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UDC 517.944
MATHEMATICS
B. D. ANNIN
ONE PROPERTY OF A SOLUTION OF THE EQUATION \(u_{xx}u_{yy}-u_{xy}^{2}=1\)
(Presented by Academician A. D. Aleksandrov on 6 IX 1965)
In the present paper, for a solution of the equation \(u_{xx}u_{yy}-u_{xy}^{2}=1\), an expression is given for the first-order derivatives on the boundary of a convex domain in terms of a single function of one variable.
Consider the Monge—Ampère equation of elliptic type
\[ u_{xx}u_{yy}-u_{xy}^{2}=1 \tag{1} \]
in a finite simply connected domain \(G\), bounded by a convex curve \(\Gamma\), given in the form
\[ x=\varphi(s),\qquad y=\psi(s),\qquad 0\le s\le s^{*}. \tag{2} \]
Here \(s\) is arc length, \(s^{*}\) is the length of the contour \(\Gamma\), and as \(s\) increases the contour \(\Gamma\) is traversed in the positive direction (the domain \(G\) remains on the left).
Let \(f(s)\) be a function defined for \(0\le s\le s^{*}\). We shall assume that: a) the functions \(\varphi(s), \psi(s), f(s)\) belong to the class\(^*\) \(C^{k,\lambda}\), \(k\ge 4\), \(0<\lambda<1\); b) at \(s=0\) and \(s=s^{*}\) the values of the functions \(\varphi(s), \psi(s), f(s)\) coincide together with their derivatives up to order \(k\) inclusive; c) the curvature at all points of \(\Gamma\) is not less than \(\chi_{0}=\mathrm{const}>0\).
There exist \((^{1,2})\) two and only two solutions of equation (1) of class \(C^{k,\mu}(G+\Gamma)\), \(0<\mu\le\lambda\), which take on \(\Gamma\) the value \(f(s)\)
\[ u|_{\Gamma}=f(s). \tag{3} \]
For one of these solutions the functions \(u_{xx}\) and \(u_{yy}\) are positive in \(G+\Gamma\), for the other they are negative.
Consider the solution for which \(u_{xx}, u_{yy}\) are positive. Let \((x_{1},y_{1})\), \((x_{2},y_{2})\) be two distinct points belonging to \(G+\Gamma\); then the inequality \((^{3})\)
\[ (x_{2}-x_{1})[u_{x}(x_{2},y_{2})-u_{x}(x_{1},y_{1})]+(y_{2}-y_{1})[u_{y}(x_{2},y_{2})-u_{y}(x_{1},y_{1})]>0 \tag{4} \]
is valid.
Introduce new variables \(\xi,\eta\) and a new function \(\Phi\) by the formulas
\[ \xi=u_{x}+x,\qquad \eta=u_{y}+y,\qquad (x,y)\in G+\Gamma; \tag{5} \]
\[ \Phi=xu_{x}+yu_{y}-u+(x^{2}+y^{2})/2. \tag{6} \]
From (4) follows the inequality
\[ (x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}<(\xi_{2}-\xi_{1})^{2}+(\eta_{2}-\eta_{1})^{2}. \tag{7} \]
Here the points \((\xi_{1},\eta_{1})\), \((\xi_{2},\eta_{2})\) of the \(\xi,\eta\)-plane correspond to the points \((x_{1},y_{1})\), \((x_{2},y_{2})\in G+\Gamma\) by virtue of formulas (5).
It follows from inequality (7) that formulas (5) effect a homeomorphic mapping of \(G+\Gamma\) onto some domain \(D+L\) of the \(\xi,\eta\)-plane. The equation of \(L\)—the image of \(\Gamma\)—can be written in the form \((0\le s\le s^{*})\)
\[ \xi_{L}(s)=u_{x}(\varphi(s),\psi(s))+\varphi(s),\qquad \eta_{L}(s)=u_{y}(\varphi(s),\psi(s))+\psi(s). \tag{8} \]
\[ \underline{\hspace{3cm}} \]
\(^*\) The derivative of order \(k\) satisfies the Hölder condition with exponent \(\lambda\).
In view of inequality (4), the first derivatives with respect to \(s\) of the functions \(\xi_L(s)\) and \(\eta_L(s)\) cannot vanish simultaneously. The Jacobian of the mapping (5), in view of (1), is positive in \(G+L\); therefore, as \(s\) increases the curve \(L\) is traversed in the positive direction.
Let \(l\) denote the length of an arc of the contour \(L\), and let \(l^*\) be the length of \(L\); in so doing, as the origin of measurement we choose the point corresponding to \(s=0\), and we shall assume that, as \(l\) increases, the curve \(L\) is traversed in the positive direction. From the equation \(l=l(s)\), \(0\le s\le s^*\), one can determine the function of class \(C^{k-1,\mu}\), \(s=s(l)\), \(0\le l\le l^*\).
The function \(\Phi(\xi,\eta)\) belongs to the class \(C^{k-1,\mu}(D+L)\) and satisfies in \(D\) the equation*
\[ \Phi_{\xi\xi}+\Phi_{\eta\eta}=1. \]
At the same time, in \(D+L\) the equalities
\[ x=\Phi_{\xi}, \qquad y=\Phi_{\eta}, \qquad u=\xi\Phi_{\xi}+\eta\Phi_{\eta}-\frac12(\Phi_{\xi}^{2}-\Phi_{\eta}^{2}). \tag{9} \]
hold. Thus, on \(L\) the equalities (8) and the equalities
\[ \partial\Phi/\partial\xi\big|_{L}=\varphi(s), \qquad \partial\Phi/\partial\eta\big|_{L}=\psi(s). \tag{10} \]
are satisfied.
Let the function \(w_1(\zeta)=\xi+i\eta\) map conformally the unit disk \(K+C: |\zeta|=1\) of the plane of the complex variable \(\zeta\) onto the domain \(D+L\), and let \(\tau\) be the polar angle in the \(\zeta\)-plane. We shall assume that the point determined by the equalities \(|\zeta|=1\), \(\tau=0\) corresponds to the point of the contour \(L\) taken as the origin for measuring the arc length \(l\), and that the point \(\zeta=0\) corresponds to the point \((\xi_0,\eta_0)\), which under the transformation (5) is the image of some point \((x_0,y_0)\in G\). The function (*) \(w_1(\zeta)\in C^{k-1,\mu}(K+C)\), and the function \(l=l(\tau)\), \(0\le\tau\le 2\pi\), belongs to the class \(C^{k-1,\mu}\).
Denote
\[ s=s(l(\tau))\equiv g(\tau), \qquad 0\le \tau\le 2\pi. \tag{11} \]
The function \(g(\tau)\) realizes a homeomorphic mapping of \([0,2\pi]\) onto \([0,s^*]\) and belongs to the class \(C^{k-1,\mu}\).
Consider the function (*) of \(\zeta\) of class \(C^{k-1,\mu}(K+C)\)
\[ (-2\Phi_{\eta}+\eta)+i(-2\Phi_{\xi}+\xi)=w_2(\zeta). \]
In view of equalities (10) and (11), for \(0\le\tau\le 2\pi\) we have
\[ \operatorname{Re}\left[\frac12 w_1(\zeta)+\frac{i}{2}w_2(\zeta)\right]_{\zeta=e^{i\tau}} =\varphi(g(\tau)), \]
\[ \operatorname{Re}\left[-\frac{i}{2}w_1(\zeta)-\frac12 w_2(\zeta)\right]_{\zeta=e^{i\tau}} =\psi(g(\tau)). \]
Hence, using the Schwarz formula, we find (*)
\[ w_1(\zeta)=\frac{1}{2\pi}\int_{-\pi}^{\pi}[\varphi(g(\omega))+i\psi(g(\omega))]\, \frac{e^{i\omega}+\zeta}{e^{i\omega}-\zeta}\,d\omega+a, \]
\[ \tag{12} \]
\[ w_2(\zeta)=-\frac{1}{2\pi}\int_{-\pi}^{\pi}[i\varphi(g(\omega))+\psi(g(\omega))]\, \frac{e^{i\omega}+\zeta}{e^{i\omega}-\zeta}\,d\omega+b. \]
Here \(a=u_x(x_0,y_0)+iu_y(x_0,y_0)\), \(b=u_y(x_0,y_0)+iu_x(x_0,y_0)\). Setting \(\zeta=e^{it}\) in the first formula (12) and taking account of formulas (11) and (8), we obtain
* The indices \(\xi,\eta\) denote partial derivatives.
representation of the boundary derivatives of first order of the function \(u(x,y)\) in terms of \(g(\tau)\)
\[ \left.\frac{\partial u}{\partial x}\right|_{\Gamma} = -\frac{1}{2\pi}\int_{-\pi}^{\pi} \psi(g(\omega))\operatorname{ctg}\frac{\tau-\omega}{2}\,d\omega +u_x(x_0,y_0), \]
\[ 0\leqslant \tau \leqslant 2\pi \tag{13} \]
\[ \left.\frac{\partial u}{\partial y}\right|_{\Gamma} = \frac{1}{2\pi}\int_{-\pi}^{\pi} \varphi(g(\omega))\operatorname{ctg}\frac{\tau-\omega}{2}\,d\omega +u_y(x_0,y_0). \]
Remark 1. As is seen from formulas (3), (11), (13), the function \(g(\tau)\) satisfies the singular integral equation \((0\leqslant \tau \leqslant 2\pi)\)
\[ f'(g(\tau))-\varphi'(g(\tau))u_x(x_0,y_0)-\psi'(g(\tau))u_y(x_0,y_0)= \]
\[ =\frac{1}{2\pi}\int_{-\pi}^{\pi} [\varphi(g(\omega))\psi'(g(\tau))-\varphi'(g(\tau))\psi(g(\omega))] \operatorname{ctg}\frac{\tau-\omega}{2}\,d\omega; \]
\[ f'(s)=df(s)/ds;\qquad \varphi'(s)=d\varphi(s)/ds,\qquad \psi'(s)=d\psi(s)/ds. \]
Remark 2. Formulas (9) and (12) give a parametric representation of the solution of the Dirichlet problem for equation (1).
Institute of Hydrodynamics
Siberian Branch of the Academy of Sciences of the USSR
Received
1 IX 1965
CITED LITERATURE
- S. N. Bernstein, Collected Works, 3, Publishing House of the Academy of Sciences of the USSR, 1960.
- I. Ya. Bakel’man, DAN, 157, No. 2, 247 (1964).
- P. Hartman, A. Wintner, Am. J. Math., 75, No. 3 (1953).
- M. A. Lavrent’ev, B. V. Shabat, Methods of the Theory of Functions of a Complex Variable, Moscow, 1958.
- L. A. Galin, Prikl. matem. i mekh., 13, issue 3, 285 (1949).