimensional Euclidean space of points \(x=(x_1,\ldots,\ldots,x_n)\); \(E_{n-1}\) is the hyperplane \(x_n=0\); \(E_n^+\) is the half-space \(x_n>0\); \(R_+\) is the set of positive real numbers; \(p=(p_1,\ldots,p_n)\) is a vector for which \(1<p_i<\infty\) \((i=1,\ldots,n)\); \(x'=(x_1,\ldots,x_{n-1})\) is a vector of the set \(E_{n-1}\).

Let \(\Omega\) be an \((n-1)\)-dimensional open set situated in \(E_{n-1}\) and having the property that for any pair of open balls \(K_1\) and \(K_2\) lying in \(\Omega\), there exists a finite number of open balls \(K^{(1)}, K^{(2)},\ldots,\ldots,K^{(m)}\) \((m\ge 2)\), also lying in \(\Omega\), such that \(K^{(1)}=K_1\), \(K^m=K_2\), and \(K^{(i)}\cap K^{(i+1)}\) is a nonempty set for all \(i=1,\ldots,m-1\). By \(\Omega \times R_+\) we denote the Cartesian product of the sets \(\Omega\) and \(R_+\). Suppose that a vector \(b(x)=(b_1(x),\ldots,b_n(x))\) is given, whose components are defined on \(\Omega \times R_+\), are measurable, positive almost everywhere, and satisfy the conditions:

1. There exists a number \(R_0\ge 0\) such that for almost all \(x'\in\Omega\)

\[ \int_{R_0}^{\infty} b_n^{1/(1-p_n)}(x',x_n)\,dx_n<\infty . \tag{1} \]

1. There exists a nonnegative measurable function \(\varphi(x_n)\), defined on \((R_0,\infty)\), for which \(\int_N^\infty \varphi(x_n)\,dx_n=\infty\) for every \(N\ge R_0\), and such that

\[ \int_K dx'\left[\int_{R_0}^{\infty}\varphi^{p_i'}(x_n)b_i^{1/(1-p_i)}(x',x_n)\,dx_n\right]<\infty, \tag{2} \]

where \(R_0\) is the number from condition 1; \(K\) is an arbitrary open ball whose closure \(\overline K\subset\Omega\), \(1\le i\le n-1\).

Denote by \(L_{p,b}^{(1)}(\Omega \times R_+)\) the set of functions for which the inequality

\[ \sum_{i=1}^{n}\left[\int_{\Omega\times R_+} b_i(x)\left|\frac{\partial u}{\partial x_i}\right|^{p_i}dx\right]^{1/p_i}<\infty, \tag{3} \]

holds, where \(\partial u/\partial x_i\) \((i=1,\ldots,n)\) is the generalized derivative in the sense of S. L. Sobolev and \(b_i(x)\ge 0\).

The following lemma is valid; it is a generalization, to spaces with a weight, of a result of S. L. Sobolev established in \((^1)\).

Lemma 1. For every function \(u(x)\in L_{p,b}^{(1)}(\Omega\times R_+)\), where \(b^{(i)}(x)\) satisfies conditions (1) and (2), there exists a constant \(C(u)\) such that for

for almost all \(x'\in\Omega\) the relation holds:
\[ u(x',x_n)=C(u)-\int_{x_n}^{\infty}\frac{\partial u}{\partial x_n}(x',t)\,dt, \tag{4} \]
where
\[ \int_{x_n}^{\infty}\frac{\partial u}{\partial x_n}(x',t)\,dt \]
converges absolutely.

Lemma 2. Let \((A,C)\) be an interval of the real line and \(-\infty\leq A<C\leq +\infty\). Let \(f(x)\) and \(b_0(x)\) be nonnegative measurable functions, with
\[ \int_A^x b_0^{1/(1-p)}(t)\,dt<\infty \]
for all \(x\in(A,C)\). Then the inequality
\[ \left\{ \int_A^C b_0(x) \left[ \frac{B_0(x)} {\left(\int_A^x B_0(t)\,dt\right)^{\frac1p\left(1+\frac{q}{p'}\right)}} \right]^p \left[\int_A^x f(t)\,dt\right]^q dx \right\}^{1/q} \leq C_0\left[\int_A^C b_0(x)f^p(x)\,dx\right]^{1/p}, \tag{5} \]
holds, where \(B_0(x)=b_0^{1/(1-p)}(x)\), \(1<p\leq q<\infty\),
\[ C_0=q^{-1/q}(p')^{1/q}\left(1+\frac{q}{p'}\right)^{1/q} \left(1+\frac{p'}{q}\right)^{1/p'}. \]

Remark. If
\[ \int_x^C B_0(t)\,dt<\infty \]
for all \(x\in(A,C)\), then with the same constant the inequality
\[ \left\{ \int_A^C b_0(x) \left[ \frac{B_0(x)} {\left(\int_x^C B_0(t)\,dt\right)^{\frac1p\left(1+\frac{q}{p'}\right)}} \right]^p \left[\int_x^C f(t)\,dt\right]^q dx \right\}^{1/q} \leq C_0\left[\int_A^C b_0(x)f^p(x)\,dx\right]^{1/p} \tag{6} \]
holds.

As a consequence of Lemmas 1 and 2, for \(A=0\), \(C=\infty\), and \(p=q=p_n\), we have the following embedding theorem.

Theorem 1. For every function \(u(x)\in L_{p,b}^{(1)}(\Omega\times R_+)\), where \(b(x)\) satisfies conditions (1) and (2), there exists a unique constant \(C(u)\) such that, with a constant \(C_1\) independent of \(u(x)\), the inequality
\[ \int_{\Omega\times R_+} b_n(x) \left[ \frac{b_n^{1/(1-p_n)}(x)} {\int_{x_n}^{\infty} b_n^{1/(1-p_n)}(x',t)\,dt} \right]^{p_n} |u(x)-C|^{p_n}\,dx \leq C_1 \int_{\Omega\times R_+} b_n(x)\left|\frac{\partial u}{\partial x_n}\right|^{p_n}dx. \tag{7} \]

The following theorem is valid on boundary values of functions from the space \(L_{p,b}^{(1)}(E_n^+)\) on the hyperplane \(E_{n-1}\), without any restriction on the behavior of the weights \(b_i(x)\) at infinity.

Theorem 2. Let \(\Omega=E_{n-1}\), \(b_i(x)=b_i(x_n)\) \((i=1,\ldots,n)\),
\[ \int_0^a b_n^{-1}(t)\,dt<\infty \]
for any \(a<\infty\), and \(p_1=\cdots=p_n=p\). Then for every function \(u(x)\in L_{p,b}^{(1)}(E_n^+)\):

1. \(\lim_{x_n\to 0}u(x',x_n)=u(x',0)\) exists and is finite for almost all
   \(x'\in E_{n-1}\).

1. The inequality
   \[ \int_0^\infty dh\, g(h)\left[\int_{E_{n-1}} \left|u(x_1,\ldots,x_i+h,\ldots,0)-u(x_1,\ldots,x_i,\ldots,0)\right|^p dx'\right]\le \]
   \[ \le C_2\left[\int_{E_n^+} b_n(x_n)\left|\frac{\partial u}{\partial x_n}\right|^p dx +\int_{E_n^+} b_i(x_n)\left|\frac{\partial u}{\partial x_i}\right|^p dx\right], \tag{8} \]
   where
   \[ g(h)=\min\left\{ b_n(h)\left[\frac{b_n^{1/(1-p)}(h)} {\displaystyle \int_0^h b_n^{1/(1-p)}(t)\,dt}\right]^p,\; b_i(h)h^{-p} \right\}, \]
   and \(C_2\) is a constant independent of the function \(u(x)\).

We shall indicate some applications of the results listed above to the theory of degenerate elliptic equations of the second order in the half-space \(E_n^+\).

Consider the equation
\[ Lu\equiv -\sum_{i,j=1}^n \frac{\partial}{\partial x_i} b_{ij}(x)\frac{\partial u}{\partial x_j} +\sum_{i=1}^n a_i(x)\frac{\partial u}{\partial x_i} +c(x)u=f(x). \tag{9} \]

The solution of equation (9) is sought in the half-space \(E_n^+\) under the condition
\[ \lim_{x_n\to 0} u(x',x_n)=0 \quad \text{a.e. on } E_{n-1}. \tag{10} \]

Let \(\sigma(x)\ge 0\) a.e. on \(E_n^+\). We shall say that \(u(x)\in \overset{0}{L}{}_{2,b,\sigma}^{(1)}(E_n^+)\) if
\[ \|u\|_{\overset{0}{L}{}_{2,b,\sigma}^{(1)}}^2 = \|u\|_{L_{2,b}^{(1)}(E_n^+)}^2 + \|u\|_{L_{2,\sigma}(E_n^+)}^2 <\infty \]
and condition (10) is satisfied for \(u(x)\).

We impose on \(b(x)\) and \(\sigma(x)\) the following restrictions:

a) on every ball \(K\) whose closure \(\overline K\subset E_n^+\), for all \(i=1,\ldots,n\):
\[ 0<\varepsilon_1(K)\le b_i(x)\le \varepsilon_2(K) \quad\text{and}\quad \sigma(x)\le \varepsilon_3(K) \]
(\(\varepsilon_1,\varepsilon_2,\varepsilon_3\) are constants);

b) there exist numbers \(N_0>\varepsilon_0>0\) such that for \(x_n<\varepsilon_0\) and \(x_n>N_0\), \(b_n(x)\) depends only on \(x_n\), and the conditions
\[ \int_0^{\varepsilon_0} b_n^{-1}(x',t)\,dt<\infty,\qquad \int_{N_0}^{\infty} b_n^{-1}(x',t)\,dt=\infty \]
are satisfied;

c) for every pair of numbers \(N>\varepsilon>0\) there is a number \(R_0(\varepsilon,N)\ge 0\) such that, for \(\varepsilon<x_n<N\), for all \(x\) such that \(|x'|>R_0\), we have \(b_i(x)=\widetilde b(|x'|)\) for each \(i<n\) \(\left(|x'|^2=\sum_{i=1}^{n-1}x_i^2\right)\).

Introduce the function
\[ \beta(x)=\sigma(x)+\frac14 b_n^{-1}(x) \left[\int_0^{x_n} b_n^{-1}(x',t)\,dt\right]^{-2}. \]
We impose on the coefficients of equation (9) the following conditions:

1) the functions \(b_{ij}(x)\) and \(a_i(x)\) are continuously differentiable on every ball \(K\) whose closure \(\overline K\subset E_n^+\);

2)
\[ \sum_{i,j=1}^n b_{ij}(x)t_i t_j \ge \sum_{i=1}^n b_i(x)t_i^2 \quad\text{for every } t=(t_1,\ldots,t_n); \]

3) there exists a \(\gamma<1\) such that a.e. on \(E_n^+\)
\[ c(x)-\sigma(x)-\frac12\sum_{i=1}^n \frac{\partial a_i}{\partial x_i} \ge -\gamma \beta(x); \]

4) there exists a constant \(\tilde c \ge 0\) such that a.e. on \(E_n^+\):

a) \(|b_{ij}(x)|^2 \le \tilde c b_i(x)b_j(x)\) \((i,j=1,\ldots,n)\);

b) \(|a_i(x)|^2 \le \tilde c b_i(x)\cdot \beta(x)\) \((i=1,\ldots,n)\);

c) \(|c(x)| \le \tilde c\beta(x)\);

5)
\[ \int_{E_n^+}\beta^{-1}(x)f^2(x)\,dx<\infty. \]

Construct the bilinear functional
\[ A(u,v)=\int_{E_n^+}\left[\sum_{i,j=1}^{n} b_{ij}(x)\frac{\partial u}{\partial x_j}\frac{\partial v}{\partial x_i} +\sum_{i=1}^{n}a_i(x)v\frac{\partial u}{\partial x_i} +c(x)uv\right]\,dx, \]
which, by condition 4), is bounded in the space \(\overset{0}{L}{}^{(1)}_{2,b,\sigma}(E_n^+)\). Therefore there exists a linear operator \(K\) \((\|K\|=\|A\|<\infty)\) such that, identically in \(u\) and \(v\) from \(\overset{0}{L}{}^{(1)}_{2,b,\sigma}(E_n^+)\), the relation
\[ A(u,v)=(Ku,v) \]
holds. From the condition
\[ \int_{E_n^+}\beta^{-1}f^2\,dx<\infty \]
there follows the existence of a function \(\tilde f(x)\in \overset{0}{L}{}^{(1)}_{2,b,\sigma}(E_n^+)\) such that, identically in \(v\in \overset{0}{L}{}^{(1)}_{2,b,\sigma}(E_n^+)\), the relation
\[ \int_{E_n^+} f(x)v(x)\,dx=(\tilde f,v) \]
is satisfied.

Theorem 3. If the coefficients and the right-hand side of equation (9) satisfy the five conditions listed above, then in the space \(\overset{0}{L}{}^{(1)}_{2,b,\sigma}(E_n^+)\) equation (9) has, moreover, a unique generalized solution; furthermore, the problem of finding the generalized solution of the equation is equivalent to the problem of finding a function minimizing the quadratic functional
\[ \mathfrak{M}_f(u)=A(u,Ku)-2A(u,\tilde f) =A(u,Ku)-2\int_{E_n^+} f(x)Ku\,dx. \]

The first boundary-value problem for a self-adjoint elliptic equation of second order in a half-space with coefficients having power-order decay at infinity was first solved by L. D. Kudryavtsev in \((^2)\).

Institute of Mathematics with Computing Center
Siberian Branch of the Academy of Sciences of the USSR

Received
4 XII 1963

CITED LITERATURE

\(^1\) S. L. Sobolev, Siberian Mathematical Journal, 4, No. 3 (1963). \(^2\) L. D. Kudryavtsev, Materials for the Joint Soviet-American Symposium on Partial Differential Equations, August, 1963, Novosibirsk.