V. R. Portnov

A Theorem on the Density of Finite Functions in Weighted Classes

(Presented by Academician S. L. Sobolev on 9 VII 1964)

Theorems on the density of the set of finite functions in spaces without weight were first proved by S. L. Sobolev in \((^1,{}^2)\). In \((^4)\) an analogous result was established for spaces with weight, when \(\Omega = E_n\). In the present article more general results are obtained both for bounded and for unbounded domains having a simple boundary in the sense of S. L. Sobolev. Applications are given to the solution of one nonlinear system of differential equations.

1. Let \(\Omega\) be a domain in the space \(E_n\) of points \(x=(x_1,x_2,\ldots,x_n)\), and suppose that the following conditions are fulfilled:

2. \(\Omega=\Omega_1\times\cdots\times\Omega_{P_0}\) \((1\le P_0\le 2)\), where \(\Omega_k\) \((k=1,2,\ldots,P_0)\) is a domain in the space \(E_{q_k}\) of the variables \(\widetilde{x}_k=(x_{n_{k-1}+1},\ldots,x_{n_k})\) \((0=n_0<n_1<\cdots<n_{P_0}=n,\ q_k=n_k-n_{k-1})\), whose boundary \(S_k\) is bounded in \(E_{q_k}\) and can be divided into a finite number of smooth manifolds \(S_k^{(1)},\ldots,S_k^{(A_k)}\) of dimensions \(q_k-s_k^{(1)},\ldots,q_k-s_k^{(A_k)}\), respectively. Here and below we introduce the notation:
   \[ \overline{x}_k=(x_1,\ldots,x_{n_{k-1}},x_{n_k+1},\ldots,x_n). \]

3. Let \(T_{\rho_{kj}}\) be the set of those points in \(\Omega_k\) which lie on the normals to \(S_k^{(j)}\) \((j=1,\ldots,A_k)\) planes of dimension \(s_k^{(j)}\) at distances \(<\rho_{kj}\). Introduce in the space of the variables \(y_1,y_2,\ldots,y_{q_k}\) cylindrical coordinates with center on the axis
   \[ y_1=y_2=\cdots=y_{s_k^{(j)}}=0. \]
   By \(\widetilde{\varphi}_{kj}\) and \(\widetilde{y}_{kj}\) we denote, respectively, the systems of angular coordinates and coordinates of the axis thereby obtained; put
   \[ \rho_{kj}^2=y_1^2+\cdots+y_{s_k^{(j)}}^2. \]
   The set \(T_{\rho_{kj}}\), starting with a sufficiently small \(\rho_{kj}^{(0)}>0\), is mapped by means of a smooth nondegenerate transformation with derivatives bounded up to order \(m_0\) onto a cylinder of the form \((0,\rho_{kj})\times\Delta_{kj}\), where \(\Delta_{kj}\) is a domain in the variables \((\widetilde{\varphi}_{kj},\widetilde{y}_{kj})\).

Consider the set of functions \(u(x)\) for which, for each
\(k=1,2,\ldots,P_0\), there exist derivatives
\[ D_{\widetilde{x}_k}^{\alpha}\left[D_{\overline{x}_k}^{\beta^{(l,k)}}u(x)\right] \quad (l=1,\ldots,\lambda_k;\ 0\le|\alpha|\le m^{(l,k)}) \]
in the sense of S. L. Sobolev. Let us write these derivatives, obtained for all \(k,l\), and \(\alpha\), in the form of the sequence
\[ D^{\alpha(1)}u,\ D^{\alpha(2)}u,\ \ldots,\ D^{\alpha(Q)}u,\ \ldots,\ D^{\alpha(R)}u \tag{1} \]
and suppose that
\[ g^p(u)=\int_{\Omega}\sum_{\gamma=1}^{Q} b_{\gamma}(x)\,\left|D^{\alpha(\gamma)}u\right|^p\,dx<\infty \qquad (1<p<\infty) \tag{2} \]
and, for every \(k=1,2,\ldots,P_0\),
\[ g^p(u)=\sum_{l=1}^{\lambda_k}\int_{\Omega} b^{(l,k)}(\widetilde{x}_k)\,a^{(l,k)}(\overline{x}_k) \sum_{|\alpha|=m^{(l,k)}} \left|D_{\widetilde{x}_k}^{\alpha}\left[D_{\overline{x}_k}^{\beta^{(l,k)}}u(x)\right]\right|^p\,dx, \tag{3} \]

where \(b_\gamma(x)\), \(b^{(l,k)}(\tilde x_k)\), \(a^{(l,k)}(\bar x_k)>0\) are continuous functions in \(\Omega\), and, if \(1\leq j\leq A_k\) and \(j'\) is such that \(s_k(j')>s_k(j)\), then for any \(\rho_{kj'}>0\) there is a \(\rho_{kj}^{(0)}>0\) such that on
\[ T_{\rho_{kj}^{(0)}}\setminus \bigcup_{s_k(j')>s_k(j)}T_{\rho_{kj'}} \]
with certain constants \(c_1(\rho_{kj'})>0\) and \(c_2(\rho_{kj'})>0\) the inequality
\[ c_1 b^{(l,k)}(x)\leq \rho_{kj}^{\gamma(l,k,j)}\Lambda_{lkj}(\tilde y_{kj},\tilde\varphi_{kj}) \leq c_2 b^{(l,k)}(x) \]
is satisfied
\[ (k=1,2,\ldots,P_0;\ l=1,2,\ldots,\lambda_k). \]

If \(\Omega_k\) is unbounded, then outside a ball of sufficiently large radius
\[ 0<\varepsilon_1\leq b^{(l,k)}(\tilde x_k)|\tilde x_k|^{-\gamma(l,k)}\leq \varepsilon_2<\infty . \]
The resulting set of functions will be denoted by \(L^{(\alpha)}_{p,\mathbf b}(\Omega)\) \((\mathbf b=(b_1,\ldots,b_Q),\ \vec\alpha=(\alpha^{(1)},\ldots,\alpha^{(R)}))\).

Let
\[ \tilde\Phi_{lkj}=m^{(l,k)}-1-\bigl[(\gamma^{(l,k,j)}+s_k(j)):p\bigr] \]
for \(j=1,\ldots,A_k\);
\[ \tilde\Phi_{lk}=m^{(l,k)}-(\gamma^{(l,k)}+q_k):p, \]
if there exists an \(r\) \((1\leq r\leq m^{(l,k)})\) such that
\[ \gamma^{(l,k)}-rp+1=0; \]
otherwise
\[ \tilde\Phi_{lk}=m^{(l,k)}-1-\bigl[(\gamma^{(l,k)}+q_k):p\bigr]. \]
Put
\[ \Phi_{lkj}=\max\bigl(-1,\min(m^{(l,k)}-1,\tilde\Phi_{lkj})\bigr),\quad \Phi_{lk}=\max\bigl(-1,\min(m^{(l,k)}-1,\tilde\Phi_{lk})\bigr). \]

Definition. We shall say that \(u(x)\in \overset{0}{L}{}^{(\alpha)}_{p,\mathbf b}(\Omega)\) if:

1) \(u(x)\in L^{(\alpha)}_{p,\mathbf b}(\Omega)\);

2)
\[ \lim_{\rho_{kj}\to 0}D_{\tilde x_k}^{\alpha} \left[D_{\tilde x_k}^{\beta(l,k)}u(\rho_{kj}\tilde y_{kj},\tilde\varphi_{kj},\bar x_k)\right]=0 \quad (0\leq|\alpha|\leq\Phi_{lkj}) \]
for almost all \((\tilde y_{kj},\tilde\varphi_{kj},\bar x_k)\), and in the case when \(\Omega_k\) is unbounded;

3)
\[ \lim_{|\tilde x_k|\to\infty}D_{\tilde x_k}^{\alpha} \left[D_{\tilde x_k}^{\beta(l,k)}u(|\tilde x_k|,\tilde\theta_k,\tilde x_k)\right]=0 \quad (\Phi_{lk}+1\leq|\alpha|\leq m^{(l,k)}-1) \]
for almost all \((\tilde\theta_k,\bar x_k)\), where \(\tilde\theta_k\) is a set of angular spherical coordinates in \(E_{q_k}\)
\[ (k=1,\ldots,P_0;\ l=1,\ldots,\lambda_k;\ j=1,\ldots,A_k). \]

Theorem 1. Suppose that for all \(k=1,2,\ldots,P_0\), \(l=1,\ldots,\lambda_k\) at least one of the following conditions is satisfied:

1. There exists a \(j(l)\) such that \(\Phi_{lkj}=m^{(l,k)}-1\).

2. \(\Omega_k\) is unbounded and \(1+\Phi_{lk}=0\).

3. \(\Omega_k\) is bounded and there exists a \(j(l)\) such that \(\Phi_{lkj}\geq\Phi_{lk}\).

Then \(C_0^{(\infty)}(\Omega)\) is dense in \(\overset{0}{L}{}^{(\alpha)}_{p,\mathbf b}(\Omega)\) in the seminorm \(g(u)\).

II. Consider the system of equations
\[ L_i(u)=\sum_{\gamma=1}^{R_i}(-1)^{|\alpha^{(i\gamma)}|} D^{\alpha^{(i,\gamma)}}\varphi_{i\gamma}\bigl(x,D^{\alpha(p,q)}u_p(x)\bigr)=f_i(x) \tag{4} \]
\[ (i;\ p=1,2,\ldots,s;\ \gamma=1,2,\ldots,R_i;\ q=1,2,\ldots,R_p; \ \text{the functions } \varphi_{i\gamma}(x,y_{pq}^{(i\gamma)}) \]
\[ (-\infty<y_{pq}^{(i,\gamma)}<\infty) \]
depend on \(R=1+\sum_{i=1}^{s}R_i\) arguments and are continuously differentiable with respect to all of them, \(u(x)=(u_1(x),\ldots,u_s(x))\) is an unknown vector-function. By \(\tilde H,\ H\) we denote the sets of vector-functions for which
\[ u_i(x)\in L^{(\vec\alpha_i)}_{2,\mathbf b_i}(\Omega),\quad \overset{0}{L}{}^{(\vec\alpha_i)}_{2,\mathbf b_i}(\Omega) \]
respectively
\[ (i=1,2,\ldots,s,\quad \mathbf b_i(x)=(b_{i1}(x),\ldots,b_{iQ_i}(x)),\quad \vec\alpha_i=(\alpha^{(i,1)},\ldots,\alpha^{(i,R_i)})). \]

In what follows, everywhere, the addition of the index \(i\) to notation encountered earlier will mean that it refers to the space of the \(i\)-th component of the vector \(u(x)\). Below we shall assume that for any \(i=1,\ldots,s\):

a) for some pair \((l_0^{(i)},k_0^{(i)})\)
\[ |\beta^{(l_0^{(i)},k_0^{(i)})}|=0 \]
and b) the condition of Theorem 1 is satisfied.

Let \(\beta_{i\gamma}(x)>0\) be such that
\[ \int_{\Omega}\beta_{i\gamma}(x)\left|D^{\alpha(i,\gamma)}u_i(x)\right|^2\,dx\leq g_i^2(u_i)\leq \]
\[ \leq \int_{\Omega}\sum_{\gamma=1}^{Q_i} b_{i\gamma}(x)\left|D^{\alpha(i,\gamma)}u_i(x)\right|^2\,dx \]
for \(u\in H\), and for \(1\leq\gamma\leq Q_i\) put \(\beta_{i\gamma}(x)=b_{i\gamma}(x)\). The functions \(\beta_{i\gamma}\) can also be obtained starting from the zero boundary conditions for the functions \(u_i(x)\) and their derivatives and the known Hardy inequality ((5), p. 296). Here we note only that the indicated functions \(\beta_{i\gamma}(x)>0\) in \(\Omega\) exist for all \(i=1,2,\ldots,s\) and \(\gamma=1,2,\ldots,R_i\).

Denote by \(H_0\) the set of vector-functions all of whose components belong to \(C_0^{(m_0)}(\Omega)\), where \(m_0\geq \max_{i\gamma}|\alpha^{(i\gamma)}|\). By \(\hat H\) we denote the set of those \(u(x)\in \widetilde H\) for which
\[ \int_{\Omega}\beta_{i\gamma}^{-1}(x)\left|\varphi_{i\gamma}\left(x,D^{\alpha(p,q)}u_p(x)\right)\right|^2\,dx<\infty. \]

Suppose that the following conditions are satisfied:

в)
\[ \sup_{y_{\rho\sigma}^{(i\gamma)}}\left|\frac{\partial\varphi_{i\gamma}}{\partial y_{pq}^{(i,\gamma)}}\left(x,y_{\rho\sigma}^{(i,\gamma)}\right)\right| \leq \widetilde c\,\beta_{i\gamma}(x)\beta_{pq}(x) \quad(1\leq \rho\leq s;\;1\leq\sigma\leq R_\rho); \]

г) there exists a matrix \(M(x)=\|m_{\theta i}(x)\|\) \((1\leq \theta,i\leq s)\), for which \(m_{\theta i}\in C^{(m_0)}(\Omega)\) \((\theta,i=1,2,\ldots,s)\) and \(|M(x)|\neq 0\) in \(\Omega\); moreover, if we denote
\[ v_i=\sum_{\theta=1}^{s}m_{\theta i}u_\theta(x)\quad (u(x)\in H_0), \]
and by \(\mu_{\rho\sigma}^{(i,\gamma)}(x)\) the functions in the relation
\[ D^{\alpha(i,\gamma)}v_i=\sum_{\rho\sigma}\mu_{\rho\sigma}^{(i,\gamma)}\cdot D^{\alpha(\rho,\sigma)}u_\rho, \]
which we shall regard as satisfied, then for all \(i,\gamma,\rho,\sigma\)
\[ \left|\mu_{\rho\sigma}^{(i,\gamma)}(x)\right|\beta_{i\gamma}(x)\leq \widetilde c\,\beta_{\rho\sigma}(x); \]

д) there exists a system of functions \(\{\Delta_{pq\rho\sigma}(x)\}\) such that
\[ \sum_{pq\sigma}\int_{\Omega}\Delta_{pq\rho\sigma}(x)\times \]
\[ \times D^{\alpha(p,q)}u_pD^{\alpha(p,\sigma)}u_\rho\,dx=0 \]
for every \(u\in H_0\), and the functions
\[ \tau_{pq\rho\sigma}= \sum_{i\gamma}\frac{\partial\varphi_{i\gamma}}{\partial y_{pq}^{(i,\gamma)}}\left(x,y_{\delta\omega}^{(i,\gamma)}\right)\left(y_{\rho\sigma}^{(i,\gamma)}(x)+\Delta_{pq\rho\sigma}\right) \quad(\delta=1,\ldots,s;\;\omega=1,\ldots,R_\delta) \]
are representable in the form
\[ \tau_{pq\rho\sigma}=\tau_{pq\rho\sigma}^{(1)}+\tau_{pq\rho\sigma}^{(2)}, \]
where
\[ \sum_{pq\rho\sigma}\tau_{pq\rho\sigma}^{(1)}t_{pq}t_{\rho\sigma}\geq \sum_{i=1}^{s}\sum_{\gamma=1}^{R_i}b_{i\gamma}(x)t_{i\gamma}^{2}, \]
\[ \left|\tau_{pq\rho\sigma}^{(2)}\right|^2\leq \gamma_{pq\rho\sigma}\beta_{pq}(x)\beta_{\rho\sigma}(x), \]
if \((p,q)\neq(\rho,\sigma)\), and
\[ -\tau_{pqpq}^{(2)}\leq \gamma_{pq p q}\beta_{pq}(x) \quad (p,\rho,i=1,2,\ldots,s;\; q=1,2,\ldots,R_p;\; \sigma=1,2,\ldots,R_\rho;\; \gamma=1,2,\ldots,R_i), \]
where \(\gamma_{pq\rho\sigma}\) are absolute constants;

е)
\[ \sum_{pq\rho\sigma}\gamma_{pq\rho\sigma}<1. \]

Theorem 2. Let \(\hat u_0\in \hat H\), \(f(x)=(f_1(x),\ldots,f_s(x))\in H^*\), and suppose that for each \(i=1,\ldots,s\) the conditions a)—e) listed above are satisfied. Then there exists a unique \(u(x)\in\hat H\) (a generalized solution of system (4)) such that
\[ \int_{\Omega}\sum_{i\gamma}\varphi_{i\gamma}\left(x,D^{\alpha(p,q)}u_p(x)\right)D^{\alpha(i\gamma)}v_i(x)\,dx = \int_{\Omega}\sum_{i=1}^{s}f_i(x)v_i(x)\,dx \]
for any \(v\in H_0\), and \(u-\hat u_0\in \hat H\).

In the proof of Theorem 2, Theorem 3 of paper (3) was used. I express my deep gratitude to S. L. Sobolev and L. D. Kudryavtsev for their attention to the work and for discussion of the results.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
4 VII 1964

CITED LITERATURE

1. S. L. Sobolev, Some applications of functional analysis in mathematical physics, L., 1950.
2. S. L. Sobolev, Sibirsk. Mat. Zhurn., 4, No. 3 (1963).
3. F. E. Browder, Materials for the Soviet-American Symposium on Partial Differential Equations, August 1963, Novosibirsk.
4. V. R. Portnov, DAN, 154, No. 3 (1964).
5. G. G. Hardy, D. E. Littlewood, G. Pólya, Inequalities, M., 1948.