Non-Classical Boundary Value Problems for Symmetric Linear Systems of Partial Differential Equations of the First Order

M. A. Galakhov

Any system of partial differential equations can be reduced to a first-order system by introducing new functions. The question of reducing boundary value problems is more complicated.

In this article a class of boundary value problems for first-order systems is singled out. Existence and uniqueness theorems are proved for generalized solutions of these problems. Many classical and non-classical boundary value problems for equations and systems are reducible to the class singled out. The initial ideas are connected with the works of K. Friedrichs [5], A. A. Dezin [1], and with the theory of the abstract Cauchy problem [3].

§ 1. Formulation of the Problem and Main Results

Consider the first-order system

\[ \sum_{i=1}^{n} A^i \frac{\partial u}{\partial x_i} + Bu = f,\quad \text{or } au=f \tag{1} \]

in the \(n\)-dimensional unit cube. Here \(u=(u_1,\ldots,u_N)\) and \(f=(f_1,\ldots,f_N)\) are vector-functions,

\[ \frac{\partial u}{\partial x_i}= \left( \frac{\partial u_1}{\partial x_i},\ldots,\frac{\partial u_N}{\partial x_i} \right), \]

\(A^i, B\) are constant square matrices. All coefficients and functions are real.

Let the matrices \(A^i\) be symmetric. We shall call the symmetric system full if there exists a set of numbers \(\alpha_i\) \((i=1,\ldots,n)\) such that

\[ \sum_{i=1}^{n} \alpha_i A^i + B + B' \]

is positive definite (the prime denotes transposition). Suppose that the \(A^i\) consist of blocks of four types, arranged on the main diagonal, and that for each block its own homogeneous boundary conditions are prescribed.

1. Positive definite block. The components \(u\) with row numbers of the block must vanish for \(x_i=0\), where \(i\) is the number of the matrix containing the block.

2. Negative definite block. The boundary conditions are the same, but for \(x_i=1\). Blocks of these two types will be called definite.

3. Zero block. The boundary conditions are free, i.e., no restrictions are imposed on \(u\).

4. We shall call a block of size \(M\times M\) separable if in it only the elements \(a_{pq}=a_{qp}\) are different from zero \((p=1,\ldots,M-L-1,\ q=M-L,\ldots,M)\), where \(L\) is some number \(0\leq L\leq M-2\). We shall assume that the components

\(u\), corresponding to the cell, have numbers from 1 to \(M\).

Consider the boundary operators

\[ \gamma_p u=\sum_{q=M-L}^{M} a_{pq}u_q,\qquad \overline{\gamma}_q u=\sum_{p=1}^{M-L-1} a_{pq}u_p. \]

Four systems of boundary conditions are possible

\[ 1)\quad \gamma_p u\big|_{x_i=0}=\gamma_p u\big|_{x_i=1}=0 \quad (p=1,\ldots,M-L-1), \]

\[ 2)\quad \gamma_p u\big|_{x_i=0}=\overline{\gamma}_q u\big|_{x_i=1}=0 \quad (p=1,\ldots,M-L-1,\ q=M-L,\ldots,M). \]

The other two systems of conditions are obtained by replacing \(\gamma_p\) by \(\overline{\gamma}_q\). We shall call a system of the cellular structure described a block system. The conditions \(\Gamma\) for system (1) define problem (1) \(\Gamma\). We shall call a complete block system simple if the following conditions are satisfied.

1. If in some rows of one matrix there lies a divisible cell, then in any other matrix either in these rows there lies a divisible cell of the same size and structure, or these rows belong to definite cells of one sign and to zero cells.

2. When two matrices are superposed, definite cells of different signs either do not intersect, or one of them belongs to the other (under superposition the cells are regarded as squares).

Theorem 1. Let system (1) be simple. Then the generalized solution of problem (1) \(\Gamma\) exists and is unique.

For the case of constant coefficients this theorem strengthens the result obtained by A. A. Dezin [1].

1) As is known [1], the distinguished class includes the Dirichlet and Neumann problems for the Laplace equation, and mixed problems for the string and heat equations.

2) Theorem 1 can also be applied to the mixed problem for the ultraparabolic equation

\[ \sum_{i=1}^{k} a_i\frac{\partial u}{\partial t_i} - \sum_{i=1}^{m} b_i\frac{\partial^2 u}{\partial x_i^2} + \sum_{i=1}^{n} c_i\frac{\partial u}{\partial x_i} +lu=f \tag{2} \]

\[ (b_i>0,\ \text{one of the } a_i \text{ is nonzero}),\quad u=0 \text{ for } t_i=0 \text{ or } 1 \text{ depending on the sign of } a_i. \]
On each piece of the boundary with respect to \(x\) there may be either a Dirichlet or a Neumann condition.

1. The class of simple systems includes a first-order hyperbolic system in two independent variables.

2. Theorem 1 is applicable to the equation

\[ \frac{\partial^n u}{\partial x_1\ldots \partial x_n} + a_{n-1}\frac{\partial^{n-1}u}{\partial x_1\ldots \partial x_{n-1}} +\ldots+a_0u=f \tag{3} \]

with the generalized \(n\)-dimensional Goursat problem.

Remark. In all cases, except for the string equation, one may assert that a classical solution of the problem for the system gives a classical solution of the problem for the corresponding equation and conversely (in the case of the string equation, additional smoothness of the solution is needed).

Thus, in the question of equivalence of equations and systems, the smoothness of generalized solutions for smooth right-hand sides becomes decisive. If smoothness is proved, then formal-

equivalence will become complete, and the results obtained for first-order systems may become a tool for studying boundary-value problems for equations and systems of higher order.

Theorem 2. Suppose that system (1) is symmetric, the matrix \(A^1\) consists of two definite cells of different signs, the remaining matrices are separable with nonzero elements in positions not occupied by the cells of the first matrix, and the matrix of lower-order terms is diagonal and positive. Then the generalized solution of problem (1) \(\Gamma\) exists and is unique.

This class includes the boundary-value problem for the Cauchy–Riemann system with lower-order terms

\[ \frac{\partial u_1}{\partial x_1}+\frac{\partial u_2}{\partial x_2}+a u_1=f_1, \]

\[ \frac{\partial u_2}{\partial x_1}-\frac{\partial u_1}{\partial x_2}-b u_2=f_2,\quad a,\ b>0. \tag{4} \]

Theorems 1 and 2 made it possible to consider a mixed problem for a symmetric system solved with respect to the derivative in time

\[ \frac{\partial u}{\partial t}+A^0\frac{\partial u}{\partial x_0}+\sum_{i=1}^{n-2} A^i \frac{\partial u}{\partial x_i}+Bu=f. \tag{5} \]

Tome [4] and a number of other authors considered the mixed problem for (5), prescribing boundary conditions only in two variables—\(t\) and \(x_0\), so that the problem was essentially two-dimensional. We are interested in conditions in all variables.

Theorem 1 immediately singles out a class of mixed problems for (5). To use Theorem 2, consider the homogeneous system (5) as an ordinary differential equation in Hilbert space

\[ \frac{\partial u}{\partial t}=Au,\quad t\in[0,\infty). \tag{6} \]

We shall call a generalized solution of the mixed problem for (5) a solution of the abstract Cauchy problem \(\mathrm{ACP}_1\) for (6) [3]. The role of the operator \(A\) is played by the closure in \(L_2\) of the operator

\[ -\left(A^0\frac{\partial u}{\partial x_0}+\sum_{i=1}^{n-2} A^i\frac{\partial u}{\partial x_i}+Bu\right) \]

on smooth functions defined in the \((n-1)\)-dimensional cube and satisfying the boundary conditions. The following assertion is valid [3].

If \(A\) generates a semigroup of class \(C_0\), then \(\mathrm{ACP}_1\) has a unique solution for \(u_0=u(0)\) belonging to the domain of \(A\).

According to one of the consequences of the Hille–Yosida–Phillips theorem [2], in order that a closed linear operator \(A\) with everywhere dense domain generate a semigroup \(T(t)\) of class \(C_0\) such that \(|T(t)|\le e^{\omega t}\) for some real \(\omega\), it is necessary and sufficient that the inequality

\[ |R(\lambda,A)|\le \frac{1}{\lambda-\omega},\quad \lambda>\omega \]

hold.

Here \(R\) denotes the resolvent of \(A\).

The estimate for the resolvent of the given operator \(A\) is obtained if Theorem 2 is applied to the system

\[ A^0\frac{\partial u}{\partial x_0} +\sum_{i=1}^{n-2} A^i\frac{\partial u}{\partial x_i} +Bu+\lambda u=f . \tag{7} \]

§ 2. PROOFS OF THE THEOREMS1

1. First-order operator. Boundary conditions. A priori inequalities.

Let \(\Omega\) be a bounded domain of \(n\)-dimensional Euclidean space with piecewise smooth boundary \(S\). (In our case \(\Omega\) is the \(n\)-dimensional unit cube.) In \(\Omega\) consider the Hilbert space \(H\) of vector functions with scalar product

\[ (u,v)=\int_{\Omega}[u,v]\,d\Omega,\qquad |u|=(u,u)^{\frac12}, \]

where

\[ [u,v]=\sum_{k=1}^{N} u_k v_k . \]

We shall say that \(u\in C^1\) if \(u\) is continuously differentiable in the closure of \(\Omega\). Consider the differential operator (1), defined on \(u\in C^1\). For \(u,v\in C^1\) the formula

\[ (au,v)=(u,a^*v)+\int_S \sum_{i=1}^{n}\sum_{k=1}^{N}\sum_{l=1}^{N} a_{kl}^i u_k v_l \cos \widehat{n x_i}\,ds \]

holds, where \(a^*\) is the operator formally adjoint to \(a\),

\[ a^*v=-\sum_{i=1}^{n} A^i\frac{\partial v}{\partial x_i}+B'v . \]

Let \(v\in C^1\) satisfy some system of homogeneous linear boundary conditions \(\Gamma^*\). In this case we shall say that \(v\in C^1_{\Gamma^*}\). Suppose that for every \(v\in C^1_{\Gamma^*}\)

\[ \int_S \sum_{i=1}^{n}\sum_{k=1}^{N}\sum_{l=1}^{N} a_{kl}^i u_k v_l \cos \widehat{n x_i}\,ds=0 . \]

Then \(u\in C^1\) must satisfy certain boundary conditions \(\Gamma\).

Let us define the strong and weak extensions of the operators \(a\) and \(a^*\) under the conditions \(\Gamma\) and \(\Gamma^*\): \(u\in H\) is a strong solution of the equation \(au=f\) if there exists a sequence \(u_i\in C^1_{\Gamma}\) such that \(|u_i-u|\to 0\), \(|au_i-f|\to 0\) as \(i\to\infty\). Thus the strong extension is the closure. \(u\in H\) is a weak solution of the equation \(au=f\) if \((u,a^*v)=(f,v)\) for all \(v\in C^1_{\Gamma^*}\). The extensions \(a^*\) are defined analogously.

It is not difficult to verify that, for operators satisfying the conditions of Theorems 1 and 2, the inequalities

\[ |u|\leq C|au|,\qquad u\in C^1_{\Gamma}, \tag{8} \]

\[ |v|\leq C|a^*v|,\qquad v\in C^1_{\Gamma^*}. \tag{9} \]

hold. (The conditions \(\Gamma^*\) are easily formulated and are analogous to the conditions \(\Gamma\).) The inequalities are obtained because of the symmetry of the system and the nonnegative-

ness of the integral over the boundary. The definite term is obtained from the completeness of the system. From inequalities (8) and (9) the following simple propositions follow.

1. The strong solution of \(au=f\) is unique.
2. The range of the closure of \(a\) is closed in \(H\), and its orthogonal complement in \(H\) consists of weak solutions of the equation \(a^{*}v=0\).
3. A weak solution of \(au=f\) exists for any right-hand side from \(H\).

These propositions are also true for \(a^{*}\), since the operators \(a\) and \(a^{*}\) are entirely analogous.

The equivalence of the weak and strong extensions will be proved. In Theorem 1 we shall prove the coincidence of the weak solution of the homogeneous equation with the strong one. In Theorem 2 we shall directly prove that the range of the closure of \(a\) fills \(H\), which is equivalent to the uniqueness of the weak solution.

2. Averaging operators. Let \(\omega(\xi)\) be an averaging kernel, i.e., an even infinitely differentiable function of one variable such that

\[ \omega(\xi)\geq 0,\qquad \omega(\xi)=0\ \text{for }|\xi|\geq 1,\qquad \int_{-1}^{1}\omega(\xi)\,d\xi=1. \]

Introduce the functions

\[ \overset{0}{\omega}_i=\frac{1}{\varepsilon_i}\, \omega\!\left(\frac{x_i-x_i'}{\varepsilon_i}\right),\qquad \overline{\omega}_i=\frac{1}{\varepsilon_i}\, \omega\!\left(\frac{x_i-x_i'-2\varepsilon_i}{\varepsilon_i}\right), \]

\[ \overset{+}{\omega}_i=\frac{1}{\varepsilon_i}\, \omega\!\left(\frac{x_i-x_i'+2\varepsilon_i}{\varepsilon_i}\right),\qquad \dot{\omega}_i=\frac{1}{\varepsilon_i}\, \omega\!\left(\frac{x_i-(1-4\varepsilon_i)x_i'-2\varepsilon_i}{\varepsilon_i}\right), \]

\[ \overset{*}{\omega}_i=\frac{1}{\varepsilon_i}\, \omega\!\left(\frac{(1-4\varepsilon_i)x_i-x_i'+2\varepsilon_i}{\varepsilon_i}\right). \]

We shall call the \(\varepsilon_i\) the radii of averaging. Multiplying \(n\) functions of the indicated form, we obtain a kernel \(W_{\varepsilon}(x,x')\) of the integral operator \(I_{\varepsilon}\), which will act on scalar functions \(u\) from \(H\) (here \(\varepsilon=\max_i \varepsilon_i\)).

Properties of \(I_{\varepsilon}\):

Lemma 1. \(I_{\varepsilon}u\) is a smooth function and \(\lvert I_{\varepsilon}u-u\rvert\to 0\) as \(\varepsilon\to 0\) (for proofs of this and of some subsequent propositions, see [1]).

Remark. The second part of the lemma remains valid if the averaging is not performed with respect to all variables.

Lemma 2. Let \(u,v\in H\). Then

\[ \left(\frac{\partial}{\partial x_i}I_{\varepsilon}u,v\right) =-(1+O(\varepsilon_i)) \left(u,\frac{\partial}{\partial x_i}I_{\varepsilon}^{*}v\right) \]

(\({}^{*}\) denotes the adjoint operator).

Lemma 3. Let \(u\in H\). Then

\[ \left|\int_{\Omega}W_{i,\varepsilon}'(x,x')u(x')\,dx'\right|\to 0 \qquad \text{as }\varepsilon\to 0, \]

where \(W_{i,\varepsilon}'\) is obtained from \(W_{\varepsilon}\) by replacing \(\omega_i\) by \(\dfrac{1}{\varepsilon_i}\omega'(\ )\) (the arguments \(\omega_i\) stand in the parentheses).

The averaging operator on vector functions \(u\) will be constructed as
\[ Iu=(I_1u_1,\ldots,I_Nu_N), \]
where \(I_k\) \((k=1,\ldots,N)\) are certain averaging operators, and we have omitted the index \(\varepsilon\). Also
\[ I^{*}v=(I_1^{*}v_1,\ldots,I_N^{*}v_N). \]
By choosing the multipliers in the kernels of the operators, one can always arrange that \(Iu\) belong to \(C_{\Gamma}^{1}\) for every \(u\) from \(H\).

1. We turn to the proof of Theorem 1. Let \(u\) be a weak solution, \(au=0\), i.e. \((u,a^*v)=0\) for every \(v\in C^1_{\Gamma^*}\). Take \(v\) in the form \(j^*v\), where \(v\in \overset{\circ}{H}\), and choose the averaging operator \(j\) so that \(j^*v\) belongs to \(C^1_{\Gamma^*}\). We apply Lemma 2 to the expression \((u,a^*I^*v)\). We obtain \(\tilde a^k j_k=0\) \((k=1,\ldots,N)\), where \(a^k\) is the operator of the \(k\)-th row of \(a\), and the tilde means that, for separable blocks, factors \(1+O(\varepsilon_i)\) have appeared. Now from \(\tilde a^k j_k u=0\) one must pass to \(|a I_\varepsilon u|\to0\) as \(\varepsilon\to0\), where \(I_\varepsilon u\in C^1_\Gamma\). For this it is necessary, in some components of \(u\), to replace “foreign” averaging operators by “own” ones (if necessary, with factors of order one). For a better understanding of the subsequent reasoning it is useful to have before one’s eyes some concrete system.

Suppose that the matrix \(A^i\) contains a separable block. In it the operators need not be replaced, by virtue of the special structure of the system (condition 1) and the choice of averaging kernels. Nor is it necessary to replace the operators in the other separable blocks lying in the given rows. We shall regard as fixed the numbers of the components of \(u\) corresponding to the given separable blocks. Consider the remaining (definite and zero) blocks to which the chosen rows belong. By the structure of the system, comparing these blocks by superposition of matrices, it is always possible to choose among them a highest one (in the matrix \(A^j\))—the one to which, under this superposition, all the others will belong. In this block, for the “own” \(I\) and “foreign” \(J\) operators (we omit indices), the \(j\)-th kernels always coincide, and the difference

\[ \frac{\partial}{\partial x_j}(J-I)u \]

can be made small by taking the remaining kernels outside the differentiation sign and letting their averaging radii tend to zero faster than \(\varepsilon_j\). Among the remaining blocks we again choose the highest one, repeat the operation, and so on. The fastest to tend to zero will be the averaging radii \(\varepsilon_i\) in the variables of the separable blocks. For this reason, in the remaining blocks one may place factors of the form \(1+O(\varepsilon_i)\). Thus we shall go through all rows where there are separable blocks.

In the remaining rows take a definite block. We shall regard as fixed the numbers of the components of \(u\) corresponding to this block. Consider the remaining blocks to which the given rows belong. By the structure of the system (condition 2), among them one can choose a highest one. Replace the operators in it as above. Among the remaining blocks again choose a highest one, and so on. At the last step all averaging radii have already been used, but all the kernels of the “own” and “foreign” operators also coincide.

In replacing the operators we used several different averaging radii. The replacement of operators in lower-order terms is trivial. Theorem 1 is proved.

1. We turn to Theorem 2. Suppose there exists \(v\in H\) such that \((au,v)=0\) for all \(u\) in the domain of definition of \(a\). We shall prove that \(v=0\).

Suppose that in the matrix \(A^1\) a positive definite block lies at the top and occupies rows from the 1st to the \(p\)-th. Denote the first variable by \(t\). Choose \(u\) of the special form:

\[ u_1=\int_0^t I v_1\,d\tau,\qquad u_{p+1}=\int_t^1 I^*v_{p+1}\,d\tau, \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

\[ u_p=\int_0^t I v_p\,d\tau,\qquad u_N=\int_t^1 I^*v_N\,d\tau, \]

\(I\) is the averaging operator over all variables except \(t\). It is easy to see that, with a suitable choice of \(I\), \(u\) will belong to the domain of definition of the closure of \(a\).

The equality \((au,v)=0\) can be written as follows:

\[ \left(A_+^1\frac{\partial \overset{+}{u}}{\partial t},\overset{+}{v}\right) + \left(A_-^1\frac{\partial \overset{-}{u}}{\partial t},\overset{-}{v}\right) + \sum_{i=2}^{n} \left(A^i\frac{\partial u}{\partial x_i},v\right) +(Du,v)=0, \tag{10} \]

where \(A_+^1\) and \(A_-^1\) are blocks of the matrix \(A^1\), and \(\overset{+}{u}\) and \(\overset{-}{u}\) are the corresponding groups of components of \(u\); \(D\) is the diagonal matrix of lower-order terms. The third term of equality (10) contains only expressions of the form

\[ \left( a_{kl}^{i}\frac{\partial}{\partial x_i}\int_t^1 I^*v_l\,d\tau,\ v_k \right) + \left( a_{lk}^{i}\frac{\partial}{\partial x_i}\int_0^t Iv_k\,d\tau,\ v_l \right), \]

which tend to zero as \(\varepsilon\to0\). Indeed,

\[ \left( \frac{\partial}{\partial x_i}\int_t^1 I^*v_l\,d\tau,\ v_k \right) = \left( \int_0^t v_k\,d\tau,\ \frac{\partial}{\partial x_i}I^*v_l \right) = \]

\[ =-(1+O(\varepsilon_i)) \left( \frac{\partial}{\partial x_i}\int_0^t Iv_k\,d\tau,\ v_l \right). \]

Here we have integrated by parts with respect to \(t\) and used Lemma 2. The term with \(O(\varepsilon_i)\) tends to zero by Lemma 3.

The first two terms of (10) give the definite part. For example,

\[ \left(A_+^1\frac{\partial \overset{+}{u}}{\partial t},\overset{+}{v}\right) = (A_+^1 I\overset{+}{v},\overset{+}{v}) = (IA_+^1\overset{+}{v},\overset{+}{v}) = \]

\[ = (A_+^1\overset{+}{v},\overset{+}{v}) + (IA_+^1\overset{+}{v}-A_+^1\overset{+}{v},\overset{+}{v}) \to (A_+^1\overset{+}{v},\overset{+}{v}). \]

The last term of (10) contains expressions \((au_i,v_i)\), where \(a>0\). Let us consider, for example, \((u_1,v_1)\).

\[ (u_1,v_1) = \left(\int_0^t Iv_1\,d\tau,\ v_1\right) = \left(I\int_0^t v_1\,d\tau,\ v_1\right) = \]

\[ = \left(\int_0^t v_1\,d\tau,\ v_1\right) + \left(I\int_0^t v_1\,d\tau-\int_0^t v_1\,d\tau,\ v_1\right). \tag{11} \]

The second term tends to zero, since \(I\) is the averaging operator:

\[ \left(\int_0^t v_1\,d\tau,\ v_1\right) = -\left(v_1,\int_0^t v_1\,d\tau\right) + \int_{\Omega} \left(\int_0^1 v_1\,dt\right)^2 dx_2\ldots dx_n, \]

i.e., the first term in (11) is nonnegative.

As a result, the whole expression (10) is represented in the form of a sum of terms definite with respect to \(v\), nonnegative, and infinitesimal. Hence it follows that \(v=0\).

Theorem 2 is proved.

The author takes this opportunity to express his gratitude to his scientific adviser A. A. Dezin.

References

1. Dezin A. A. Boundary value problems for some symmetric linear systems of first order. Mat. sb., 49, no. 4, 1959.
2. Dunford N. and Schwartz J. T. Linear operators. General theory. IL, 1962.
3. Hille E. and Phillips R. Functional analysis and semigroups. IL, 1962.
4. Thomée V. Archive for Rational Mechanics and Analysis. 13, no. 2, 1963.
5. Friedrichs K. Symmetric hyperbolic linear differential equations. Comm. Pure and Appl. Math. 7, no. 2, 1954.

Received by the editors
May 3, 1965

Moscow Institute of Physics and Technology

1. The exposition in items 1 and 2 largely follows [1]. ↩