L. N. SLOBODETSKII

ESTIMATES OF SOLUTIONS OF ELLIPTIC AND PARABOLIC SYSTEMS

(Presented by Academician V. I. Smirnov, 18 I 1958)

MATHEMATICS

1. In the article \((^1)\) the spaces \(W_{x(1), \ldots, x(r), p}^{(l_1,\ldots,l_r)}(Q)\) were introduced for \(p=2\) and arbitrary nonnegative \(l_1,\ldots,l_r\). Keeping the notation adopted there, we introduce the indicated spaces for arbitrary \(p\) \((1<p<+\infty)\).

Let first \(l_k\) be a nonnegative integer. We shall say that
\(f=f(x)\in W_{x(k),p}^{(l_k)}(Q)\) if \(f\) has generalized derivatives with respect to \(x^{(k)}\), summable over \(Q\) to the power \(p\), up to order \(l_k\). In this case we set:

\[ \|f\|_{W_{x(k),p}^{(l_k)}(Q)} = \left[ \sum_{q\le l_k}\int_Q |D_{x(k)}^q f|^p\,dx \right]^{1/p}. \]

Let now \(l_k=l_k'+\lambda_k\). We shall say that \(f\in W_{x(k),p}^{(l_k)}(Q)\) if
\(f\in W_{x(k),p}^{(l_k')}(Q)\) and if for all \(q\le l_k'\)

\[ L_{k,p}(D_{x(k)}^q f) = \int_{Q\times \Omega^{(k)}} \left| \Delta(x^{(k)},y^{(k)})D_{x(k)}^q f \right|^p \frac{dx\,dy^{(k)}}{|x^{(k)}-y^{(k)}|^{n_k+p\lambda_k}} <+\infty . \]

In this case

\[ \|f\|_{W_{x(k),p}^{(l_k)}(Q)} = \left\{ \|f\|_{W_{x(k),p}^{(l_k')}(Q)}^p + \sum_{q\le l_k'} L_{k,p}(D_{x(k)}^q f) \right\}^{1/p}. \]

We shall further say that
\(f\in W_{x(1),\ldots,x(r),p}^{(l_1,\ldots,l_r)}(Q)\) if
\(f\in W_{x(k),p}^{(l_k)}(Q)\) for all \(k=1,2,\ldots,r\). In this case

\[ \|f\|_{W_{x(1),\ldots,x(r),p}^{(l_1,\ldots,l_r)}(Q)} = \left\{ \sum_{k=1}^r \|f\|_{W_{x(k),p}^{(l_k)}(Q)}^p \right\}^{1/p}. \]

For \(r=1\) and \(Q=\Omega\subset E_n\), the space \(W_{x,p}^{(l)}(\Omega)\) will be denoted simply by \(W_p^{(l)}(\Omega)\). Likewise, when
\(l_1=l_2=\cdots=l_r=l\), we shall denote
\(W_{x(1),\ldots,x(r),p}^{(l,\ldots,l)}(Q)\) by \(W_p^{(l)}(Q)\).

Theorem 1. Let the domain \(Q\) be bounded by sufficiently smooth surfaces and let
\(f\in W_{x(1),\ldots,x(r),p}^{(l_1,\ldots,l_r)}(Q)\). Then for any integral and nonnegative
\(m_1,\ldots,m_r\), satisfying the inequality

\[ \mu=1-\sum_{k=1}^r \frac{m_k}{l_k}\ge 0, \]

there exist generalized derivatives \(D_{x(1)}^{m_1}\cdots D_{x(r)}^{m_r} f\), belonging to
\(W_{x(1),\ldots,x(r),p}^{(\bar l_1,\ldots,\bar l_r)}(Q)\) with \(\bar l_k=l_k\mu\) \((k=1,2,\ldots,r)\). Moreover

\[ \left\|D_{x(1)}^{m_1}\cdots D_{x(r)}^{m_r} f\right\|_ {W_{x(1),\ldots,x(r),p}^{(\bar l_1,\ldots,\bar l_r)}(Q)} \leq C\|f\|_{W_{x(1),\ldots,x(r),p}^{(l_1,\ldots,l_r)}(Q)} \tag{1} \]

with \(C\) depending only on \(Q\).

It follows from this theorem that, for integer \(l\), the space \(W_p^{(l)}(Q)\) coincides with the corresponding space of S. L. Sobolev.

It would be interesting to obtain, for arbitrary \(p\), embedding theorems analogous to those given in \((1)\) for \(p=2\). Unfortunately, this has not yet been accomplished.

2. Let \(\Omega\) be a finite or infinite domain of \(n\)-dimensional space \(E_n\), whose boundary is a sufficiently smooth \((n-1)\)-dimensional finite or infinite surface without edge \(S\). Consider in \(\Omega\) the linear differential operator

\[ L\left(x,\frac{\partial}{\partial x}\right)u = \sum_{r=1}^{2k}\sum_{i_1,\ldots,i_r=1}^{n} A^{(i_1,\ldots,i_r)}(x) \frac{\partial^r u}{\partial x_{i_1}\cdots \partial x_{i_r}} + A(x)u, \]

where \(A^{(i_1,\ldots,i_r)}(x)\) and \(A(x)\) are square matrices of order \(N\), and \(u=u(x)\) is a vector function with \(N\) components. Denote by \(L_0(x,\partial/\partial x)\) the principal part of \(L(x,\partial/\partial x)\), and by \(L_0(x,i\alpha)\) the matrix obtained from \(L_0(x,\partial/\partial x)\) by replacing the symbols \(\partial/\partial x_s\) by the expressions \(i\alpha_s\) \((s=1,2,\ldots,n)\). Following I. G. Petrovskii \((2)\), we shall say that \(L(x,\partial/\partial x)\) is elliptic in \(\Omega\) if, for every \(x\in\Omega\) and any real \(\alpha_1,\ldots,\alpha_n\), the determinant \(|L_0(x,i\alpha)|\) of the matrix \(L_0(x,i\alpha)\) satisfies the inequality:

\[ \bigl||L_0(x,i\alpha)|\bigr| \geq \delta |\alpha|^{2k} \quad (\delta>0,\quad |\alpha|=\sqrt{\alpha_1^2+\cdots+\alpha_n^2}). \]

Suppose that \(S\) can be covered by a finite number of overlapping surfaces \(\sigma_s\) \((s=1,2,\ldots,q)\) such that each \(\sigma_s\) can be specified by a sufficiently regular parametric equation \(x=x(\gamma')\) \((\gamma'=(\gamma_1,\ldots,\gamma_{n-1}))\). Denote by \(\nu=\nu(\gamma')\) the unit normal vector to \(S\) at the point \(x'=x(\gamma')\), and assume that

\[ x=x(\gamma')+\nu(\gamma')\gamma_n \tag{2} \]

establishes a one-to-one and sufficiently regular correspondence between some domain of the space of points \(\gamma=(\gamma_1,\ldots,\gamma_n)\) and some \(n\)-dimensional neighborhood of the surface \(\sigma_s\). Specify on \(S\) \(k\) linear differential operators \(R_\mu(x',\partial/\partial x)\) of orders \(m_\mu\) \((\mu=1,2,\ldots,k)\). Using (2), transform \(L(x,\partial/\partial x)\) and \(R_\mu(x',\partial/\partial x)\) to the new variables \(\gamma_1,\ldots,\gamma_n\). Denote the results of this transformation by \(L(\gamma,\partial/\partial\gamma)\) and \(R_\mu(\gamma,\partial/\partial\gamma)\), and their principal parts by \(L_0(\gamma,\partial/\partial\gamma)\) and \(R_\mu^{(0)}(\gamma',\partial/\partial\gamma)\) \((\mu=1,2,\ldots,k)\).

Denote by \(D(\gamma';\beta')\) \((\beta'=(\beta_1,\ldots,\beta_{n-1}))\) the matrix of order \(kN\):

\[ D(\gamma';\beta') = \left\| \int_{C(\beta')} R_\mu^{(0)}(\gamma',i\beta) L_0^{-1}(\gamma',i\beta) \beta_n^{\lambda-1}\,d\beta_n \right\|_{\lambda,\mu=1,2,\ldots,k} \quad (\beta=(\beta_1,\ldots,\beta_n)), \]

where \(C(\beta')\) is a positively oriented contour, situated in the upper half-plane of the complex \(\beta_n\)-plane and enclosing all roots lying there of the equation \(|L_0(\gamma',i\beta)|=0\) for fixed \(\gamma'\) and \(\beta'\).

We shall say that \(L\) and \(R_\mu\) satisfy on \(S\) condition (L) (of B. Ya. Lopatinskii \((^3)\)) if, for every \(x'\in S\) and any real \(\beta_1,\ldots,\beta_{n-1}\), for which
\[ |\beta'|=\sqrt{\beta_1^2+\cdots+\beta_{n-1}^2}=1, \]
the inequality
\[ \|D(\gamma',\beta')\|\ge \delta' \qquad (\delta'>0) \]
holds.

Theorem 2. Suppose: 1) \(L\) is elliptic in \(\Omega\), and \(L\) and \(R_\mu\) \((\mu=1,2,\ldots,k)\) satisfy condition (L) on \(S\); 2) \(l\ge 2k\), \(m_\mu<l-\tfrac12\) \((\mu=1,2,\ldots,k)\). Then, for every vector-function \(u\in W_2^{(l)}(\Omega)\), the inequality
\[ C_1\left[ \|Lu\|_{W_2^{(l-2k)}(\Omega)} + \sum_{\mu=1}^k \|R_\mu u\|_{W_2^{(l-m_\mu-\frac12)}(S)} \right] \le \|u\|_{W_2^{(l)}(\Omega)} \le \]
\[ \le C_2\left[ \|Lu\|_{W_2^{(l-2k)}(\Omega)} + \sum_{\mu=1}^k \|R_\mu u\|_{W_2^{(l-m_\mu-\frac12)}(S)} + \|u\|_{L_2(\Omega)} \right], \tag{3} \]
where \(C_1\) and \(C_2\) are positive constants depending only on \(\Omega\) and on the differential operators \(L\) and \(R_\mu\).

Theorem 3. Suppose that condition 1) of Theorem 2 is satisfied and \(l\ge 2k\). Then, for every \(u\in W_p^{(l)}(\Omega)\) satisfying the boundary conditions
\[ R_\mu u\big|_S=0 \qquad (\mu=1,2,\ldots,k), \]
the inequality
\[ C_1\|Lu\|_{W_p^{(l-2k)}(\Omega)} \le \|u\|_{W_p^{(l)}(\Omega)} \le C_2\left[\|Lu\|_{W_p^{(l-2k)}(\Omega)}+\|u\|_{L_p(\Omega)}\right]. \tag{4} \]

Theorems 2 and 3 generalize the corresponding results of O. V. Guseva \((^4)\), F. Browder \((^5)\), A. I. Koshelev \((^6)\), and L. Nirenberg \((^{7,8})\).

1. Let us now consider, in the cylinder \(Q=\Omega\times[0,T]\) \((x\in\Omega;\ 0\le t\le T\le+\infty)\), the parabolic differential operator
   \[ L\left(t,x,\frac{\partial}{\partial t},\frac{\partial}{\partial x}\right)u= \]
   \[ = \frac{\partial u}{\partial t} - \sum_{r=1}^{2k}\sum_{i_1,\ldots,i_r=1}^{n} A^{(i_1,\ldots,i_r)}(t,x) \frac{\partial^r u}{\partial x_{i_1}\cdots \partial x_{i_r}} - A(t,x)u . \]

The operator
\[ L_0\left(t,x,\frac{\partial}{\partial t},\frac{\partial}{\partial x}\right) = \frac{\partial}{\partial t} - \sum_{i_1,\ldots,i_{2k}=1}^{n} A^{(i_1,\ldots,i_{2k})}(t,x) \frac{\partial^{2k}}{\partial x_{i_1}\cdots \partial x_{i_{2k}}} \]
will be called the principal part of the operator \(L\).

On \(\Gamma=S\times[0,T]\) let \(k\) linear differential operators be given:
\[ R_\mu(t,x',\partial/\partial t,\partial/\partial x) \qquad (\mu=1,2,\ldots,k). \]
We shall call the principal part of \(R_\mu\) the collection of all its terms of the form \(B(t,x')D_x^{s_1}D_t^{s_2}\), for which the quantity \(s_1+2ks_2\) is greatest. In this case
\[ m_\mu=\max\{s_1+2ks_2\} \]
will be called the order of \(R_\mu\).

Proceeding as before, we transform \(L\) and \(R_\mu\) to the variables \(t,\gamma_1,\ldots,\gamma_n\). Denote the results of the transformation by
\[ L(t,\gamma,\partial/\partial t,\partial/\partial\gamma) \]
and
\[ R_\mu(t,\gamma',\partial/\partial t,\partial/\partial\gamma). \]
Introduce the matrix
\[ D(t,\gamma',\beta_0,\beta')= \]
\[ = \left\| \int_{C(\beta_0,\beta')} R_\mu^{(0)}(t,\gamma',i\beta_0,i\beta) L_0^{-1}(t,\gamma',i\beta_0,i\beta) \beta_n^{\lambda-1}\,d\beta_n \right\|_{\mu,\lambda=1,2,\ldots,k}, \]
where \(C(\beta_0,\beta')\) is a positively oriented contour lying in the upper half-plane \(\beta_n\) and enclosing all the roots situated there of the equation
\[ |L_0(t,\gamma',i\beta_0,i\beta)|=0 \]
for fixed \(t,\gamma',\beta_0\), and \(\beta'\).

We shall say that \(L\) and \(R_\mu\) \((\mu=1,2,\ldots,k)\) are connected on \(\Gamma\) by condition \((\mathcal L)\), if for every point \((t,x')\in\Gamma\) and any real \(\beta_0,\beta_1,\ldots,\beta_{n-1}\) for which \(\beta_0^2+|\beta'|^2=1\), the inequality
\[ \|D(t,\gamma',\beta_0,\beta')\|\geq \delta' \quad (\delta'>0) \]
holds.

Theorem 4. Let: 1) \(l_1=2kl_2,\ l_2\geq 1,\ L\) be a parabolic differential operator, and let \(L\) and \(R_\mu\) \((\mu=1,2,\ldots,k)\) be connected on \(\Gamma\) by condition \((\mathcal L)\); 2) \(m_\mu<l_1-\frac12\) \((\mu=1,2,\ldots,k)\). Then for any vector-function \(u\in W_{x,t,2}^{(l_1,l_2)}(Q)\) the inequality
\[ C_1\left[ \|Lu\|_{W_{x,t,2}^{(l_1-2k,\;l_2-1)}(Q)} +\|u\|_{W_2^{(l)}(\Omega)} +\sum_{\mu=1}^{k}\|R_\mu u\|_{W_{x,t,2}^{(l_\mu^{(1)},\;l_\mu^{(2)})}(\Gamma)} \right]\leq \]
\[ \leq \|u\|_{W_{x,t,2}^{(l_1,l_2)}(Q)} \leq C_2\left[ \|Lu\|_{W_{x,t,2}^{(l_1-2k,\;l_2-1)}(Q)} +\right. \]
\[ \left. +\|u\|_{W_2^{(l)}(\Omega)} +\sum_{\mu=1}^{k}\|R_\mu u\|_{W_{x,t,2}^{(l_\mu^{(1)},\;l_\mu^{(2)})}(\Gamma)} +\|u\|_{L_2(Q)} \right], \tag{5} \]
where
\[ l=l_1\left(1-\frac{1}{2l_2}\right)=l_1-k=(2l_2-1)k,\qquad l_\mu^{(1)}=l_1-m_\mu-\frac12, \]
\[ l_\mu^{(2)}= l_2\left(1-\frac{m_\mu}{l_1}-\frac{1}{2l_1}\right) \quad (\mu=1,2,\ldots,k), \]
and \(C_1\) and \(C_2\) are positive constants depending only on \(Q,\ L\), and \(R_\mu\) \((\mu=1,2,\ldots,k)\).

Theorem 5. Suppose that condition 1) of Theorem 4 is satisfied. Then for any \(u\in W_{x,t,p}^{(l_1,l_2)}(Q)\) satisfying the conditions \(u|_{t=0}=R_\mu u|_\Gamma=0\) \((\mu=1,2,\ldots,k)\), the inequality
\[ C_1\|Lu\|_{W_{x,t,p}^{(l_1-2k,\;l_2-1)}(Q)} \leq \|u\|_{W_{x,t,p}^{(l_1,l_2)}(Q)} \leq \]
\[ \leq C_2\left[ \|Lu\|_{W_{x,t,p}^{(l_1-2k,\;l_2-1)}(Q)} +\|u\|_{L_p(Q)} \right]. \tag{6} \]

Theorems 4 and 5 admit generalization to arbitrary parabolic systems of I. G. Petrovskii \((^9)\) with derivatives with respect to \(t\) of arbitrary orders. These theorems generalize results of O. A. Ladyzhenskaya \((^{10})\).

Leningrad State Pedagogical Institute
named after A. I. Herzen

Received
16 I 1958

CITED LITERATURE

1. L. N. Slobodetskii, DAN, 118, No. 2 (1958).
2. I. G. Petrovskii, Uspekhi Matem. Nauk, 1, no. 3–4 (1946).
3. B. Ya. Lopatinskii, Ukr. Matem. Zhurn., 5, No. 2 (1953).
4. O. V. Guseva, DAN, 102, No. 6 (1955).
5. E. Browder, Comm. Pure and Appl. Math., 9, 351 (1956).
6. A. I. Koshelev, DAN, 116, No. 4 (1957).
7. L. Nirenberg, Comm. Pure and Appl. Math., 8, 649 (1955).
8. L. Nirenberg, Comm. Pure and Appl. Math., 9, 509 (1956).
9. I. G. Petrovskii, Bull. MGU, ser. A, 1, 7 (1938).
10. O. A. Ladyzhenskaya, DAN, 97, No. 3 (1954).