UDC 517.946

MATHEMATICS

N. V. ZHITARASHU

A PRIORI ESTIMATES AND SOLVABILITY OF GENERAL BOUNDARY VALUE PROBLEMS FOR GENERAL ELLIPTIC SYSTEMS WITH DISCONTINUOUS COEFFICIENTS

(Presented by Academician I. N. Vekua, 29 III 1965)

1. Here we set forth the result of a study of general boundary value problems for systems elliptic in the sense of Douglis—Nirenberg with discontinuous coefficients. These results were obtained by means of a method developed in recent years in a number of works \(^{(1-6)}\), and are a generalization to systems of the investigations of M. Schechter \(^{(12)}\), Ya. A. Roitberg and Z. G. Sheftel’ \(*\) \(^{(1,3)}\).

2. Notation, statement of the problem. Let \(G\) be a bounded domain of the space \(E_n\) with boundary \(\Gamma\), divided by a surface \(\gamma\) into two subdomains \({}^{**}\) \(G_1\) and \(G_2\), with \(\gamma\) and \(\Gamma\) having no common points. Let, for definiteness, \(\gamma\) be the boundary of \(G_1\).

We shall consider the spaces (see \(^{(1-3)}\)) \(W_p^l(G)=W_p^l(G_1)+W_p^l(G_2)\) (\(l\ge 0\) an integer, \(p>1\)) and the spaces \(C^{l+\alpha}(G)=C^{l+\alpha}(G_1)+C^{l+\alpha}(G_2)\) (\(0<\alpha<1\)) of piecewise smooth functions with norm (see \(^{(4)}\))

\[ |v|_{l+\alpha}=|v|_{l+\alpha}^{G}=|v|_{l+\alpha}^{G_1}+|v|_{l+\alpha}^{G_2}. \]

By \(C^{l+\alpha}(\gamma)\) and \(C^{l+\alpha}(\Gamma)\) we shall mean the spaces of Hölder functions on \(\gamma\) and \(\Gamma\) (see \(^{(4)}\)), and by \(v^{(m)}(x)\) the value of the function \(v(x)\) for \(x\in G_m\) \((m=1,2)\).

We define the operators

\[ \mathscr{L}(x,\partial/\partial x)= \begin{cases} \mathscr{L}^{(1)}(x,\partial/\partial x), & x\in G_1,\\ \mathscr{L}^{(2)}(x,\partial/\partial x), & x\in G_2; \end{cases} \tag{1} \]

\[ B^{(1)}(x,\partial/\partial x)\ \text{and}\ B^{(2)}(x,\partial/\partial x)\ \text{on } \gamma; \tag{2} \]

\[ \mathscr{H}(x,\partial/\partial x)\ \text{on } \Gamma. \tag{3} \]

\(\mathscr{L}^{(m)}(x,\partial/\partial x)\), \(B^m(x,\partial/\partial x)\) \((m=1,2)\), \(\mathscr{H}(x,\partial/\partial x)\) are matrices of respective sizes \(N\times N\), \(2r\times N\), \(r\times N\) (the numbers \(r\) and \(N\) will be determined below), whose elements \(l_{\sim m)}(x,\partial/\partial x)\), \(b_{qj}^{(m)}(x,\partial/\partial x)\) \((m=1,2)\), \(h_{kj}(x,\partial/\partial x)\) are linear differential operators of orders \(s_i+t_j\), \(\sigma_q+t_j\), \(m_k+t_j\). The \(s_i,t_j,\sigma_q,m_k\) are integers, with \(s_i\le 0\), \(t_j\ge 0\) \((i,j=1,\ldots,N)\). Denote by \(\mathscr{L}_0(x,\partial/\partial x)\), \(B_0^{(m)}(x,\partial/\partial x)\) \((m=1,2)\), \(\mathscr{H}_0(x,\partial/\partial x)\) the operators obtained from the operators (1), (2), (3) by discarding in \(l_{ij}(x,\partial/\partial x)\), \(b_{qj}^{(m)}(x,\partial/\partial x)\), \(h_{kj}(x,\partial/\partial x)\) all terms containing differentiations of orders lower than \(s_i+t_j\), \(\sigma_q+t_j\), \(m_k+t_j\); and by \(\mathscr{L}_0(x,\xi)\), \(B_0^{(m)}(x,\xi)\), \(\mathscr{H}_0(x,\xi)\) the matrices obtained from the corresponding operators by replacing \(\partial/\partial x=(\partial/\partial x_1,\ldots,\partial/\partial x_n)\) by \(\xi=(\xi_1,\ldots,\xi_{n-1},\tau)=(\xi',\tau)\). The operator

* After the present work had been completed, it became known to the author that analogous results had been obtained by Z. G. Sheftel’.

** All results are also valid in the case of a partition of \(G\) into a finite number of domains.

\(\mathcal L(x,\partial/\partial x)\) is called properly elliptic in \(\overline G\) (see (8)) if, for any \(x\in \overline G_m\) and any real vector \((\xi',\tau)\ne 0\),

\[ L^{(m)}(x,\xi',\tau)=\det\bigl(\mathcal L_0^{(m)}(x,\xi',\tau)\bigr)\ne 0, \tag{4} \]

and the roots \(\tau\) of the polynomials (4), for any real \(\xi'\ne 0\), are equally distributed in the upper and lower half-planes. From proper ellipticity it follows that the order of \(L(x,\xi',\tau)\) is an even number, \(\sum(s_i+t_i)=2r\).

Below we shall consider only properly elliptic operators.

The coefficients entering into \(l_{ij}, b_{qj}^{(m)}, h_{kj}\) belong respectively to the spaces

\[ C^{l-s_i+\alpha}(G),\ C^{l-\sigma_q+\alpha}(\gamma),\ C^{l-m_k+\alpha}(\Gamma),\quad \text{where } l>l_0=\max_{q,k}(|\sigma_q|,|m_k|). \]

Consider the problem of finding a solution of the system

\[ \mathcal L(x,\partial/\partial x)u(x)=f(x),\quad x\in G-\gamma, \tag{5} \]

satisfying on \(\gamma\) the conjugation conditions

\[ B^{(1)}(x,\partial/\partial x)u^{(1)}(x)-B^{(2)}(x,\partial/\partial x)u^{(2)}(x)=\varphi(x),\quad x\in\gamma, \tag{6} \]

and on \(\Gamma\) the boundary conditions

\[ \mathcal H[x,\partial/\partial x)u(x)=g(x),\quad x\in\Gamma. \tag{7} \]

For vector-functions \(u(x)=(u_1,\ldots,u_N)\), \(f(x)=(f_1,\ldots,f_N)\), introduce the spaces \(\mathfrak U^{(l)}\) and \(\mathfrak F^{(l)}\) in \(G\) as direct products of the spaces

\[ \mathfrak U^{(l)}=\prod_{j=1}^{N} W_p^{\,l+t_j}(G),\quad \mathfrak F^{(l)}=\prod_{i=1}^{N} W_p^{\,l-s_i}(G)\quad (l\ge l_0+1) \tag{8} \]

with norms

\[ \|u,\mathfrak U^{(l)}\|=\sum_{j=1}^{N}\|u_j\|_{l+t_j},\quad \|f,\mathfrak F^{(l)}\|=\sum_{i=1}^{N}\|f_i\|_{l-s_i}. \tag{9} \]

By \(\Phi^{(l-1/p)}\) and \(\mathcal G^{(l-1/p)}\) we denote the direct products of the spaces of L. N. Slobodetskii (see (7)) on \(\gamma\) and \(\Gamma\)

\[ \Phi^{(l-1/p)}=\prod_{q=1}^{2r} W_p^{\,l-\sigma_q-1/p}(\gamma),\quad \mathcal G^{(l-1/p)}=\prod_{k=1}^{r} W_p^{\,l-m_k-1/p}(\Gamma) \tag{10} \]

with norms

\[ \|\varphi,\Phi^{(l-1/p)}\|=\sum_{q=1}^{2p}\|\varphi_q\|_{l-\sigma_q-1/p},\quad \|g,\mathcal G^{(l-1/p)}\|=\sum_{k=1}^{r}\|g_k\|_{l-m_k-1/p}. \tag{11} \]

By \(\mathfrak U_c^{(l+\alpha)}\), \(\mathfrak F_c^{(l+\alpha)}\), \(\Phi_c^{(l+\alpha)}\), \(\mathcal G_c^{(l+\alpha)}\) we shall understand the analogous spaces composed of \(C^{l+t_j+\alpha}(G)\), \(C^{l-s_i+\alpha}(G)\), \(C^{l-\sigma_q+\alpha}(\gamma)\), \(C^{l-m_k+\alpha}(\Gamma)\), with the corresponding norms.

Denote by
\[ B(x,\partial/\partial x)u(x) = B^{(1)}(x,\partial/\partial x)u^{(1)} - B^{(2)}(x,\partial/\partial x)u^{(2)} \]
on \(\gamma\), and associate with problem (5)—(7) the operator
\(\mathfrak A=(\mathcal L,B,\mathcal H)\), acting boundedly from

\[ \mathfrak U_c^{(l+\alpha)}\to H_c^{(l+\alpha)} =\mathfrak F_c^{(l+\alpha)}\times\Phi_c^{(l+\alpha)}\times\mathcal G_c^{(l+\alpha)} \quad \text{for } l\ge l_0, \]

\[ \mathfrak U^{(l)}\to H^{(l)} =\mathfrak F^{(l)}\times\Phi^{(l-1/p)}\times\mathcal G^{(l-1/p)} \quad \text{for } l\ge l_0+1. \]

3. Condition of joint covering of operators.
Let \(t=\max t_j\), let \(l\ge l_0\) be an integer, and let the surface \(\gamma(\Gamma)\) belong to the class

$C^{l++\alpha}$. For any point $P \in \gamma(\Gamma)$, write all operators in such a local coordinate system $\tilde{x}$ that the axis $\tilde{x}_n$ is directed along the inward normal to $\gamma(\Gamma)$, and the remaining axes lie in the tangent plane. Simplifying the notation, we again denote the local coordinates by $x$. Denote by $\widehat{\mathscr L}_0(x,\xi',\tau)$ the matrix adjoint to $\mathscr L_0(x,\xi',\tau)$, and

\[ C_m(x,\xi',\tau)=B_0^{(m)}(x,\xi',\tau)\cdot \widehat{\mathscr L}_0^{(m)}(x,\xi',\tau) \quad (m=1,2). \tag{12} \]

It follows from proper ellipticity that

\[ L^m(x,\xi',\tau)=L_+^{(m)}(x,\xi',\tau)\cdot L_-^{(m)}(x,\xi',\tau) \quad (m=1,2) \tag{13} \]

(the $\tau$-roots of $L_+^{(m)}$ $(L_-^m)$ have $\operatorname{Im}\tau>0$ $(\operatorname{Im}\tau<0)$ for any real $\xi'\ne 0$).

Let $C_1'(x,\xi',\tau)$ and $C_2'(x,\xi',\tau)$ be the matrices composed of the remainders from division (with respect to $\tau$) of $C_1(x,\xi',\tau)$ and $C_2(x,\xi',\tau)$ respectively by $L_+^{(1)}$ and $L_-^{(2)}$. Following the terminology of $(^{1-3})$, we introduce the notion of joint covering of operators.

Definition. The operators (2) and (3) jointly cover the properly elliptic operator $\mathscr L(x,\partial/\partial x)$ if:
1) for any point $x\in\gamma$ and any real vector $\xi'\ne 0$, the rows of the matrix
\[ C'=\bigl(C_1'(x,\xi',\tau),\,C_2'(x,\xi',\tau)\bigr) \]
are linearly independent with respect to $\tau$;
2) the operator $\mathscr H(x,\partial/\partial x)$ on $\Gamma$ satisfies the Lopatinskii condition $(^{5,6})$.

4. Estimates of Schauder type. Let $l\ge l_0$.

Theorem 1. Let the vector-function $u(x)\in \mathfrak U^{(l+\alpha)}$ be a solution of problem (5)—(7), $\mathscr Lu\in \mathfrak F_c^{(l+\alpha)}$, $Bu\in \Phi_c^{(l+\alpha)}$, $\mathscr Hu\in \mathscr G^{(l+\alpha)}$.

The joint covering condition is necessary and sufficient for $u(x)\in \mathfrak U_c^{(l+\alpha)}$,

\[ |u,\mathfrak U_c^{(l+\alpha)}| \le A_1\bigl(|f,\mathfrak F_c^{(l+\alpha)}| + |\varphi,\Phi_c^{(l+\alpha)}| + |g,\mathscr G^{(l+\alpha)}| + |u|_0\bigr), \tag{14} \]

where
\[ |u|_0=\sum_{j=1}^{N}|u_j|_0, \]
and $A_1$ does not depend on $f,\varphi,g$.

The following assertion holds (see $(^{11})$):

Theorem 2. 1) The range $R(\mathfrak A)$ of the operator $\mathfrak A$ is closed in $H_c^{(l+\alpha)}$:
\[ R(\mathfrak A)=\overline{R(\mathfrak A)}. \]

2) The homogeneous problem (5)—(7) has in $H_c^{(l+\alpha)}$ a finite number of linearly independent solutions.

5. A priori estimates in $L_p$. In this section we shall assume that all the conditions imposed above are fulfilled, with $l\ge l_1=l_0+1$.

Theorem 3. Let $u(x)\in \mathfrak U^{(l)}$ be a solution of problem (5)—(7),
\[ \mathscr Lu\in \mathfrak F^{(l)},\quad Bu\in \Phi^{(l-1/p)},\quad \mathscr Hu\in \mathscr G^{(l-1/p)}. \]

The joint covering condition is necessary and sufficient for $u(x)\in \mathfrak U^{(l)}$ and

\[ \|u,\mathfrak U^{(l)}\| \le A_2\bigl(\|f,\mathfrak F^{(l)}\| + \|\varphi,\Phi^{(l-1/p)}\| + \|g,\mathscr G^{(l-1/p)}\| + \|u\|_0\bigr), \tag{15} \]

where
\[ \|u\|_0=\sum_{j=1}^{N}\|u_j\|_0, \]
and $A_2$ does not depend on $f,\varphi,g$.

We shall call the operator $\mathfrak A$ of problem (5)—(7) elliptic if $\mathscr L(x,\partial/\partial x)$ is properly elliptic and the operators (2), (3) jointly cover it.

The following theorem is valid (see $(^{9,11})$):

Theorem 4. The following assertions are equivalent:

1) the operator $\mathfrak A$ is elliptic;

2) if $u\in \mathfrak U^{(l)}$, $\mathscr Lu\in \mathfrak F^{(l)}$, $Bu\in \Phi^{(l-1/p)}$, $\mathscr Hu\in \mathscr G^{(l-1/p)}$, then $u(x)\in \mathfrak U^{(l)}$ and the estimate (15) holds;

3) the operator $\mathfrak A$ is a $\Phi$-operator in the sense of (10).

In conclusion I express my deep gratitude to S. D. Eidelman for formulating the problem, for his constant attention to the present work, and for valuable advice.

Voronezh Polytechnic
Institute

Received
25 III 1965

CITED LITERATURE

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