V. R. PORTNOV

THE FIRST BOUNDARY-VALUE PROBLEM FOR ONE CLASS OF QUASILINEAR SYSTEMS OF EQUATIONS HAVING DIVERGENT FORM

(Presented by Academician S. L. Sobolev on 24 III 1970)

1°. Introduction. The principal results on the solvability of the first boundary-value problem for quasilinear strongly elliptic systems of differential equations having divergent form were obtained in a number of works by M. I. Vishik, F. E. Browder, Yu. A. Dubinskii, and other authors. For a survey of these works see \((^{1})\).

In the present note, on the basis of one theorem \(((^{1}),\) Theorem 12), due to F. E. Browder and Yu. A. Dubinskii, sufficient conditions are given for solvability and unique (up to a vector-function from a certain space \(\dot S\)) solvability of the first boundary-value problem for systems of equations of the form (2) in spaces of the S. L. Sobolev type. In the particular case these are strongly elliptic systems of differential equations having divergent form and degenerating on the boundary of the domain and at infinity, as well as certain nonelliptic quasilinear systems of differential equations with or without degeneration \((^{3,4})\).

2°. Auxiliary definitions and notation. \(R, n, s\) are natural numbers; \(E_n\) is \(n\)-dimensional Euclidean space of points \(x=(x_1,\ldots,x_n)\), \(E_R\) is \(R\)-dimensional Euclidean space, whose points will subsequently be denoted as follows: \(\xi=(\xi_1,\ldots,\xi_R)\), \(\eta=(\eta_1,\ldots,\eta_R)\), \(\zeta=(\zeta_1,\ldots,\zeta_R)\); \(\Omega\) is a domain in \(E_n\); \(T(\Omega)\) is the totality of measurable and almost everywhere finite functions on \(\Omega\); \(\mathcal K'(\Omega)\) is the totality of additive and homogeneous functionals on \(C_0^{(\infty)}(\Omega)\);

\[ T(\Omega;s)=\underbrace{T(\Omega)\times\ldots\times T(\Omega)}_{s};\quad C_0^{(\infty)}(\Omega;s)=\underbrace{C_0^{(\infty)}(\Omega)\times\ldots\times C_0^{(\infty)}(\Omega)}_{s}. \]

Further, if \(\mathscr P:C_0^{(\infty)}(\Omega)\to T(\Omega)\) is an additive homogeneous operator, and
\[ \Gamma_{\mathscr P}=\{u\in T(\Omega):u\,\mathscr P v\in L_1(\Omega),\ \forall v\in C_0^{(\infty)}(\Omega)\}, \]
then the operator \(\mathscr P^*:\Gamma_{\mathscr P}\to\mathcal K'(\Omega)\) is defined on each function \(u\in\Gamma_{\mathscr P}\) by the equality
\[ \langle \mathscr P^*u,v\rangle=\int_{\Omega}u\mathscr P v\,dx,\qquad \forall v\in C_0^{(\infty)}(\Omega). \tag{1} \]

Consider the system of equations
\[ \sum_{k=1}^{R}\mathscr L^{(k,j)*}\bigl(\varphi^{(k)}(x,\mathscr L^{(1)}u,\ldots,\mathscr L^{(R)}u)\bigr)=f_j,\qquad j=1,\ldots,s. \tag{2} \]

In system (2): \(\varphi^{(k)}(x,\xi)\) is a function defined on \(\Omega\times E_R\), \(f_j\in\mathcal K'(\Omega)\); \(\mathscr L^{(k)}\) is an additive and homogeneous operator, defined on some linear space \(\mathfrak U_k\subset T(\Omega;s)\) and mapping it into \(T(\Omega)\), where
\[ C_0^{(\infty)}(\Omega;s)\subset \mathfrak U=\bigcap_{1\le k\le R}\mathfrak U_k;\quad u=(u_1(x),\ldots,u_s(x))\in\mathfrak U— \]

unknown vector-function. Further, the operator \(\mathscr L^{(k)}\) on \(C_0^{(\infty)}(\Omega;s)\) can be written in the form

\[ \mathscr L^{(k)}u=\sum_{j=1}^{s}\mathscr L^{(k,j)}u_j,\quad u=(u_1,\ldots,u_s)\in C_0^{(\infty)}(\Omega;s), \]

where \(\mathscr L^{(k,j)}:C_0^{(\infty)}(\Omega)\to T'(\Omega)\). The operator \(\mathscr L^{(k,j)*}:\Gamma_{\mathscr L^{(k,j)}}^{*}\to \mathscr K'(\Omega)\) is defined as was indicated above (see equality (1)).

Put \(\mathscr Lu=(\mathscr L^{(1)}u,\ldots,\mathscr L^{(R)}u)\), \(\forall u\in \mathfrak U\).

The collection of functionals \(f=(f_1,\ldots,f_s)\) will be called the right-hand side of system (2). Obviously \(f\) may be regarded as an additive and homogeneous functional on \(C_0^{(\infty)}(\Omega;s)\).

Definition 1. A vector-function \(u\in\mathfrak U\) is called a solution of system (2) if, for all \(v\in C_0^{(\infty)}(\Omega;s)\),

\[ \sum_{k=1}^{R}\varphi^{(k)}(x,\mathscr Lu)\mathscr L^{(k)}v\in L_1(\Omega) \quad\text{and}\quad \int_{\Omega}\left(\sum_{k=1}^{R}\varphi^{(k)}(x,\mathscr Lu)\mathscr L^{(k)}v\right)\,dx=\langle f,v\rangle. \]

Below we shall need the notions of a variable \(N\)-function \(M(x,w)\), a variable \(N\)-function \(M^*(x,w)\), the Orlicz space \(L_M(\Omega)\), and the \(\Delta_2'\)-condition for \(M(x,w)\). Definitions of these notions are available, for example, in \((^2)\).

3°. Formulation of the first boundary-value problem. Let \(M_1(x,w),\ldots,M_R(x,w)\) be variable \(N\)-functions satisfying the \(\Delta_2'\)-condition on \(\Omega\). Put

\[ Q=\{u\in\mathfrak U:\mathscr L^{(k)}u\in L_{M_k}(\Omega),\ k=1,\ldots,R\}. \]

Let \(C_0^{(\infty)}(\Omega;s)\subset Q\). Introduce a pair of linear spaces \(V\) and \(\dot V\) such that

\[ C_0^{(\infty)}(\Omega;s)\subset \dot V\subset V\subset Q. \]

The first boundary-value problem for system (2) is posed as follows. A vector-function \(u^{(0)}\in V\) is given. It is required to find a solution \(u\in V\) of system (2) such that \(u-u^{(0)}\in\dot V\).

4°. Conditions for solvability and unique solvability of the first boundary-value problem for system (2) in the space \(V\).

Assume that one more collection of variable \(N\)-functions is given: \(\mathscr E_1(x,w),\ldots,\mathscr E_R(x,w)\), and that, for any natural \(k\le R\), both functions \(\mathscr E_k(x,w)\) and \(\mathscr E_k^*(x,w)\) satisfy the \(\Delta_2'\)-condition on \(\Omega\) and, moreover, \(\mathscr L^{(k)}(\dot V)\subset L_{\mathscr E_k}(\Omega)\).

Introduce on the space \(\dot V\) the seminorm

\[ p(u)=\sum_{k=1}^{R}\|\mathscr L^{(k)}u\|_{L_{\mathscr E_k}(\Omega)}. \tag{3} \]

Assume further the existence of such a decomposition of the space \(\dot V\) into a direct sum of subspaces \(\dot X\) and \(\dot S\), that on the space \(\dot X\) the seminorm (3) is a norm, and \(\dot X\) is a Banach space with respect to this norm. In what follows the decomposition \(\dot V=\dot X\oplus\dot S\) will be regarded as fixed.

We now proceed to formulate the conditions for solvability and unique solvability of the first boundary-value problem for system (2).

I. Each function \(\varphi^{(k)}(x,\xi)\) satisfies the Carathéodory condition and

\[ \sum_{k=1}^{R}\mathscr E_k^*\bigl(x,\varphi^{(k)}(x,\xi+\eta)\bigr) \le B(x)+C\left(\sum_{k=1}^{R}\mathscr E_k(x,\xi_k)+\sum_{k=1}^{R}M_k(x,\eta_k)\right), \]

where \(B(x)\in L_1(\Omega)\), \(C\) is a constant.

II. For some natural number \(A \leq R\) the inequalities hold:

a)
\[ \sum_{k=1}^{R}\varphi^{(k)}(x,\xi+\eta)\xi_k \geq \gamma\left(\sum_{k=1}^{A} M_k(x,\xi_k) -\delta\sum_{k=1}^{R}\mathscr E_k(x,\xi_k)\right) - C\sum_{k=1}^{R} M_k(x,\eta_k)-B(x); \]

b)
\[ \sum_{k=1}^{R}\int_{\Omega} M_k\bigl(x,\mathscr L^{(k)}u\bigr)\,dx \geq \varepsilon\sum_{k=1}^{R}\int_{\Omega}\mathscr E_k\bigl(x,\mathscr L^{(k)}u\bigr)\,dx - K, \qquad \forall u\in \dot X, \]

where \(\gamma,\delta,\varepsilon,C,K\) are constants, \(\gamma>0,\ \varepsilon>\delta\). \(B(x)\in L_1(\Omega)\).

III. The inequality
\[ \sum_{k=1}^{R}\bigl(\varphi^{(k)}(x,\xi)-\varphi^{(k)}(x,\zeta)\bigr)(\xi_k-\zeta_k)\geq 0 \]
holds.

IV. a) For some set \(\mathfrak P\subset\{1,\ldots,R\}\), each operator \(\mathscr L^{(k)}\), \(k\in\mathfrak P\), either maps \(\dot X\) into \(L_{\mathscr E_k}(\Omega)\) completely continuously, or there exists a sequence of measurable sets
\[ \Omega^{(k)}_1\subset\Omega^{(k)}_2\subset\cdots,\quad \ldots,\quad \bigcup_{\nu=1}^{\infty}\Omega^{(k)}_\nu=\Omega,\quad \operatorname{mes}\bigl(\Omega\setminus\Omega^{(k)}_\nu\bigr)>0, \]
such that \(\mathscr L^{(k)}\) maps \(\dot X\) into \(L_{\mathscr E_k}(\Omega^{(k)}_\nu)\) completely continuously;

b) the inequality
\[ \sum_{k=1}^{R} \bigl(\varphi^{(k)}(x,\eta+\xi)-\varphi^{(k)}(x,\eta+\zeta)\bigr)(\xi_k-\zeta_k) \geq \varkappa(x,\eta,\xi^{(0)},\zeta), \]
holds, in which the function \(\varkappa\) is representable in the form
\[ \varkappa(x,\eta,\xi^{(0)},\zeta) = \omega(x,\eta,\xi^{(0)},\zeta) + \sum_{k=1}^{R}\Lambda_k(x,\eta,\xi^{(0)},\zeta)\xi_k, \]
where \(\xi^{(0)}\) is the collection of variables from the space \(E_R\) with indices from the set \(\mathfrak P\), and the functions \(\omega\) and \(\Lambda_k\) satisfy the Carathéodory condition;

c)
\[ \lim_{t\to 0} t^{-1}\varkappa(x,\eta,\xi^{(0)},\xi-t\zeta)=0; \]

d) for the functions \(\omega\) and \(\Lambda_k\) the inequality
\[ |\omega|+\sum_{k=1}^{R}\mathscr E_k^*(x,\Lambda_k) \leq B(x)+C\left( \sum_{k=1}^{R} M_k(x,\eta_k) + \sum_{k=1}^{R}\mathscr E_k(x,\xi_k) + \sum_{k\in\mathfrak P}\lambda_k(x)\mathscr E_k(x,\xi_k) \right), \]
holds, where \(B(x)\in L_1(\Omega)\), \(C\) is a constant, \(\lambda_k(x)\in L_\infty(\Omega)\), \(\lambda_k(x)\geq 0\), and, moreover, if it is known that \(\mathscr L^{(k)}\) maps \(\dot X\) into \(L_{\mathscr E_k}(\Omega)\) completely continuously, then \(\lambda_k(x)\equiv 1\) on \(\Omega\); otherwise
\[ \lim_{\nu\to\infty}\left(\operatorname{vrai\,sup}_{x\in\Omega\setminus\Omega^{(k)}_\nu}\lambda_k(x)\right)=0. \]

V.
\[ \int_{\Omega}\left(\sum_{k=1}^{R}\varphi^{(k)}(x,\mathscr L u)\mathscr L^{(k)}v\right)\,dx=0, \qquad \forall u\in V,\ v\in \dot S. \]

VI.

\[ \int_\Omega \left(\sum_{k=1}^{R} \varphi^{(k)}(x,\mathcal L u+\mathcal L w)\mathcal L^{(k)}v\right)\,dx = \int_\Omega \left(\sum_{k=1}^{R} \varphi^{(k)}(x,\mathcal L u)\mathcal L^{(k)}v\right)\,dx, \]

\[ \forall u\in V,\quad w\in \dot S,\quad v\in C_0^{(\infty)}(\Omega;s). \]

VII. \(C_0^{(\infty)}(\Omega;s)\) is dense in \(\dot V\) with respect to the seminorm (3).

VIII. For all \(u,v\in V\) such that \(u\ne v\) and \(u-v\in \dot X\):

\[ \int_\Omega \left(\sum_{k=1}^{R}\bigl(\varphi^{(k)}(x,\mathcal L u)-\varphi^{(k)}(x,\mathcal L v)\bigr)\mathcal L^{(k)}(u-v)\right)\,dx>0. \]

5°. Theorems on solvability and unique solvability of the first boundary-value problem for system (2) in the space \(V\).

Definition 2. We shall call the right-hand side \(f\) of system (2) admissible if

\[ f_j=\sum_{k=1}^{R}\mathcal L^{(k,j)*}F_k(x),\quad \forall j=1,\ldots,s, \]

where \(F_k\in L_{\mathscr E_k*}(\Omega)\). If, in addition,

\[ \sum_{k=1}^{R}\int_\Omega F_k\mathcal L^{(k)}v\,dx=0,\quad \forall v\in \dot S, \]

then we shall write \(f\perp \dot S\).

Theorem 1 (on solvability of the first boundary-value problem). Let: a) conditions I, II, V be satisfied; b) at least one of the two conditions III or IV be satisfied; c) the right-hand side \(f\) of system (2) be admissible, and \(f\perp \dot S\). Then the first boundary-value problem for system (2) has at least one solution \(u\in V\).

In the following theorem the necessity of the condition \(f\perp S\) for solvability of the first boundary-value problem is considered.

Theorem 2. Let: a) conditions I, V, VII be satisfied; b) the right-hand side \(f\) of system (2) be admissible; c) there exist at least one solution \(u\in V\) of the first boundary-value problem for system (2). Then \(f\perp \dot S\).

Theorem 3 (on unique solvability up to a vector-function from \(\dot S\) of the first boundary-value problem). Let: a) conditions I, II, V—VIII be satisfied; b) the right-hand side \(f\) of system (2) be admissible and \(f\perp \dot S\). Then there exists at least one solution \(u\in V\) of the first boundary-value problem for system (2), and, moreover, the totality of all solutions of the first boundary-value problem can be written in the form \(u+\dot S\).

In conclusion, we note that the density of the set \(C_0^{(\infty)}(\Omega;s)\) in the space \(\dot V\) with respect to the seminorm (3) is a necessary condition for the unique (up to a vector-function in \(\dot S\)) solvability of the first boundary-value problem.

Theorem 4. Let: a) conditions I, II, V be satisfied; b) at least one of the two conditions III or IV be satisfied; c) the right-hand side \(f\) of system (2) be admissible and \(f\perp \dot S\); d) the totality of all solutions of the first boundary-value problem for system (2) can be written in the form: \(u+\dot S\), where \(u\in V\). Then \(C_0^{(\infty)}(\Omega;s)\) is dense in the space \(\dot V\) with respect to the seminorm (3).

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

Received
12 III 1970

CITED LITERATURE

1. Yu. A. Dubinskii, UMN, 23, no. 1 (139) (1968).
2. V. R. Portnov, DAN, 175, no. 2 (1967).
3. V. R. Portnov, Candidate’s dissertation, Novosibirsk State University, 1967.
4. V. R. Portnov, in: Embedding Theorems and Their Applications, Baku, 1966; Moscow, 1970.