UDC 517.919

MATHEMATICS

I. V. MAIOROV

ON A NONLINEAR SYSTEM OF EQUATIONS OF MIXED TYPE

(Presented by Academician I. G. Petrovskii, March 6, 1968)

Problem T. In the domain \(D\), find a regular solution of the system

\[ E(z_i)=y\,\partial^2 z_i/\partial x^2+\partial^2 z_i/\partial y^2 =f_i(x,y,z_1,\ldots,z_n), \tag{1} \]

taking the prescribed values:

\[ z_i\big|_{AC}=\psi_i(t),\qquad z_i\big|_{\Gamma}=\varphi_i(S),\qquad i=1,2,\ldots,n, \tag{2} \]

where the \(\varphi_i\) are continuously differentiable and the \(\psi_i\) are twice continuously differentiable functions.

The domain \(D\) is bounded by a Jordan curve \(\Gamma\) in the half-plane \(y>0\), with endpoints at the points \(A(0,0)\), \(B(1,0)\), and by arcs of the characteristics \(AC\) and \(BC\) of the equation \(E(z)=0\), if \(y<0\). The functions \(f_i\) are assumed to be continuous and to have second derivatives in the domain \(x,y\in\overline D\), \(|z_i|\le c\), and moreover \(\partial f_i/\partial z>0\).

We first establish some properties of solutions of the linear system

\[ y z_{xx}+z_{yy}-cz=0, \tag{3} \]

where \(z=(z_1,z_2,\ldots,z_n)\) is a vector, and \(c(x,y)\) is a positive definite \(n\times n\) matrix \(c=\|c_{ik}\|\), whose components satisfy the conditions

\[ (n-1)(c_{ik}+c_{ki})\le 2\sqrt{c_{ii}c_{kk}},\qquad i\ne k. \tag{4} \]

Let

\[ R(x,y)=\left(\sum_{k=1}^n z_k\right)^{1/2}; \]

\(D_1,D_2\) are the parts of the domain \(D\) in the half-planes respectively \(y>0\), \(y<0\).

Lemma 1. The function \(R(x,y)\) cannot attain a positive maximum at an interior point of the domain \(D_1\) (see \((^1)\)).

Supposing that \(R(x,y)\) attains a positive maximum at an interior point \(P\) of the domain \(D_1\), we obtain

\[ E(R(P))\le 0. \tag{5} \]

On the other hand, by virtue of (3),

\[ E(R)=\frac{1}{R}\left[y(z_x)^2+(z_y)^2+zcz-y(R_x)^2-(R_y)^2\right]. \tag{6} \]

Consequently, at the point \(P\),

\[ E(R)>0. \tag{7} \]

The contradiction between inequalities (5) and (7) proves the lemma.

Lemma 2. If at some point \(x=x_0\) of the segment \((0,1)\) of the axis \(y=0\), \(R(x,y)\) assumes its greatest positive value, and if the values of \(R(x,y)\) on \(\Gamma\) are less than \(R(x_0,0)\), then

\[ \overline R(x_0)=\lim_{y\to 0}\partial R(x_0,y)/\partial y<0 \tag{8} \]

provided that this limit exists.

For degenerating elliptic equations this proposition is given in work \((^2)\).

Obviously, \(\overline{R}(x_0)>0\) cannot occur. Suppose that \(\overline{R}(x_0)=0\). Let \(\mu>0\), let \(d\) be the diameter of the domain \(D_1\), and let \(R(x_0,0)=1\). Consider the function

\[ u=\varepsilon R/(e^{\mu d}-\varepsilon e^{\mu y}). \tag{9} \]

It is easy to see that the function \(u(x,y)\) must have a positive maximum at an interior point \(P\) of the domain \(D_1\), and, by virtue of (9),

\[ E(R(P))=\frac{1}{\varepsilon}\left[ E(u)-\frac{2\varepsilon e^{\mu y}}{e^{\mu d}-\varepsilon e^{\mu y}}u_y -\frac{\varepsilon \mu e^{\mu y}}{e^{\mu d}-\varepsilon e^{\mu y}}u \right]<0. \tag{10} \]

On the other hand, at the point \(P\), taking (9) into account, we obtain \(R_x=0\), \(R_y=-\mu e^{\mu y}u\), and, by virtue of (3) and (4), from (6) we obtain the inequality \(E(R(P))>0\), contradicting (10), which proves the lemma.

Lemma 3. In the domain \(D_2\) there exists a unique continuously differentiable solution of system (3), assuming on the boundary the prescribed values

\[ z_i\big|_{AC}=\psi_i(t), \qquad z_i\big|_{AB}=\tau_i(x). \tag{11} \]

Let \(z_i^{(0)}\) be continuous solutions of the equation \(E(z_i)=0\) in the domain \(\overline{D}_2\), assuming on the boundary the values (11). Such solutions are known \((^2)\).

Let

\[ u_i=z_i-z_i^{(0)}. \tag{12} \]

Then the functions \(u_i\) satisfy the equation

\[ E(u_i)-\sum_{k=1}^{n} c_{ik}(u_k+z_k^{(0)})=0 \tag{13} \]

and the homogeneous boundary conditions (11). Setting \(u_i^{(0)}=0\), and

\[ u_i^{(m)}=\lambda\int_{0}^{\xi} d\xi' \int_{\xi'}^{\eta} \frac{V(\xi,\eta;\xi',\eta')}{(\eta'-\xi')^{2/3}} \sum_{k=1}^{n} c_{ik}\bigl(u_k^{(m-1)}+z_i^{(0)}\bigr)\,d\eta', \tag{14} \]

we find that

\[ \left|u_i^{(m+1)}-u_i^{(m)}\right| \leq 2cM^{m+1}(\eta-\xi)^{1/3}\eta^{1/3+(m+1)/3}\frac{\xi^m}{m!}, \tag{15} \]

where \(c=\max\{|z_k^{(0)}|\}\), \(M=6\lambda Nn\), \(N=\max|c_{ik}|\) for all \(i,k=1,2,\ldots,n\).

From the estimates (15) follows the uniform convergence of the sequence \(u_i^{(m)}\) to the solution of equation (13).

Lemma 4. If the function \(R(\xi,\eta)\) is equal to zero on \(AC\) and assumes its greatest positive value at some point \((x_0,0)\) on \(AB\), then there exists in \(D_2\) a neighborhood of this point in which

\[ R(\xi,\eta)<R(x_0,0). \tag{16} \]

From (12) we have

\[ z_i=k\int_{0}^{\xi}\tau_i(t) \frac{(\eta-\xi)^{2/3}\,dt}{(\eta-t)^{5/6}(\xi-t)^{5/6}} +u_i(\xi,\eta), \tag{17} \]

where \(u_i=\lim\limits_{m\to\infty}u_i^{(m)}\).

Assuming that \(z_i\) on \(AB\) attains its greatest positive value at the point \((x_0,0)\), and taking into account the estimates (15), we obtain

\[ z_i \leq \tau_i(x_0)\left\{ 1-\left(\frac{\eta-\xi}{\eta}\right)^{2/3} \left[{}^3/2\,k-2cM\overline{M}\eta^{1/3}\right] \right\}, \tag{18} \]

where \(\overline{M}=\max e^{\xi}\eta^{1/3}\).

It follows from inequality (18) that if \(\eta<(3k/4cM\overline{M})^{3/4}\), then \(z_i(\xi,\eta)<\tau_i(x_0)\). Since \(z_i(\xi,\eta)\) in some neighborhood of the point \((x_0,0)\), by virtue of continuity, as well as \(\tau(x_0)\), is positive, we obtain inequality (16) in an obvious way.

Lemma 5. If the functions \(z_i(x,y)\) satisfy equation (3) in the domain \(D\), and are equal to zero on the characteristic \(AC\), then the norm \(R(x,y)\) on the segment \(AB\) cannot assume a greatest positive value.

By virtue of the first lemma, \(R(x,y)\) in \(D_1\) cannot have a positive maximum. Assuming that this maximum is attained on \(AB\), by virtue of (8) and (16) we arrive at a contradiction.

Theorem 1. The solution of problem \(T\) for system (3) is unique in \(D\).
The proof follows easily from Lemmas 1, 3, and 5.

Theorem 2. If

\[ (n-1)\left(\frac{\partial f_i}{\partial z_k}+\frac{\partial f_k}{\partial z_i}\right) \leqslant 2\left(\frac{\partial f_i}{\partial z_i}\frac{\partial f_k}{\partial z_k}\right)^{1/2}, \]

then in the domain \(D\) there exists a solution of problem \(T\) for equation (1).

Supposing that problem \(T\) for system (1) has two solutions \(z_i\) and \(w_i\), we obtain that the difference \(u_i=z_i-w_i\) satisfies the system

\[ E(u_i)=\sum_{k=1}^{n}\frac{\partial f_i}{\partial u_k}u_k, \]

which possesses all the properties of system (3) and, consequently, by virtue of the first theorem has the unique zero solution.

We preface the proof of the existence of a solution by two lemmas.

Lemma 6. In the domain \(D_1\), for system (1) there exists a unique twice continuously differentiable solution which assumes the prescribed values on the boundary

\[ z_i\big|_{\Gamma}=\varphi_i(s),\qquad dz_i/dy=\nu_i(x). \tag{19} \]

The uniqueness of the solution follows from Lemmas 1 and 2. Let \(z_i^{(0)}\) be the solution of the equation \(E(z_i)=0\), satisfying conditions (19).

Putting

\[ v_i=z_i-z_i^{(0)}, \tag{20} \]

we find that \(v_i\) satisfy the equation \(E(v_i)=f_i(x,y,v_1+z_1^{(0)},\ldots,\)
\(\ldots,v_n+z_n^{(0)})\) and the homogeneous conditions (19). Replacing this equation by an integral one and solving it by the method of successive approximations, where \(v_i^{(0)}=0\), and

\[ v_i^{(m+1)} = \iint_{D_1} f_i\!\left(\xi,\eta,v_1^{(m)}+z_1^{(0)},\ldots,v_n^{(m)}+z_n^{(0)}\right) G(x,y;\xi,\eta)\,d\xi\,d\eta, \]

we obtain that the functions \(v_i=\lim\limits_{m\to\infty}v_i^{(m)}\) solve the posed problem.

For \(v_i\) the estimates

\[ \left|dv_i(x,0)/dx\right|\leqslant c,\qquad c=\mathrm{const}. \tag{21} \]

are valid.

Lemma 7. In the domain \(D_2\) there exists a unique continuously differentiable solution of system (1), which assumes on the boundary the values

\[ z_i\big|_{AC}=\psi_i(t),\qquad z_i\big|_{AB}=\nu_i(x) \tag{22} \]

and has the form

\[ z_i=u_i+z_i^{(0)},\qquad u_i=\lim_{m\to\infty}u_i^{(m)}, \tag{23} \]

where

\[ z_i^{(0)}(\xi,\eta) = k\int_{0}^{\xi}\nu_i(t)(\xi-t)^{-1/6}(\eta-t)^{-1/6}\,dt + \int_{0}^{b}\left(\psi_i' + \frac{\psi_i}{6t}\right) V(\xi,\eta;0,t)\,dt, \tag{24} \]

\[ u_i^{(m+1)}(\xi,\eta)=\lambda\int_0^\xi d\xi'\int_\xi^\eta \frac{V(\xi,\eta;\xi',\eta')}{(\eta'-\xi')^{2/3}}\, f_i\bigl(\xi,\eta,u_1^{(m)}+z_1^{(0)},\ldots,u_n^{(m)}+z_n^{(0)}\bigr)\,d\eta'. \]

For \(u_i(\xi,\eta)\), when \(\eta=\xi=x\), we have the estimates

\[ \left|\,du_i(x,x)/dx\,\right|\leq c,\qquad c=\mathrm{const}. \tag{25} \]

Now from equations (20) and (23), along the segment \(AB\), one can form a system of equations with respect to \(\tau_i(x)\) and \(\nu_i(x)\).

Eliminating \(\tau_i(x)\) from this system, we reduce the solution of problem T to the solution of a singular integral equation with respect to \(\nu_i(x)\) (see (2)).

By virtue of the estimates (21) and (25), this singular equation is reduced in the usual way to a Fredholm equation, whose solvability follows from the uniqueness of the solution proved above.

Volgograd Pedagogical Institute

Received
28 II 1968

REFERENCES

\(^{1}\) A. V. Bitsadze, Boundary-Value Problems for Second-Order Elliptic Equations, “Nauka,” 1966.
\(^{2}\) K. I. Babenko, On the Theory of Equations of Mixed Type, Dissertation, Moscow, 1951.