UDC 517.944

ON SOME SYSTEMS OF DIFFERENTIAL EQUATIONS OF MIXED AND MIXED-COMPOUND TYPES

V. P. Didenko

In the present paper we consider systems of second-order partial differential equations of mixed type
\(G(y)v_{xx}-v_{yy}+Av_x+Bv_y+Cv=h\). For such systems, under certain conditions on the coefficients in domains of a special form, a boundary-value problem is posed; the existence and uniqueness are proved of a solution which has first derivatives in the domain, square-integrable, and which satisfies the boundary conditions in the mean.

The paper also gives an example of a system of differential equations of mixed-compound type (the terminology is from [3]). For this system analogous questions are analyzed. It is characteristic that, in posing the first boundary-value problem for a system of mixed-compound type, the boundary conditions for part of the components of the solution vector are prescribed on the whole boundary, while for the remaining components the boundary conditions are prescribed not on the whole boundary; whereas, in posing the first boundary-value problem for a mixed system, the boundary conditions are prescribed at once for all components of the solution vector not on the whole boundary.

In § 1 a proof is given of an inequality for a differential operator of mixed type. This inequality, as well as the assertions of § 4, is then used in § 2 to prove existence and uniqueness of the solution of the first boundary-value problem for systems of differential equations of mixed type.

In § 3 analogous questions are considered for one example of a system of differential equations of mixed-compound type.

§ 1. PROOF OF THE BASIC INEQUALITY

Consider the system of second-order partial differential equations in the plane of variables \((x,y)\):

\[ Lu = G(y)v_{xx}-v_{yy}-C(x,y)v=h(x,y), \tag{1.1} \]

where \(G(y)\), \(C(x,y)\) are given symmetric square matrices satisfying the conditions:
\(C(x,y)\leqslant 0,\ \dfrac{\partial C}{\partial x}\geqslant 0,\ \dfrac{\partial C}{\partial y}\geqslant 0;\quad G(y)>0\)

for \(y>0\), \(G(y)<0\) for \(y<0\); \(\dfrac{\partial G(y)}{\partial y}>0\), \(h(x,y)\) is a given vector,

\(v=\{v_1,\ldots,v_n\}\) is the unknown vector.

Matrix inequalities here and below are understood as inequalities for the corresponding quadratic forms constructed on arbitrary vectors of nonzero length.

In the upper half-plane the characteristic polynomial for system (1.1)

\[ \det |G(y)-\lambda^{2}E| \tag{1.2} \]

has \(n\) real roots \(\lambda_1^{2},\ldots,\lambda_n^{2}\). Thus, from each point of the axis \(y=0\) into the half-plane \(y>0\) there issue two families of characteristics: \(n\) “nonnegative” ones (denote them by \(\lambda_1^{+},\ldots,\lambda_n^{+}\)) and \(n\) “nonpositive” ones (denote them by \(\lambda_1^{-},\ldots,\lambda_n^{-}\)).

In the lower half-plane the characteristic polynomial for system (1.1) has no real roots. By a mixed domain we shall mean a domain containing within itself an interval of the axis \(y=0\).

On the axis \(y=0\) take two arbitrary points \(A(x,0)\) and \(B(x_2,0)\), and consider in the half-plane \(y>0\) twice continuously differentiable curves: one passing through the point \(A\) so that along it

\[ \frac{dx}{dy}>\sup_{\lambda_i^{+}}\{\lambda_1^{+},\ldots,\lambda_n^{+}\}=\Lambda^{+}(y), \tag{1.3} \]

and another passing through the point \(B\) so that along it

\[ 0>\frac{dx}{dy}>\sup_{\lambda_i^{-}}\{\lambda_1^{-},\ldots,\lambda_n^{-}\}. \tag{1.4} \]

We choose the directions of these lines so that they intersect. Denote their point of intersection by \(C(x_0,y_0)\).

In the lower half-plane \((y<0)\), from the point \(A\) to the point \(B\) draw a twice continuously differentiable curve \(\sigma\) so that along it

\[ \left|\frac{dx}{dy}\right|>\sup_{0<y<y_0}\Lambda^{+}(y), \tag{1.5} \]

where \(y_0\) is the ordinate of the point \(C\).

Denote by \(\Omega\) the simply connected mixed domain of the plane of the variables \((x,y)\), bounded in the lower half-plane \(y<0\) by the line \(\sigma\) with endpoints at \(A\) and \(B\), and in the upper half-plane \((y>0)\) by the lines \(AC\) and \(BC\).

By \(\dot C^s\) denote the set of \(s\)-times continuously differentiable vectors in the closed domain \(\overline{\Omega}\) satisfying the condition

\[ v\big|_{AC+\sigma}=0. \tag{1.6} \]

By \(W_2^s\) we shall denote the space (of S. L. Sobolev) with norm

\[ |v|_{W_2^s} = \left\{ \int_{\Omega}\sum_{k=0}^{s}[D^k v\cdot D^k v]\,d\Omega \right\}, \]

where \(D^k\) denotes differentiation of order \(k\). \(W_2^0\) will denote the Hilbert space of square-integrable vectors (with scalar product in it

\[ (v\cdot u)=\int_{\Omega}[v_1u_1+\cdots+v_nu_n]\,d\Omega \]
).

Theorem 1. For vectors \(v\in \dot C^2\) the inequality

\[ |Lv|_{W_2^0}\ge C_1 |v|_{W_2^1}. \tag{1.7} \]

holds.

Proof. We multiply scalarly the vector \(Lv\) by the vector
\(\left\{b\dfrac{\partial v}{\partial x}+c\dfrac{\partial v}{\partial y}\right\}\), where \(b>0\), \(c>0\) (\(b,c\) are functions to be defined below), and integrate over the domain \(\Omega\). After grouping terms and applying Green’s formula we obtain

\[ \begin{aligned} \iint\limits_{\Omega}\left\{b\frac{\partial v}{\partial x}+c\frac{\partial v}{\partial y}\right\}Lu\,d\Omega ={}&\frac12\int\limits_{\Gamma}(bn_x+cn_y)^{-1} \left\{n_y\frac{\partial v}{\partial x}-n_x\frac{\partial v}{\partial y}\right\}\times\\ &\times\{Eb^2-c^2G\}\left\{n_y\frac{\partial v}{\partial x}-n_x\frac{\partial v}{\partial x}\right\}\,d\Gamma\\ &+\frac12\int\limits_{\Gamma}-\{bn_x+cn_y\}^{-1}\times\\ &\times\left\{b\frac{\partial v}{\partial x}+c\frac{\partial v}{\partial y}\right\} \{En_y^2-n_x^2G\} \left\{b\frac{\partial v}{\partial x}+c\frac{\partial v}{\partial y}\right\}\,d\Gamma\\ &+\int\limits_{\Omega}\left\{ \frac12\frac{\partial v}{\partial y}\frac{\partial c}{\partial y}\frac{\partial v}{\partial y} +\frac{\partial v}{\partial x}\frac{\partial b}{\partial y}\frac{\partial v}{\partial y} +\frac12\frac{\partial v}{\partial x}\frac{\partial(cG)}{\partial y}\frac{\partial v}{\partial x}\right.\\ &\left.\qquad -\frac12\frac{\partial v}{\partial x}\frac{\partial(bG)}{\partial x}\frac{\partial v}{\partial x} -\frac{\partial v}{\partial x}\frac{\partial(cG)}{\partial x}\frac{\partial v}{\partial x} -\frac12\frac{\partial v}{\partial y}\frac{\partial b}{\partial x}\frac{\partial v}{\partial y} \right\}d\Omega\\ &-\int\limits_{\Omega}(vbC,v)_x\,d\Omega -\int\limits_{\Omega}(vcC,v)_y\,d\Omega +\int\limits_{\Omega}v(bC)_xv\,d\Omega +\int\limits_{\Omega}v(cC)_yv\,d\Omega . \end{aligned} \tag{1.8} \]

Here \(\{n_x,n_y\}\) denotes the unit vector of the outward normal to the boundary \(\Gamma\) of the domain \(\Omega\). We now choose \(b\) and \(c\) so that

\[ \frac{dx}{dy}>\frac{b}{c}>\sup_{0\le y\le y_0}\Lambda^{+}(y), \tag{1.3'} \]

and along \(\sigma\) one has

\[ \frac{b}{c}<\left|\frac{dx}{dy}\right|. \]

Then, by virtue of (1.3′), we obtain that \(Eb^2-c^2G(y)>0\) on \(\Gamma\), while on \(AC+\sigma\) the inequality \(bn_x+cn_y<0\) holds. On the boundary \(BC\), since \(n_x>0,\ n_y\ge 0\), we have \(bn_x+cn_y>0\). Moreover, on \(BC\), by (1.4), the matrix \(En_y^2-n_x^2G(y)\) is nonpositive, and on \(AC+\sigma\), by (1.3), the matrix \(En_y^2-n_x^2G(y)\) is nonnegative. Therefore the first and second integrals on the right-hand side of (1.8) are nonnegative, with \(v|_{AC+\sigma}=0\). In order to obtain nonnegativity of the remaining integrals on the right-hand side of (1.8), we put \(b=b_0-2\sigma x,\ c=c_0-\delta y\), where \(b_0,c_0,\delta\) are positive constants. It is easy to verify that, if \(\delta\) is sufficiently small, then the third integral is estimated from below by the quantity

\[ c_1\iint\limits_{\Omega} \left\{\left(\frac{\partial v}{\partial x}\right)^2+ \left(\frac{\partial v}{\partial y}\right)^2\right\}\,d\Omega, \]

where \(c_1\) is some positive constant. The remaining integrals on the right-hand side of (1.8) are nonnegative. Thus we obtain

\[ \int_{\Omega}\left\{\,b\frac{\partial v}{\partial x}+c\frac{\partial v}{\partial y},\,Lv\right\}d\Omega \geq c_2\int_{\Omega}\left\{\left(\frac{\partial v}{\partial x}\right)^2+ \left(\frac{\partial v}{\partial y}\right)^2\right\}d\Omega . \tag{1.9} \]

It is easy to verify that, for continuously differentiable functions satisfying condition (1.6), the inequality

\[ \int_{\Omega}\left(\frac{\partial v}{\partial x}\right)^2d\Omega \geq c_3\int_{\Omega}v^2\,d\Omega \tag{1.10} \]

holds.

Indeed, enclose the domain \(\Omega\) in a rectangle, and extend the vector \(u\) continuously outside the domain \(\Omega\), but inside the rectangle, as follows: a) in the curvilinear triangle bounded by the line \(CB\) and the sides of the rectangle, in a manner constant with respect to \(x\); b) in the remaining parts, by zero. Obviously, we have

\[ v=\int_a^x \frac{\partial v}{\partial x}\,dx . \]

Squaring both sides of this inequality and applying Schwarz’s inequality to the right-hand side, we have

\[ v^2 \leq \operatorname{const}\int_a^b\left(\frac{\partial v}{\partial x}\right)^2dx \]

(here \(a\) and \(b\) are, respectively, the width and the length of the rectangle). Finally, integrating both sides of the last inequality over the domain \(\Omega\),

\[ \int_{\Omega}v^2\,d\Omega \leq \operatorname{const}\int_{\Omega}\left\{\int_a^b\left(\frac{\partial v}{\partial x}\right)^2dx\right\}d\Omega . \tag{1.11} \]

Using the fact that the integral of \(\left(\dfrac{\partial v}{\partial x}\right)^2\) over the domain \(\Omega\) is equal to the integral of \(\left(\dfrac{\partial v}{\partial x}\right)^2\) over the rectangle, from (1.11) we obtain the required result. Taking into account inequality (1.10) and the obvious inequality

\[ \|v\|_{W_2^1}\geq \operatorname{const}\,\|v\|_{W_2^0}, \]

from (1.9) we obtain

\[ \int_{\Omega}\left\{\,b\frac{\partial v}{\partial x}+c\frac{\partial v}{\partial y},\,Lv\right\}d\Omega \geq c_4\|v\|_{W_2^1}. \tag{1.12} \]

Applying Schwarz’s inequality to the left-hand side of (1.12) and the inequality

\[ \left\|b\frac{\partial v}{\partial x}+c\frac{\partial v}{\partial y}\right\|_{W_2^0} \leq \operatorname{const}\,\|v\|_{W_2^1}, \]

and then canceling by \(\|v\|_{W_2^1}\), we obtain the desired inequality. Theorem 1 is completely proved.

§ 2. Statement of the Problem, Existence, Uniqueness

Consider system (1.1) in a mixed domain \(\Omega\), for which Theorem 1 of § 1 holds. We divide the boundary \(\Gamma\) of the domain \(\Omega\) into two parts: \(\Gamma=\Gamma_*\cup\Gamma_+\), where by \(\Gamma_*\) we denote that part of the boundary \(\Gamma\) on which the boundary conditions will be prescribed, and by \(\Gamma_+\) the remaining part of the boundary.

Problem 1. In the domain \(\Omega\), find a vector \(u\in W_2^1\) which is the limit, in the metric of \(W_2^1\), of a sequence of vectors \(\{u^i\}\in W_2^2\) satisfying the boundary condition

\[ \left.u^i\right|_{\Gamma_*}=0, \]

and such that \(f_i \equiv Lu^i\) tends to \(f\) in the metric of \(W_2^0\).

In what follows, by a solution of the problem \(Lu=f\), \(\left.u\right|_{\Gamma_*}=0\) we shall mean a solution in the above sense, and shall call it a strong solution. In § 4 it will be proved that

Theorem 2. In a mixed domain \(\Omega\) with boundary \(\Gamma=\Gamma_*\cup\Gamma_+\) (\(\Omega\) and \(\Gamma\) are defined precisely in § 4) there exists a unique solution \(u\in W_2^2\) of the problem

\[ Tu=f,\qquad \left.u\right|_{\Gamma_*}=0, \]

if \(f\in W_2^1\). (Here by \(Tu=f\) is denoted system (1.1) with \(C=0\).)

It is easy to verify that for the domain \(\Omega\) described in § 4, Theorem 1 holds. Relying on Theorems 1 and 2, we prove the following assertion.

Theorem 3. In a mixed domain \(\Omega\), the problem

\[ Lu=f,\qquad \left.u\right|_{\Gamma_*}=0 \tag{2.1} \]

has, moreover, a unique strong solution. Here by \(Lu=f\) is denoted system (1.1), with \(f\in W_2^0\).

Before proving Theorem 3, we formulate some evident lemmas.

Lemma 1. Let \(u^i\) be a strong solution of the problem

\[ Lu^i=f_i,\qquad \left.u^i\right|_{\Gamma_*}=0, \]

where \(f_i\in W_2^1\), and let the sequence \(f_i\) converge in the metric of \(W_2^0\) to \(f\). Then there exists a strong solution of the problem

\[ Lu=f,\qquad \left.u\right|_{\Gamma_*}=0. \]

The proof of this lemma is obtained by applying Theorem 1 and passing to the limit.

Lemma 2. A strong solution of the problem

\[ Tu=f-Cu,\qquad \left.u\right|_{\Gamma_*}=0 \]

is also a strong solution of the problem

\[ Tu+Cu=f,\qquad \left.u\right|_{\Gamma_*}=0. \]

The proof is obvious.

Proof of Theorem 3. Using Theorem 2 and Lemma 1, we conclude that there exists a unique solution of the problem

\[ Tu_0=f,\qquad u_0\big|_{\Gamma_*}=0\quad (f\in W_2^0). \tag{2.2} \]

Let us now consider the problem

\[ Tu-\lambda Cu=f,\qquad u\big|_{\Gamma_*}=0, \tag{2.3} \]

where \(\lambda\) is a parameter, \(0\le \lambda\le 1\). We apply the process of iteration. The problem
\(Tu'=f+\lambda Cu^0,\; u'\big|_{\Gamma_*}=0\), where \(u_0\) is the solution of problem (2.2), by Theorem 2 and Lemma 1 has a strong solution. Continuing the process further, we conclude that there exists a strong solution of the problem

\[ Tu^i=f+\lambda Cu^{i-1},\qquad u^i\big|_{\Gamma_*}=0. \]

We shall show that if \(\lambda\) is sufficiently small, then the sequence \(\{u^i\}\) converges in the metric of \(W_2^1\). Indeed, we have

\[ \left|Tu^i-Tu^{i-1}\right|_{W_2^0} = \left|\lambda C\left(u^{i-1}-u^{i-2}\right)\right|_{W_2^0} \le \lambda^2 C_5\left|u^{i-1}-u^{i-2}\right|_{W_2^1}. \]

Using Theorem 1, we obtain

\[ c_1\left|u^i-u^{i-1}\right|_{W_2^1} \le \lambda^2 C_5\left|u^{i-1}-u^{i-2}\right|_{W_2^1}. \tag{2.4} \]

Here the positive constant \(c_1\) does not depend on the matrix \(C\) from (1.1).

We now fix \(\lambda^2\), taking it equal to \(\lambda_0^2<\dfrac{c_1}{c_5}\). Using Lemmas 1, 2, we obtain the proof of Theorem 3 for problem (2.3). Suppose now that there exists a strong solution of the problem

\[ Tu-\lambda_0(n-1)Cu=f,\qquad u\big|_{\Gamma_*}=0. \]

Next consider the problem

\[ Tu-\lambda_0 nCu=f\qquad Tu-\lambda_0(n-1)Cu=f+\lambda_0Cu, \]

\[ u\big|_{\Gamma_*}=0 \tag{2.5} \]

(here \(n\) is an integer, \(>0\)). As before, iterating and using Theorem 1, we obtain the inequality

\[ c_1\left|u^i-u^{i-1}\right|_{W_2^1} \le \lambda_0^2 c_5\left|u^{i-1}-u^{i-2}\right|_{W_2^1}. \]

Here \(c_1, c_5\) are the same as in inequality (2.4). Thus the sequence \(\{u^i\}\) converges in the metric of \(W_2^1\) and, by Lemmas 1, 2, determines a strong solution of problem (2.5). Since \(\lambda_0\) is fixed, by choosing \(n\) appropriately we obtain a strong solution of problem (2.1). Thus Theorem 3 is completely proved. Let us now consider, in the same domain \(\Omega\), the system

\[ G(y)u_{xx}-u_{yy}+A(x,y)u_x+B(x,y)u_y+C(x,y)u=f(x,y), \tag{2.6} \]

where \(G\) satisfies all the conditions listed at the beginning of § 1, the matrices \(A, B, C\) are square and sufficiently “small” (the “smallness” will be defined precisely below), and \(f\in W_2^0\).

Introducing the notation \(Tu=Gu_{xx}-u_{yy}\), \(\widehat T u=Au_x+Bu_y+Cu\), the system (2.6) can be rewritten in the form \(Tu+\widehat T u=f\). Using the solution \(u_0\) of problem (2.2), we consider the problem

\[ Tu=f-\widehat T u_0,\qquad u\big|_{\Gamma_*}=0. \tag{2.7} \]

Applying Theorem 2 and Lemma 1, we conclude that this problem has a unique strong solution. Iterating further, we obtain a strong solution of the problem

\[ Tu^i=f-\widehat T u^{i-1},\qquad u^i\big|_{\Gamma_*}=0. \]

We shall show that the sequence \(\{u^i\}\) converges in the metric \(W_2^1\). Indeed, applying Theorem 1, we have

\[ c_1\left|u^i-u^{i-1}\right|_{W_2^1} \le \left|T\left(u^i-u^{i-1}\right)\right|_{W_2^0} = \]

\[ = \left|\widehat T\left(u^{i-1}-u^{i-2}\right)\right|_{W_2^0} \le c_6\left|u^{i-1}-u^{i-2}\right|_{W_2^1}. \]

Now, if the matrices \(A,B,C\) are so “small” that the inequality \(c_6<c_1\) is satisfied, then the sequence \(\{u^i\}\) converges and has only one limit. Using the preceding inequality, we obtain uniqueness. Thus we have proved

Theorem 4. Problem (2.7) in the domain \(\Omega\) has, moreover, a unique strong solution if the matrices \(A,B,C\) are sufficiently “small.”

§ 3. An Example of a System of Mixed-Compound Type

In a domain \(\Omega\) of the \((x,y)\)-plane, consider the system of differential equations of second order

\[ Lu\equiv G(y)u_{xx}-u_{yy}+Au_x+Bu_y+Cu=f. \tag{3.1} \]

Here \(G(y)\), \(A(x,y)\), \(B(x,y)\), \(C(x,y)\) are square matrices of size \(m\), \(f=\{f_1(x,y),\ldots,f_m(x,y)\}\). The system (3.1) is called a system of mixed-compound type in the domain \(\Omega\) (by analogy with [3], where third-order equations are considered) if its characteristic polynomial \(\det(G(y)-\lambda^2)\) has no real roots in one part of the domain \(\Omega\), while in the remaining part of the domain, along with real roots, it also has complex roots. Let us consider one such example of a system of mixed-compound type.

Suppose that the matrix \(G(y)\) of the system (3.1) has the form

\[ G(y)= \begin{pmatrix} G_1 & 0\\ 0 & G_2 \end{pmatrix}, \]

where \(G_1(y)\) is a square matrix of size \(n\), has the same properties as the matrix \(G(y)\) at the beginning of § 1, and the matrix \(G_2(y)\), square of size \(m-n\), is negative definite. The vector \(f\) is square integrable in \(\Omega\).

We shall consider the system (3.1) in the domain \(\Omega\) with boundary \(\Gamma=\Gamma_*\cup\Gamma_+\), which is described in detail in § 4.

Problem II. In the domain \(\Omega\), find a vector \(u\in W_2^1\) which is the limit, in the sense of the metric \(W_2^1\), of a sequence of vectors \(\{u^i\}\in \overset{\circ}{W}{}_2^2\), satisfying the boundary conditions:

\[ u_j^i\big|_{\Gamma_*}=0\quad (j=1,\ldots,n),\qquad u_j^i\big|_{\Gamma}=0\quad (j=n+1,\ldots,m), \]

and such that \(f_i \equiv Lu^i\) tends to \(f\) in the metric of \(W_2^0\). In what follows, by a solution of the problem
\[ Lu=f,\quad u_j\big|_{\Gamma_*}=0\quad (j=1,\ldots,n),\quad u_j\big|_{\Gamma}=0\quad (j=n+1,\ldots,m) \]
we shall mean a solution in the above sense, and shall call it strong.

Theorem 4. Let the matrix \(G(y)\) be continuously differentiable in the closed domain \(\overline{\Omega}\), let \(f\in W_2^0\), and let the matrices \(A,B,C\) be sufficiently “small.” The “smallness” will be specified below. Then the problem

\[ Lu=f,\quad u_j\big|_{\Gamma_*}=0\quad (j=1,\ldots,n),\quad u_j\big|_{\Gamma}=0\quad (j=n+1,\ldots,m) \tag{3.2} \]

has, moreover, a unique strong solution.

Proof. Using Theorem 2, Lemma 1 of § 2, and the solvability of the first boundary-value problem for strongly elliptic systems, we obtain that the problem

\[ G(y)u_{xx}-u_{yy}=f,\quad u_j\big|_{\Gamma_*}=0\quad (j=1,\ldots,n), \]

\[ u_j\big|_{\Gamma}=0\quad (j=n+1,\ldots,m) \tag{3.3} \]

has a unique strong solution. Introducing the notation \(Tu=Gu_{xx}-u_{yy}\), \(\widehat{T}u=Au_x+Bu_y+Cu\), we rewrite system (3.1) as: \(Tu+\widehat{T}u=f\).

Consider now the problem

\[ Tu+\widehat{T}u^0=f,\quad u_j\big|_{\Gamma_*}=0\quad (j=1,\ldots,n), \]

\[ u_j\big|_{\Gamma}=0\quad (j=n+1,\ldots,m), \tag{3.4} \]

where \(u^0\) is the strong solution of problem (3.3). Since \(f-\widehat{T}u^0\in W_2^0\), problem (3.4) has a strong solution (for the same reasons as problem (3.3)). Iterating further, we obtain that there exists a strong solution of the problem

\[ Tu^i=f-\widehat{T}u^{i-1}, \]

\[ u_j^i\big|_{\Gamma_*}=0\quad (j=1,\ldots,n),\qquad u_j^i\big|_{\Gamma}=0\quad (j=n+1,\ldots,m). \]

Let us show that the sequence \(\{u^i\}\) converges in the metric \(W_2^1\). Indeed, obviously, we have

\[ \left|Tu^i-Tu^{i-1}\right|_{W_2^0} = \left|\widehat{T}u^{i-1}-\widehat{T}u^{i-2}\right|_{W_2^0} \le c_7\left|u^{i-1}-u^{i-2}\right|_{W_2^1}. \]

Applying Theorem 1, we obtain

\[ c_8\left|u^i-u^{i-1}\right|_{W_2^1} \le c_7\left|u^{i-1}-u^{i-2}\right|_{W_2^1}. \]

Let now the matrices \(A,B,C\) be so small that the inequality \(c_7<c_8\) is satisfied; then the sequence \(\{u^i\}\) converges in the metric \(W_2^1\). Further, using the same means as in § 2, we obtain that the sequence \(\{u^i\}\) determines the unique solution of problem (3.2). The theorem is completely proved.

§ 4. THE FIRST BOUNDARY-VALUE PROBLEM FOR A SPECIAL SYSTEM OF DIFFERENTIAL EQUATIONS OF MIXED TYPE

In the \((x,y)\)-plane consider a domain \(\Omega\) with boundary \(\Gamma=\Gamma_0\cup\Gamma_-\cup\Gamma_+\), where

\[ \Gamma_0:\ (x-y)^2-R^2=0,\quad R\text{ is a positive number}, \qquad x_0\le x\le x_+, \]

\[ \Gamma_-:\ x=x_0, \]

\[ \Gamma_+:\ x=x_+, \]

and we require that \(x_0<-R,\ En_2^2-n_1^2G>0\) on \(\Gamma_0\). Here \(E\) is the identity matrix, \(n=\{n_1,n_2\}\) is the vector of the exterior normal to \(\Gamma\). Denote \(\Gamma_*=\Gamma_0\cup\Gamma_-\).

Problem. Find a vector \(u\in W_2^2\) satisfying system (1.1) in the domain \(\Omega\) for \(C=0\) and vanishing on \(\Gamma_*\).

We shall show that the existence and uniqueness of this problem can be proved by reduction to Friedrichs’ theory [4]. To this end we give the result of [4] as applied to the present case.

In a domain \(\Omega\) of the \((x^1,x^2)\)-plane, consider the system of first-order differential equations

\[ Ku=2\sum_\rho \alpha^\rho \frac{\partial u}{\partial x^\rho}+\gamma u=f, \]

where \(\alpha^\rho,\gamma\) are given \(n\)-dimensional square matrices depending on \((x^1,x^2)\), and \(f\) is a given \(n\)-dimensional vector depending on \((x^1,x^2)\). The matrices \(\alpha^\rho,\gamma\) satisfy the following conditions.

I. The matrices \(\alpha^\rho\) are symmetric.

II. The symmetric part of the matrix

\[ k=\gamma-\sum_\rho \frac{\partial \alpha^\rho}{\partial x^\rho} \]

is positive definite: \(k+k'>0\).

III\(_0\). The matrix \(\beta\) can be represented in the form \(\beta=\beta_+ + \beta_-\) in such a way that the matrix \(\mu=\beta_+-\beta_-\) satisfies the inequality \(\mu+\mu'>0\).

III\(_1\). Every vector \(u\) can be represented as \(u=u_+ + u_-\), with \(\beta_+u_-=\beta_-u_+=0\).

Denote \(M=\mu-\beta\). The boundary condition \(Mu=0\) on \(\Gamma\) is called “admissible.” It is also required that on \(\Gamma_+\) the matrix \(\beta\) be nonnegative, and on \(\Gamma_-\) nonpositive. We construct the matrix \(\mu\) by setting, respectively, \(\mu=+\beta\) on \(\Gamma_+\).

The boundary condition on \(\Gamma_-\) will be \(Mu=(\mu-\beta)u=-2\beta u=0\), or \(\beta u=0\). Next the domain \(\Omega\) is divided into pieces \(P_\rho\), and in each of them “tangential” first-order differential operators are introduced:

\[ D_\sigma=d_\sigma^\tau\frac{\partial}{\partial x^\tau}+d_\sigma \quad \text{(summation over } \tau), \]

where \(d_\sigma^\tau\) are functions and \(d_\sigma\) are matrices, with \(d_\sigma^\tau,d_\sigma\) continuously differentiable. The operators \(D_\sigma\) are subject to the following conditions:

1. \(d_\sigma^2=0\) on \(\Gamma_0\) (if the given piece \(P_\rho\) contains part of \(\Gamma_0\)).

2. The operators \(D_\sigma\) form a “complete” system, i.e., every operator of the form

\[ d^\tau\frac{\partial}{\partial x^\tau}+d, \]

where \(d^\tau,d\) are continuous and \(d^2=0\) on \(\Gamma_0\), can be represented as a linear combination of the operators \(D_\sigma\) with continuous coefficients.

1. There exist continuous matrices \(p_\sigma^\tau\) and continuously differentiable matrices \(t_\sigma\) such that the commutator \([D_\sigma,K]=D_\sigma K-KD_\sigma\) can be represented in the form
   \[ [D_\sigma,K]=p_\sigma^\tau D_\tau+t_\sigma K. \]

2. There exists a matrix \(q_\sigma^\tau\) such that the commutator \([D_\sigma,M]=D_\sigma M-MD_\sigma\) can be represented in the form
   \[ [D_\sigma,M]=q_\sigma^\rho D_\rho+t_\sigma^\Gamma M, \]
   where the matrix \(t_\sigma^\Gamma\) is related to \(t_\sigma\) by the formula
   \[ t_\sigma^\Gamma=t_\sigma+\hat d_\sigma,\qquad \hat d_\sigma=-\frac{\partial}{\partial x^2}d_\sigma^2 . \]

To formulate the following conditions, introduce a system \(u=\{u_\sigma\}_1\) of functions \(u_0,u_1,\ldots\) with the same indices as for the operator \(D_\sigma\), and introduce an operator \(K_1\) acting on this system as follows:
\[ K_1u_1=\{Ku_\sigma+p_\sigma^\tau u_\tau\}. \]

For the operator \(K_1\) introduce the boundary operator \(M_1\) as follows:
\[ M_1u_1=\{Mu_\sigma+q_\sigma^\tau u_\tau\}. \]

For the operators \(K_1\) and \(M_1\) the systems are constructed:
\[ \alpha_1^\rho u_1=\{\alpha^\rho u_\sigma\},\qquad \beta_1 u_1=\{\beta u_\sigma\}, \]
\[ k_1u_1=\{ku_\sigma+p_\sigma^\tau u_\tau\},\qquad \mu_1u_1=\{\mu u_\sigma+q_\sigma^\tau u_\tau\}. \]

1. The symmetric part of the matrix \(k_1\) is positive definite.

2. The symmetric part of the matrix \(\mu_1\) is nonnegative.

If these conditions are satisfied, then for each vector \(f\in W_2^1\) there exists a strong solution \(u\in W_2^1\) of the problem
\[ Ku=f,\qquad Mu=0 \]
(a strong solution is defined on p. 22). Let us verify the satisfaction of the formulated conditions for the problem posed at the beginning of the paragraph.

For this purpose introduce the vector \(u=\{u_1,u_2\}\), where
\[ u_1=-\frac{\partial v}{\partial x},\qquad u_2=\frac{\partial v}{\partial y}. \]

In the new notation, system (1.1) with \(C=0\) is written as
\[ Lu=\begin{pmatrix} G&0\\ 0&E \end{pmatrix}\frac{\partial u}{\partial x} - \begin{pmatrix} 0&E\\ E&0 \end{pmatrix}\frac{\partial u}{\partial y} =f_1, \tag{4.1} \]
where \(E\) is the identity matrix, and \(f_1=(f,0)\).

Multiplying system (4.1) on the left by the matrix
\[ Z=2\begin{pmatrix} Eb&cG\\ Ec&Eb \end{pmatrix}, \]
we obtain the symmetric operator
\[ K=ZL =2\begin{pmatrix} bG&cG\\ cG&Eb \end{pmatrix}\frac{\partial}{\partial x} - 2\begin{pmatrix} cG&Eb\\ Eb&Ec \end{pmatrix}\frac{\partial}{\partial y}. \tag{4.2} \]

Here \(b=1-2\sigma x,\quad c=1-\sigma(x+y),\quad \sigma\) is a positive constant. The matrix \(k\) at the origin will have the form

\[ k=\begin{pmatrix} G_y & 0\\ 0 & E\sigma \end{pmatrix}. \]

Obviously, the matrix \(k\) will be positive definite if the domain \(\Omega\) is taken in a sufficiently small neighborhood of the origin (recall that \(G(0)=0\)). Define boundary conditions for the vector \(u\) as follows:

\[ n_2u_1-n_1u_2=0\quad \text{on } \Gamma_*=\Gamma_0\cup\Gamma_- . \tag{4.3} \]

Let us show that this boundary condition is “admissible.” The matrices \(\mu,\beta,\beta_+,\beta_-\) on the boundary \(\Gamma_*\) will have the form

\[ u\beta u=(bn_1+cn_2)^{-1}\big[(n_2u_1-n_1u_2)(Eb^2-c^2G)(n_2u_1-n_1u_2)- \]

\[ -(bu_1+cu_2)(En_2^2-n_1^2G)(bu_1+cu_2)\big], \]

\[ u\beta u=(bn_1+cn_2)^{-1} \left\{ \underbrace{ \begin{pmatrix} Eb^2-c^2G & 0\\ 0 & Eb^2-c^2G \end{pmatrix} \begin{pmatrix} En_2^2 & -En_1n_2\\ -En_1n_2 & En_1^2 \end{pmatrix}}_{\beta_-} -\right. \]

\[ \left. -\underbrace{ \begin{pmatrix} En_2^2-n_1^2G & 0\\ 0 & En_2^2-n_1G \end{pmatrix} \begin{pmatrix} Eb^2 & Ebc\\ Ebc & Ec^2 \end{pmatrix}}_{\beta_+} \right\}u,u, \]

\[ u\beta_+u=-(bn_1+cn_2)^{-1}(bu_1+cu_2)(En_2^2-n_1^2G)(bu_1+cu_2), \]

\[ u\beta_-u=(bn_1+cn_2)^{-1}(n_2u_1-n_1u_2)(Eb^2-c^2G)(n_2u_1-n_1u_2), \]

\[ u\mu u=-(bn_1+cn_2)^{-1}\big[(n_2u_1-n_1u_2)(Eb^2-c^2G)(n_2u_1-n_1u_2)+ \]

\[ +(bu_1+cu_2)(En_2^2-n_1^2G)(bu_1+cu_2)\big]. \]

The symmetric part of the matrix \(\mu\) is nonnegative on \(\Gamma_*\), since there the inequalities

\[ En_2^2-Gn_1^2>0 \]

hold (on \(\Gamma_0\) by assumption, and on \(\Gamma_-\) we have \(G(y)<0\));

\[ b^2-c^2G>0 \]

(since \(G(0)=0\), this inequality is satisfied if the domain \(\Omega\) is taken in a sufficiently small neighborhood of the axis \(y=0\));

\[ bn_1+cn_2<0 \]

(here \(b=1-2\sigma x,\ c=1-\sigma(x+y)\)); on \(\Gamma_-\) (the validity of this inequality is obvious); on the line \(x-y+R=0,\ x_0\le x\le x_+\), we have

\[ bn_1+cn_2=-\frac{1}{\sqrt{2}}\sigma R<0; \]

on the line \(x-y-R=0,\ x_0\le x\le x_+\), we have

\[ bn_1+cn_2=-\frac{1}{\sqrt{2}}\sigma R<0. \]

On \(\Gamma_+\) put \(\mu=\beta\). The symmetric part of the matrix \(\beta\) on \(\Gamma_+\) is nonnegative, since on \(\Gamma_+\) the inequalities

\[ bn_1+cn_2>0,\qquad En_2^2-n_1^2G<0,\qquad Eb^2-c^2G>0 \]

are satisfied.

The last inequality is valid throughout the entire domain \(\Omega\). Thus, all conditions of “admissibility” of the boundary conditions (4.3) on \(\Gamma\) are fulfilled.

It is easy to verify that one can introduce operators \(D_\delta\) which satisfy requirements 1, 2, 3, 4, 6. The fulfillment of these conditions is sufficient for the existence of strong solutions in the domain \(\Omega\) of the problem

\[ Ku=f,\qquad Mu=0. \]

By a strong solution is meant a vector \(u\in W_2^0\) which is the limit, in the sense of \(W_2^0\), of a sequence of vectors \(\{u^i\}\) satisfying the condition \(Mu^i=0\) on \(\Gamma_*\), and such that \(Ku^i=f_i\) tends to \(f\) in the metric of \(W_2^0\).

We shall now verify condition 5. To this end, consider system (4.1) first for \(y<0\) (i.e., where it is elliptic). Since the vector \(u=(u_1,u_2)\), by virtue of (4.1), satisfies the equations

\[ \frac{\partial u_1}{\partial y}-\frac{\partial u_2}{\partial x}=0 \]

in the strong, and hence also in the weak, sense, and the boundary conditions \(n_2u_1-n_1u_2=0\) on \(\Gamma_*\), there exists a vector \(v\) which has strong derivatives

\[ \frac{\partial v}{\partial x}=u_1,\qquad \frac{\partial v}{\partial y}=u_2 \]

and vanishes on \(\Gamma_*\). Using the theory of strongly elliptic systems, we conclude that the vector \(v\) satisfies system (1.1) for \(C=0\) (inside \(\Omega\), with \(y<0\)) strongly and has strong derivatives up to some order, provided only that \(G\) and \(f\) are sufficiently smooth.

Denote by \(C^\infty\) the class of infinitely differentiable functions. Define the function \(\xi(x)\ge 0\) from \(C^\infty\) as follows:

\[ \xi(x)= \begin{cases} 0 & \text{for } x\le x_0,\\ 1 & \text{for } x_0+\delta\le x\le x_0+4\delta,\\ 0 & \text{for } x_0+5\delta\le x \end{cases} \]

(where \(\delta<-\dfrac{1}{2}(R+x_0)\) so that \(y<0\) on the segment \(x=x_0+5\delta\), \(|x-y|\le R\)), we may assert that \(\xi v\) has strong derivatives up to the second order inclusive, at least in the interval \(x_0+\delta\le x\le x_0+4\delta\).

Define now the function \(\eta(x)\) from \(C^\infty\) as follows:

\[ \eta(x)= \begin{cases} 0, & \text{for } x_0+\delta\le x\le x_0+2\delta,\\ 1, & \text{for } x_0+3\delta\le x. \end{cases} \]

Construct the operator

\[ \widetilde L = \eta \begin{pmatrix} G & 0\\ 0 & E \end{pmatrix} \left( \frac{\partial}{\partial x}+\frac{\partial}{\partial y} \right) - \begin{pmatrix} G & E\\ E & E \end{pmatrix} \frac{\partial}{\partial y} +\widetilde\gamma, \]

where the matrix \(\widetilde\gamma\in C^\infty\) is such that \(\widetilde\gamma=0\) for \(x_0+3\delta\le x\), while for \(x\le x_0+3\delta\) the matrix \(\widetilde\gamma\) will be defined below.

We shall apply the operator \(\widetilde L\) to the vector \(\tilde u=\eta u\). Obviously, the vector \(\tilde f=\widetilde L\tilde u\) belongs to \(W_2^1\) (for \(x\ge x_0+3\delta\) this follows from \(f=\tilde f_*\), while for \(x_0+\delta\le x\le x_0+4\delta\) it follows from the differentiability of \(u\); for \(x\le x_0+2\delta\) we have \(\tilde f=0\)). On the boundary \(|y-x|=R\) the matrix \(\beta\) for \(\widetilde L\)

has the same form as for the operator \(L\), since the boundary condition \(Mu=0\) is “admissible” there. On the segment \(x_0=0\) the matrix \(\tilde\beta\) vanishes.

We shall show that \(\tilde L\tilde u=\tilde f\) in the domain \(\Omega\) satisfies all the conditions listed above. To this end we define the matrix \(\tilde\gamma\) so that the matrix \(\tilde k+\tilde k'\) is positive definite. Consequently, the solution \(\tilde u=\eta u\) of the system \(\tilde L\tilde u=\tilde f\) is unique (by virtue of the fulfillment of conditions I, II, III\(_0\), III\(_1\); see [4]).

Let us now introduce the operators:
\[ \tilde D_0=1,\qquad \tilde D_1=\varepsilon_1\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}\right),\qquad \tilde D_2=\varepsilon_2\bigl[R^2-(x-y)^2\bigr]\frac{\partial}{\partial y}. \]

It is clear that they are “tangential” on \((x-y)=R\), and the commutators \([\tilde D_1,\tilde K]\) and \([\tilde D_2,\tilde K]\) can be expressed in terms of the operators \(\tilde D_0,\tilde D_1\), and \(\tilde K\). Our task is to show that the matrix \(\tilde k+\tilde k'\) can be made positive definite if \(\tilde\gamma,\varepsilon_1,\varepsilon_2\) and the domain \(\Omega\) are chosen in a suitable way.

Computing the matrix
\[ \tilde k=\|\tilde k_{ij}\|,\qquad i,j=(0,1,2), \]
we obtain that \(\tilde k_{00}=\tilde k_{22}=\tilde k\), while the matrices \(\tilde k_{ij}\) for \(i\ne j\) are either zero or proportional to the quantities \(\varepsilon_1,\varepsilon_2\). Thus the matrix \(\tilde k+\tilde k'\) can be made positive definite (by choosing \(\varepsilon_1,\varepsilon_2\) sufficiently small, respectively) if the matrix \(\tilde k_{11}+\tilde k'_{11}\) is positive definite.

The matrix \(\tilde k_{11}+\tilde k'_{11}\) can be made positive definite for
\[ x<x_0+3\delta, \]
by choosing \(\tilde\gamma\) and \(\delta\) in a suitable way. Note that the derivatives of the matrix \(\tilde\gamma\) do not enter \(\tilde k_{11}\).

We shall now show that also for \(x>x_0+3\delta\) the matrix \(\tilde k_{11}+\tilde k'_{11}\) can be made positive definite. To this end, consider the commutator \([\tilde D_1,K]\) up to a term that is a multiple of \(L\) or \(K\) (this fact will be denoted by the symbol \(\equiv\)):
\[ [\tilde D_1,K]\equiv Z[\tilde D_1,L] =\varepsilon_1 Z \begin{pmatrix} G'_y & 0\\ 0 & 0 \end{pmatrix} \frac{\partial}{\partial x}. \]

Using system (4.1), we obtain
\[ \frac{\partial}{\partial y} \equiv \begin{pmatrix} 0 & E\\ G & 0 \end{pmatrix} \frac{\partial}{\partial x}, \]
whence
\[ \tilde D_1 \equiv \varepsilon_1\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}\right) \equiv \varepsilon_1 \begin{pmatrix} E & E\\ G & E \end{pmatrix} \frac{\partial}{\partial x} \]
or
\[ \frac{\partial}{\partial x} \equiv \frac{1}{\varepsilon_1} \begin{pmatrix} E & E\\ G & E \end{pmatrix}^{-1} \tilde D_1. \]

Therefore
\[ [\tilde D_1,K]=\lambda \tilde D_1, \]
where \(\lambda\) at the origin has the form

\[ \lambda(0)=2 \begin{pmatrix} E & 0\\ E & E \end{pmatrix} \begin{pmatrix} G'_y & 0\\ 0 & 0 \end{pmatrix} \begin{pmatrix} E & -E\\ 0 & E \end{pmatrix}. \]

Thus, we have \(k_{11}=k+\lambda\). At the origin of coordinates we obtain

\[ \underset{1}{k_{11}}(0)= \begin{pmatrix} 3G'_y & -2G'_y\\ 2G'_y & \sigma-2G'_y \end{pmatrix} \]

and, consequently,

\[ \underset{1}{k_{11}}(0)+\underset{1}{k'_{11}}(0) = 2 \begin{pmatrix} 3G'_y & 0\\ 0 & E\sigma-2G'_y \end{pmatrix}. \]

We now choose \(\sigma>2G'_y(0)\) (if necessary, by reducing the domain \(\Omega\), we can make the matrix \(\underset{1}{k_{11}}+\underset{1}{k'_{11}}\) positive definite for \(x>x_0+3\delta\). Then, by what was said above, the matrix \(\underset{1}{k}+\underset{1}{k'}\) will be positive definite in the domain \(\Omega\).

Conditions 4) and 6) are checked easily, since the commutators \([\widetilde D_1,\widetilde K]\), \([\widetilde D_2,\widetilde K]\) can be expressed in terms only of the operators \(D_0\), \(\widetilde D_1\), \(\widetilde K\), \(M\) (i.e., without the operator \(\widetilde D_2\)). Thus, all the conditions listed at the beginning of the paragraph are fulfilled.

Hence, by virtue of [4] we conclude that \(u\) has strong first-order derivatives throughout the domain \(\Omega\).

Thus, the existence and uniqueness of problem I, posed at the beginning of § 2, have been proved.

References

1. Bitsadze A. V. Equations of mixed type. Moscow, 1959.
2. Sobolev S. L. Some applications of functional analysis in mathematical physics. Novosibirsk, 1962.
3. Bitsadze A. V.; Salakhtdinov M. S. Siberian Mathematical Journal, vol. II, No. 1, 1961.
4. Friedrichs K. O. Comm. Pure Appl. Math. XI, No. 3, 1958.
5. Didenko V. P. DAN SSSR, 144, No. 4, 1962.

Received by the editors
September 30, 1965.

Novosibirsk State University