Abstract Generated abstract
The paper studies a Tricomi boundary value problem for a class of mixed type equations with parameter \(\lambda\), aiming to obtain estimates without assuming smallness of the coefficient or of the transition interval. By introducing an exponential substitution based on a Riccati equation and applying Protter’s extremum theorem, the authors derive an a priori bound for the solution in terms of the prescribed boundary data and its derivative on a characteristic. This estimate implies uniqueness and continuous dependence for fixed \(\lambda\). The result is then applied to a three-dimensional mixed type equation by Fourier transformation in one variable, yielding sufficient decay conditions on transformed boundary functions that ensure existence of a solution represented by an inverse Fourier integral.
Full Text
UDC 517.946
MATHEMATICS
A. M. EZHOV, S. P. PUL'KIN
AN ESTIMATE OF THE SOLUTION OF THE TRICOMI PROBLEM FOR A CLASS OF EQUATIONS OF MIXED TYPE
(Presented by Academician I. N. Vekua on 30 I 1970)
§ 1. The equation
\[ \operatorname{sgn} y \cdot |y|^{m} u_{xx}+u_{yy}-\lambda^{2}\operatorname{sgn} y \cdot |y|^{m}u=0 \qquad (m>0) \tag{1} \]
will be considered in a domain \(D\) bounded by a smooth curve \(\Gamma_0\), lying in the half-plane \(y\ge 0\), with endpoints at the points \(A(0,0)\), \(B(1,0)\), and by the characteristics \(\Gamma_1,\Gamma_2\) of equation (1) issuing from the points \(A\) and \(B\).
The Tricomi problem. In the domain \(D\), find a solution \(u(x,y,\lambda)\) of equation (1) satisfying the boundary conditions
\[ u\big|_{\Gamma_0}=\varphi(x,\lambda),\qquad 0\le x\le 1,\qquad -\infty<\lambda<+\infty, \]
\[ u\big|_{\Gamma_1}=\psi(x,\lambda),\qquad 0\le x\le \tfrac12,\qquad -\infty<\lambda<+\infty, \tag{2} \]
and the gluing conditions
\[ u\big|_{y=0-0}=u\big|_{y=0+0},\qquad \partial u/\partial y\big|_{y=0-0}=\partial u/\partial y\big|_{y=0+0}. \]
The uniqueness of the solution of the Tricomi problem for the equation
\[ y u_{xx}+u_{yy}+c(x,y)u=0 \tag{F} \]
was proved in \((^{1,2})\). In \((^3)\) a principle of absolute extremum was established for equation (F) when \(c(x,y)=-\lambda^2 y\). In all these works the proof of uniqueness of the solution of the Tricomi problem was carried out under the condition that the coefficient \(c(x,y)\) is sufficiently small, or when the length of the transition line is sufficiently small. By means of a certain device it turns out to be possible to remove this restriction for equation (1).
Introduce a new unknown function \(v(x,y,\lambda)\) by the formula
\[ u(x,y,\lambda)=v(x,y,\lambda)\exp\left[\int \omega(x)\,dx\right], \tag{3} \]
where \(\omega(x)\) is a continuously differentiable and nonnegative solution on \([0;1]\) of the differential equation
\[ d\omega/dx+\omega^2-\lambda^2=0. \tag{4} \]
The function \(v(x,y,\lambda)\) is a solution of the equation
\[ \operatorname{sgn} y \cdot |y|^{m}v_{xx}+v_{yy} +2\operatorname{sgn} y \cdot |y|^{m}\omega(x)v_x=0. \tag{5} \]
As a function \(\omega(x)\) with the required properties one may take the function \(\omega(x)=\lambda\,\operatorname{th}\lambda x\). For the coefficients of equation (5), certain conditions established in \((^4)\) are fulfilled. The function
\[ z(x,y,\lambda)=\pm v(x,y,\lambda)+My, \]
where \(M\) is a constant, is also a solution of equation (5). Choose \(M\) so that \(z(x,y,\lambda)\) is a nondecreasing function of \(y\) on the characteristic
\(\Gamma_1\). For this it is enough to set
\[ M=\max_{\Gamma_1}\left|v_y-(y)^{m/2}v_x\right|. \tag{6} \]
Then the function \(z(x,y,\lambda)\) satisfies all the conditions of Protter’s theorem (see (4), § 5, Theorem 4), on the basis of which we can write
\[ |v(x,y,\lambda)|\leq \max_{\Gamma_0}|v(x,y,\lambda)|+MM_1, \tag{7} \]
where \(M_1=\max (y_0-y),\quad y_0=\max_{\overline D}|y|\).
Set
\[ g(x)=\exp\left[-\int \omega(x)\,dx\right]. \]
Taking here \(\omega(x)=\lambda \operatorname{th}\lambda x\), we obtain \(g(x)=1/2\operatorname{ch}\lambda x\). Since \(v(x,y,\lambda)=g(x)u(x,y,\lambda)\), passing in (6) to the function \(u(x,y,\lambda)\), we find
\[ M=\max_{\Gamma_1}\left|g(x)\left[u_y-(-y)^{m/2}u_x\right]-(-y)^{m/2}u g'(x)\right|. \]
Hence we obtain the estimate
\[ M\leq M_2\left[\max_{\Gamma_1}g(x)|\psi_x'(x,\lambda)|+|\lambda|\max_{\Gamma_1}g(x)|\psi(x,\lambda)|\right], \tag{8} \]
where \(M_2=\max_{\Gamma_1}(-y)^{m/2}\).
Now we rewrite estimate (7), taking (8) into account, as follows:
\[ |u(x,y,\lambda)|g(x)\leq \max_{\Gamma_0}g(x)|\varphi(x,\lambda)|+ \]
\[ +M_1M_2\left[\max_{\Gamma_1}g(x)|\psi_x'(x,\lambda)|+|\lambda|\max_{\Gamma_1}g(x)|\psi(x,\lambda)|\right]. \]
Taking further into account that
\[ \max_{\Gamma_1}g(x)=\max_{\Gamma_0}g(x)=1/2, \]
we finally obtain the estimate of the solution of problem (1), (2):
\[ |u(x,y,\lambda)|\leq \operatorname{ch}\lambda x\left\{\max_{\Gamma_0}|\varphi(x,\lambda)|+ \right. \]
\[ \left. +M_1M_2\left[\max_{\Gamma_1}|\psi_x'(x,\lambda)|+|\lambda|\max_{\Gamma_1}|\psi(x,\lambda)|\right]\right\}. \tag{9} \]
From (9) follows the uniqueness of the solution of problem (1), (2), as well as the continuous dependence of this solution on the boundary functions for any fixed \(\lambda\).
§ 2. Consider the equation
\[ \operatorname{sgn}y\cdot |y|^m(u_{xx}+u_{zz})+u_{yy}=0,\quad m>0. \tag{10} \]
The result obtained in § 1 makes it possible to solve the following boundary-value problem. Let \(G\) be a domain of three-dimensional space \((x,y,z)\), bounded by the surfaces:
\[ S_0:\ x(1-x)=\frac{4}{(m+2)^2}y^{m+2},\quad y\geq 0,\quad -\infty<z<+\infty, \]
\[ S_1:\ x-\frac{2}{m+2}(-y)^{(m+2)/2}=0,\quad 0\leq x\leq 1/2,\quad -\infty<z<+\infty, \]
\[ S_2:\ x+\frac{2}{m+2}(-y)^{(m+2)/2}=0,\quad 1/2\leq x\leq 1,\quad -\infty<z<+\infty. \]
Problem TS. In the domain \(G\), find a solution \(w(x,y,z)\) of equation (10), continuous in \(\overline{G}\) and satisfying the conditions:
\[ \left.w\right|_{S_0}=\Phi(x,z),\qquad \left.w\right|_{S_1}=\Psi(x,z), \]
\[ \lim_{z\to\infty} w(x,y,z)=\lim_{z\to\infty}\partial w/\partial z=0. \tag{11} \]
This problem was considered for \(m=1\) in \((^3)\). However, in that work the existence of a solution of problem (1), (2) for arbitrary \(\lambda\) was not proved, and no conditions were obtained ensuring the existence of a solution in the form of a Fourier integral. Our estimates for the solution of problem (1), (2) make it possible to fill this gap. Let \(w(x,y,z)\) be a solution of problem TS. Its Fourier transform with respect to the variable \(z\),
\[ u(x,y,\lambda)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} w(x,y,z)e^{i\lambda z}\,dz, \]
is a solution in the domain \(D\) of equation (1), satisfying the conditions
\[ \left.u\right|_{\Gamma_0}=\varphi(x,\lambda),\qquad -\infty<\lambda<+\infty, \]
\[ \left.u\right|_{\Gamma_1}=\psi(x,\lambda),\qquad -\infty<\lambda<+\infty, \]
where \(D\) is the plane domain bounded by the arcs \(\Gamma_0,\Gamma_1,\Gamma_2\), which are the lines of intersection of the plane \(z=0\) with the surfaces \(S_0,S_1,S_2\), respectively. Here
\[ \varphi(x,\lambda)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\Phi(x,z)e^{i\lambda z}\,dz, \]
\[ \psi(x,\lambda)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\Psi(x,z)e^{i\lambda z}\,dz. \]
Thus, problem TS is reduced to problem (1), (2). Conversely, if the function \(u(x,y,\lambda)\) is a solution of the Tricomi problem (1), (2), then the solution of problem TS is obtained by means of the inverse Fourier transform:
\[ w(x,y,z)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}u(x,y,\lambda)e^{-iz\lambda}\,d\lambda. \tag{12} \]
Taking into account the estimate (9) for the function \(u(x,y,\lambda)\), we can impose on the boundary functions \(\Phi(x,z)\) and \(\Psi(x,z)\) conditions ensuring the existence of the integral (12). Let the functions \(\Phi(x,z)\), \(\Psi(x,z)\) be such that, for the functions \(\varphi(x,\lambda)\), \(\psi(x,\lambda)\), the estimates
\[ \varphi(x,\lambda)=O(1/|\lambda|^\alpha \operatorname{ch}\lambda),\qquad \psi(x,\lambda)=O(1/|\lambda|^{1+\alpha}\operatorname{ch}\lambda), \]
\[ \psi'_x(x,\lambda)=O(1/|\lambda|^\alpha \operatorname{ch}\lambda),\qquad \alpha>1. \]
hold. Then from (9) we have the estimate \(u(x,y,\lambda)=O(1/|\lambda|^\alpha)\), which ensures the existence of the integral (12), giving a solution of problem TS.
Kuibyshev State University
Received
14 I 1970
References
- K. I. Babenko, On the theory of equations of mixed type, Dissertation, Moscow, 1951.
- V. F. Volkodavov, L. M. Nevostruev, Volga Mathematical Collection, issue 4 (1966).
- A. M. Nakhushev, Differential Equations, 4, no. 1 (1968). S. Agmon, L. Nirenberg, M. Protter. Comm. Pure and Appl. Math., 6, 4 (1953).