MATHEMATICS

S. I. POKHOZHAEV

AN ANALOGUE OF SCHMIDT’S METHOD FOR A NONLINEAR EQUATION

(Presented by Academician M. A. Lavrent’ev, 22 VIII 1960)

Schmidt’s method \((^{1})\) makes it possible to reduce the solution of linear integral equations to the solution of such equations with a fixed degenerate kernel. One can propose a certain analogue of this method for nonlinear equations as well. Thus the questions of existence and uniqueness of the solution of a nonlinear problem are reduced to the investigation of a certain system of finite equations. The possibility of applying this method is connected with obtaining certain a priori estimates.

In the present note we shall show this on the example of the equation

\[ u(P)=\iint_D K(P,Q)u^2(Q)\,dQ+\psi(P), \tag{1} \]

which arises in the boundary-value problem

\[ \Delta u+u^2=0,\qquad u\big|_{\Gamma}=f(s), \tag{2} \]

where \(\Gamma\) is the boundary of a two-dimensional bounded domain \(D\), for which there exists the Green’s function \(K(P,Q)\) of the Dirichlet problem. Problem (2) is considered in the class of functions \(C^{(2)}\). The only condition imposed on the function \(f(s)\) is continuity.

It will be proved that, for the given domain \(D\) and function \(f(s)\), this problem reduces to a certain nonlinear integral equation with a fixed degenerate kernel.

Lemma 1. Let a function \(v(P)\) be given which has continuous first derivatives in \(D+\Gamma\) and continuous second derivatives in \(D\), is everywhere positive in \(D\), and satisfies \(v|_{\Gamma}=0\). Then, for any solution \(U(P)\) of the boundary-value problem (2), the inequality

\[ \iint_D v(Q)U^2(Q)\,dQ\le A_v^2, \tag{3} \]

holds, where

\[ A_v=\frac12\left[\iint_D \frac{(\Delta v(Q))^2}{v(Q)}\,dQ\right]^{1/2} +\frac12\left[\iint_D \frac{(\Delta v(Q))^2}{v(Q)}\,dQ+4\int_{\Gamma}|fv_{\nu}|\,ds\right]^{1/2} \]

and \(v_{\nu}\) is the derivative of the function \(v\) in the direction of the outward normal \(\nu\).

This estimate is obtained from Green’s formula for the functions \(v(P)\) and the solution \(U(P)\) of the boundary-value problem (2). Estimate (3) can also be obtained directly from equation (1).

It is clear that there exist functions \(v(P)\) for which the quantity \(A_v\) is finite. For example, as the function \(v(P)\) one may take the first eigenfunction of the boundary-value problem

\[ \Delta\varphi+\lambda\varphi=0,\qquad \varphi\big|_{\Gamma}=0. \tag{4} \]

Lemma 2. For any solution \(U(P)\) of the integral equation (1), the estimate

\[ \max_D |U(P)|\le C_v+a, \tag{5} \]

holds.

where \(C_\nu\) is the positive root of the equation:

\[ x^3=B_\nu(x+a)^2, \]

\[ B_\nu=A_\nu^4\max_D\iint_D \frac{K^3(P,Q)}{\nu^2(Q)}\,dQ,\qquad a=\max_\Gamma |f(s)|. \]

Proof. From the integral equation (1) and the estimate (3) it follows that

\[ U(P)-\psi(P)\leq A_\nu\left[\iint_D \frac{K^2(P,Q)U^2(Q)}{\nu(Q)}\,dQ\right]^{1/2}. \]

Now, using the integral equation (1) once more, we obtain:

\[ \max_D (U(P)-\psi(P))\leq C_\nu; \]

whence the required estimate (5) follows.

Let us now consider the question of finding all solutions of the boundary-value problem (2), which is equivalent to finding all solutions of the integral equation (1).

Represent the kernel \(K(P,Q)\) of the integral equation (1) in the form of the sum of two kernels:

\[ K(P,Q)=M_n(P,Q)+\Gamma_n(P,Q), \tag{6} \]

where

\[ M_n(P,Q)=\sum_{k=1}^n \frac{\Phi_k(P)\Phi_k(Q)}{\lambda_k}, \]

and \(\Phi_k(P)\), \(\lambda_k\) are the eigenfunctions and eigenvalues of the boundary-value problem (4).

Put

\[ \gamma_n=\max_D\iint_D |\Gamma_n(P,Q)|\,dQ. \]

Theorem. There exist a number \(n\) and a nonlinear operator \(R[V(P)]\) such that all solutions \(U(P)\) of the integral equation (1) are representable in the form

\[ U(P)=V(P)+R[V(P)], \]

where \(V(P)\) is a solution of the integral equation with fixed degenerate kernel \(M_n(P,Q)\):

\[ V(P)=\iint_D M_n(P,Q)(V(Q)+R[V(Q)])^2\,dQ, \tag{7} \]

satisfying the condition

\[ \gamma_n\max_D |V(P)|<1/4. \tag{8} \]

Proof. Equation (1) is equivalent to the system of equations

\[ V(P)=\iint_D M_n(P,Q)(V(Q)+W(Q))^2\,dQ+\psi(P); \tag{9} \]

\[ W(P)=\iint_D \Gamma_n(P,Q)(V(Q)+W(Q))^2\,dQ. \tag{10} \]

Consider separately equation (10) for fixed \(n\). Suppose a continuous function \(V(Q)\) is given, satisfying condition (8). Then equation (10) has a continuous solution \(W(P)\), which is found by the method of successive Picard approximations according to the scheme

\[ W(P)=\lim_{k\to\infty} W_k(P), \]

where

\[ W_k(P)=\iint_D \Gamma_n(P,Q)\bigl(V(Q)+W_{k-1}(Q)\bigr)^2\,dQ \quad (k=2,3,\ldots), \tag{11} \]

\[ W_1(P)=\iint_D \Gamma_n(P,Q)V^2(Q)\,dQ. \]

In this case the solution \(W(P)\) satisfies the inequality

\[ |W(P)|\leq \frac{1}{2\gamma_n}q-b,\quad \text{where } q=1-\sqrt{1-4b\gamma_n},\quad b=\max_D |V(P)|. \]

The maximum deviation of the function \(W_k(P)\), obtained at the \(k\)-th step in solving equation (11), from the exact solution \(W(P)\) does not exceed the quantity

\[ |W_k(P)-W(P)|\leq \frac{1}{\gamma_n}q^{k-1} \left(\frac{q}{2}-\gamma_n b\right)(q-\gamma_n b) \quad (k=1,2,3,\ldots). \]

Consequently, under condition (8), equation (10) admits a representation of the solution \(W(P)\) in terms of the function \(V(P)\) in the form \(W(P)=R[V(P)]\), where the operator \(R[V]\) is given by the formulas

\[ R[V(P)]=\lim_{k\to\infty} R_k[V(P)], \]

\[ R_k[V(P)]=\iint_D \Gamma_n(P,Q_1) \left( V(Q_1)+\iint_D \Gamma_n(Q_1,Q_2) \left( V(Q_2)+\cdots \right.\right. \]

\[ \left.\left. \cdots+ \left(\iint_D \Gamma_n(Q_{k-1},Q_k)V^2(Q_k)\,dQ_k\right)^2 \cdots \right)^2 \right)\,dQ_1 \quad (k=1,2,\ldots). \]

In the class of functions \(W(P)\) satisfying the condition

\[ 2\gamma_n \max_D |V(P)+W(P)|<1, \tag{12} \]

the solution of equation (10) is unique.

Let us now consider jointly the system of equations (9), (10). For any two functions \(V(P)\) and \(W(P)\) satisfying this system, the a priori estimates

\[ \max_D |V(P)|\leq M_1,\qquad \max_D |V(P)+M(P)|\leq M_2 \tag{13} \]

hold.

For example, by virtue of Lemma 2, as \(M_1\) and \(M_2\) one may take the numbers

\[ M_1=(C_v+a)^2 \max_D \left[ S\iint_D K^2(P,Q)\,dQ_1 \right]^{-1/2}+a,\qquad M_2=C_v+a, \]

where \(S\) is the area of the domain \(D\).

By virtue of the a priori estimates (13), the assertion of the theorem follows from the fact that \(\gamma_n\to 0\) as \(n\to\infty\), and is valid for every \(n\geq n_0\), where \(n_0=\max\{n_1,n_2\}\), and \(n_1\) and \(n_2\) are chosen from the conditions \(4\gamma_{n_1}M_1<1\), \(2\gamma_{n_2}M_2<1\).

Corollary. The number of solutions of the boundary-value problem (2) is determined by the number of solutions of equation (7) satisfying condition (8).

Remark. The operator \(R[V]\) depends on the number \(n\), and

\[ \max_D |R[V(P)]|\to 0 \]

as \(n\to\infty\) for a fixed function \(V(P)\).

In conclusion I express my deep gratitude to L. V. Ovsyannikov for valuable comments.

Institute of Hydrodynamics
Siberian Branch of the Academy of Sciences of the USSR

Received
18 VIII 1960

References

1. E. Schmidt, Math. Ann., 64, 161 (1907).