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}^{(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}^{(l_k)}(Q)
=
\left{
|f|{W^p}^{(l_k')}(Q)
+
\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}^{(l_1,\ldots,l_r)}(Q)
=
\left{
\sum_{k=1}^r
|f|{W^p}^{(l_k)}(Q)
\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}^{(\bar l_1,\ldots,\bar l_r)}(Q)
\leq
C|f|{W}^{(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^k
|R_\mu u|{W_2^{(l-m\mu-\frac12)}(S)}
\right]
\le
|u|{W_2^{(l)}(\Omega)}
\le
]
[
\le
C_2\left[
|Lu|}(\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|}(\Omega)
\le
|u|{W_p^{(l)}(\Omega)}
\le
C_2\left[|Lu|\right].}(\Omega)}+|u|_{L_p(\Omega)
\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