MATHEMATICS

E. P. ZHIDKOV, G. I. MAKARENKO

SOLUTION OF THE DIRICHLET PROBLEM

FOR A NONLINEAR ELLIPTIC EQUATION

BY INTRODUCING A CONTINUOUS PARAMETER

(Presented by Academician N. N. Bogolyubov, 23 XII 1968)

In [1] the following was proved.

Theorem 1. Let \(y=\Phi(x)\) be a continuous function mapping a Banach space \(X\) into a Banach space \(Y\). Suppose that the equation

\[ \Phi(x)=0 \]

has a unique solution \(x^*\) in the domain \(D\):

\[ \|x-x^*\|\leq M. \]

Assume that in \(D\) there exist continuous Fréchet derivatives \(\Phi'(x)\) and \(\Phi''(x)\), as well as the inverse operator \(\Phi'(x)^{-1}\), for which the inequality

\[ \|\Phi'(x)^{-1}\|\leq B \]

holds.

Then there exists a sphere \(S:\ \|x-x^*\|\leq \varepsilon\), belonging to the domain \(L\), such that for any \(x_0\in S\) the differential equation

\[ d\bar{x}(t)/dt=-\Phi'(x)^{-1}\Phi(x), \]

where \(t\) is a real parameter, with the initial condition

\[ x(0)=x_0 \]

has a unique solution \(x(t)\) on the interval \(0\leq t<+\infty\), and

\[ \lim_{t\to\infty}\|x(t)-x^*\|_X=0. \]

In [1–4] Theorem 1 was used in solving boundary-value problems for nonlinear ordinary differential equations of the second and \(n\)-th orders, in solving nonlinear integral equations, and also in solving an inverse problem in scattering theory. In the present paper the method of stabilization with respect to a parameter is applied to the solution of a boundary-value problem for a nonlinear elliptic differential equation of second order. The formulation of the problem is given, a theorem on the introduction of a continuous parameter is formulated, and a scheme of an approximate method for solving the stated problem is presented.

I. Consider the nonlinear differential equation

\[ \Delta z(x,y)+f(x,y,z)=0, \tag{1} \]

where \(\Delta\) is the Laplace operator, and \(f(x,y,z)\) is a function twice continuously differentiable in its arguments. We seek a solution of equation (1), given in a bounded domain \(G\), satisfying on the boundary \(\Gamma\) the condition

\[ z(x,y)\big|_{\Gamma}=0. \tag{2} \]

We assume that the boundary \(\Gamma\) of the domain \(G\) is sufficiently smooth.

Consider the complete linear normed space \(X(G,H)\) of functions twice continuously differentiable in the closed domain \(\bar{G}\)

\(z(x,y)\), vanishing on the boundary and such that the second derivatives of \(z(x,y)\) satisfy, in \(\overline G\), the Hölder condition with exponent \(\lambda\) \((0<\lambda<1)\). The norm of \(z(x,y)\) in the space \(X(G,H)\) is defined as follows:

\[ \|z(x,y)\|_X = \sum_{l=0}^{2}\sum_{m=0}^{l} \max_{\overline G} \left| \frac{\partial^l z(x,y)}{\partial x^m \partial y^{\,l-m}} \right| + H_{z''_{x^2}}+H_{z''_{xy}}+H_{z''_{y^2}}, \tag{3} \]

where by \(H_{z''_{x^2}}, H_{z''_{xy}}, H_{z''_{y^2}}\) we denote the lower bounds of the Hölder constants with exponent \(\lambda\), respectively, for the functions \(\partial^2 z/\partial x^2\), \(\partial^2 z/\partial x\partial y\), \(\partial^2 z/\partial y^2\) in the domain \(\overline G\). Introduce the Banach space \(Y(G,H)\) of functions \(w(x,y)\) continuous in \(\overline G\) and satisfying the Hölder condition with exponent \(\lambda\). We define the norm in \(Y(G,H)\) by the relation

\[ \|w(x,y)\|_Y=\max_{\overline G}|w(x,y)|+H_w, \tag{4} \]

where \(H_w\) is the lower bound of the Hölder constants for the function \(w(x,y)\) in the domain \(\overline G\).

Theorem 2. Let, in the domain \(D\)

\[ \|z-z^*\|_X\le M \tag{5} \]

of the space \(X(G,H)\), the boundary-value problem (1)—(2) have, and moreover uniquely, a solution \(z^*(x,y)\). Suppose that for any \(z(x,y)\in D\) and any \(w(x,y)\in Y(G,H)\) the boundary-value problem

\[ \Delta v(x,y)+f'_z(x,y,z)v(x,y)=w(x,y), \tag{6} \]

\[ v(x,y)|_\Gamma=0 \tag{7} \]

has a unique solution \(v(x,y)\in X(G,H)\).

Then there exists a sphere \(S\)

\[ \|z-z^*\|_X\le \varepsilon,\qquad \varepsilon<M, \tag{8} \]

such that for any \(z_0\in S\) in the cylinder \(\Omega=G\times[0\le t<+\infty)\) there exists a unique solution of the system of equations

\[ \partial^2 v/\partial x^2+\partial^2 v/\partial y^2+f'_z(x,y,z)v(x,y,t) = -\bigl[\Delta z+f(x,y,z)\bigr], \]

\[ \partial z(x,y,t)/\partial t=v(x,y,t), \tag{9} \]

satisfying the conditions:

\[ v(x,y,t)|_B=0, \]

\[ z(x,y,t)|_{t=0}=z_0(x,y),\qquad z_0(x,y)|_\Gamma=0 \tag{10} \]

(\(B\) is the lateral surface of the cylinder \(\Omega\)), and, moreover,

\[ \lim_{t\to\infty}\|z(x,y,t)-z^*(x,y)\|_X=0. \tag{11} \]

A detailed proof of Theorem 2 is given in the authors’ paper \((^5)\); here we merely note that for the proof one must verify that all the conditions of Theorem 1, formulated for a Banach space, are fulfilled in the case of our particular problem. In doing so, the theorem on boundedness of the inverse operator is used essentially (see, for example, \((^6)\), p. 157), as well as the theorems on Hölder continuity of the solution of an elliptic equation and on uniform estimation of the quantity \(H_{v''}\) for the solution of the Dirichlet problem (6)—(7) (see \((^7,^8)\)).

II. Scheme of the numerical solution of problem (9)—(10). Choose the step of motion with respect to the parameter \(t\), denote it by \(\tau\). Divide the domain \(\Omega_T=G\times[0\le t\le T]\) into \(n\) parts by planes parallel to the plane \(XOY\):

\[ t_0=0,\quad t_1=\tau,\quad t_2=2\tau,\ldots,t_n=n\tau=T. \]

Having an initial value \(z(x,y,0)=z_0(x,y)\), \(z_0(x,y)\in S\), and substituting it into the first equation of system (9), we obtain a linear elliptic equation with respect to \(v(x,y,0)\)

\[ \Delta v(x,y,0)+f'_z(x,y,z_0(x,y))v(x,y,0) =-[\Delta z_0(x,y)+f(x,y,z_0(x,y))] \tag{12} \]

with the boundary condition

\[ v(x,y,0)\big|_{\Gamma}=0. \tag{13} \]

Solving the boundary-value problem (12)—(13) by any known method (for example, by the grid method), we obtain the value of the function \(v(x,y,t)\) on the layer \(t_0=0\).

We now replace the second equation of system (9) by the difference relation

\[ [z(x,y,t_1)-z_0(x,y)]/\tau=v(x,y,0), \]

whence we find the value of the function \(z(x,y,t)\) on the layer \(t=t_1\).

In general, if the function \(z(x,y,t)\) is known on the layer \(t=t_k\), then the function \(v(x,y,t)\) on this layer is determined by solving the linear boundary-value problem with respect to \(v(z,y,t_k)\) as a function of \(x\) and \(y\):

\[ \Delta v(x,y,t_k)+f'_z(x,y,z(x,y,t_k))v(x,y,t_k)= \]
\[ =-[\Delta z(x,y,t_k)+f(x,y,z(x,y,t_k))], \tag{14} \]

\[ v(x,y,t_k)\big|_{Б}=0. \tag{15} \]

Having thereby found \(v(x,y,t_k)\), we determine the function \(z(x,y,t)\) on the next layer \(t=t_{k+1}\):

\[ z(x,y,t_{k+1})=z(x,y,t_k)+\tau v(x,y,t_k) \tag{16} \]

and so on.

In conclusion, we note that, on the basis of Theorem 2 of \((^1)\), one can draw the following conclusion:

If it is assumed that the solution of the boundary-value problem (14)—(15) is carried out exactly, then as \(\tau\to 0\) we obtain convergence of the approximate solution to the exact one.

Joint Institute for Nuclear Research
Dubna, Moscow Region

Received
3 XII 1968

REFERENCES

\(^1\) E. P. Zhidkov, I. V. Puzynin, DAN, 180, No. 1, 18 (1968).
\(^2\) E. P. Zhidkov, G. A. Ososkov, DAN, 180, No. 6, 1279 (1968).
\(^3\) E. P. Zhidkov, Choi Zai Khen, Preprint of the Joint Institute for Nuclear Research, 11—3427, Dubna, 1967.
\(^4\) Ya. Wizner, E. P. Zhidkov, V. Lelek, Preprint of the Joint Institute for Nuclear Research, P5-3895, Dubna, 1968.
\(^5\) E. P. Zhidkov, G. I. Makarenko, Preprint of the Joint Institute for Nuclear Research, P5-4128, Dubna, 1968.
\(^6\) L. A. Lyusternik, V. I. Sobolev, Elements of Functional Analysis, “Nauka,” 1965.
\(^7\) L. Bers, F. John, M. Schechter, Equations with Partial Derivatives, Moscow, 1966.
\(^8\) C. Miranda, Equations with Partial Derivatives of Elliptic Type, IL, 1957.