Reports of the Academy of Sciences of the USSR
1970. Volume 190, No. 2

UDC 577.93

MATHEMATICS

D. K. GVAZAVA

ON THE UNIQUENESS OF THE SOLUTION OF THE TRICOMI PROBLEM FOR ONE CLASS OF NONLINEAR EQUATIONS

(Presented by Academician M. A. Lavrent’ev on 2 VI 1969)

Let \(D\) be a simply connected bounded domain in the plane of the variables \(x, y\) with boundary \(\Gamma=\sigma+AC+CB\), where \(\sigma\) is an open Jordan curve in the half-plane \(y>0\) with endpoints at \(A(0,0)\), \(B(1,0)\). The curves \(AC\) and \(CB\) lie in the lower half-plane and are given by the equations

\[ x-\frac{2}{m+2}(-y)^{(m+2)/2}=0,\qquad x+\frac{2}{m+2}(-y)^{(m+2)/2}=1 \]

respectively. The number \(m\) is odd and positive.

In this domain we consider the equation

\[ T[u]\equiv y^m u_{xx}+u_{yy}=f(x,y,u), \tag{1} \]

where \(f(x,y,u)\) is a function which, for all \((x,y)\in \overline D\) and for all \(|u|<\infty\), satisfies the following conditions: \(f(x,y,u)\) and \(f_u(x,y,u)\) are continuous,

\[ f_u(x,y,u)>0, \tag{2} \]

\[ f_u(x,y,u)\leq m(m+4)4^{-(m+4)/(m+2)}(m+2)^{-2m/(m+2)}y^{-2} \quad \text{for } y<0. \tag{3} \]

Equation (1) is elliptic for \(y>0\), degenerates parabolically for \(y=0\), is hyperbolic for \(y<0\), and the curves \(AC\) and \(CB\) are its characteristics.

By \(\varphi(s)\) we denote a continuous function prescribed on \(\sigma\) of the arc length \(s\), measured from the point \(B(1,0)\) in the positive direction. The function \(\psi(x,y)\) is defined for \((x,y)\in \overline D,\ y\leq 0\) and is continuously differentiable once. In addition, we introduce the notation \(D_1=D\cap\{y>0\}\); \(D_2=D\cap\{y<0\}\). By \(D^\varepsilon\) we denote the subdomain of \(D\) obtained by cutting off from \(D\) a piece by the curves \(\sigma_\varepsilon\) and \(B_\varepsilon C_\varepsilon\), where

\[ \sigma_\varepsilon:\ (x-1)^2+\frac{4}{(m+2)^2}y^{m+2}=\varepsilon^2,\qquad B_\varepsilon C_\varepsilon:\ x+\frac{2}{m+2}(-y)^{(m+2)/2}=1-\varepsilon. \]

Accordingly, we shall denote

\[ D_1^\varepsilon=D^\varepsilon\cap\{y>0\},\qquad D_2^\varepsilon=D^\varepsilon\cap\{y<0\}. \]

Tricomi problem. It is required to find a solution \(u(x,y)\) of equation (1), regular in \(D\) and continuous in \(\overline D\), satisfying the conditions

\[ u|_\sigma=\varphi, \tag{4} \]

\[ u|_{AC}=\psi|_{AC}, \tag{5} \]

and moreover its first derivatives may have integrable singularities at the point \(A\) and on the closed arc \(BC\).

Theorem 1. If conditions (2), (3) are satisfied, then problem (1), (4), (5) can have only one solution.

The validity of this assertion follows directly from the following theorem.

Theorem 2. If conditions (2), (3) are satisfied, then any solution of problem (1), (4), (5), if it exists, satisfies the estimate

\[ |u|\leq kK, \tag{6} \]

where \(k>0\) is a constant depending on the diameter of the domain \(D\), and the number \(K\) is determined

is determined as follows:

\[ K=\max_{\overline D}|f(x,y,0)|+\max_{\overline\sigma}|\varphi(s)|+\max_{\overline{AC}}|\psi(x,y)|+\max_{\overline{AC}}|x^{m/(m+2)}\psi_\eta|, \tag{7} \]

where

\[ \psi_\eta=-\frac12\psi_x+\frac12(-y)^{-m/2}\psi_y. \]

First consider the equation

\[ T[u]=-c(x,y)u+F(x,y) \tag{8} \]

and prove the validity of estimate (6) for it.

Theorem 3. If the function \(z(x,y)\) satisfies the conditions

\[ z(x,y)\in C(\overline D)\cap C^1(D\cup AC)\cap C^2(D), \]

\[ T[z]+C(x,y)z=E(x,y)\ge 0, \tag{9} \]

\[ z_\eta(x,y)\big|_{AC}\ge 0, \tag{10} \]

then the greatest positive value of the function \(z(x,y)\), if it exists, is attained on \(\overline\sigma\).

To prove this assertion we shall use the following known theorems.

Theorem (see \((^1,{}^2)\)). If conditions (2), (3), (9), (10) are satisfied, then the function \(z(x,y)\) can attain its greatest positive value in the closed domain \(\overline D^{\,\varepsilon}\) only on the segment \(y=0,\ 0\le x\le 1-\varepsilon\).

Theorem (see \((^2)\)). Under the conditions of the preceding theorem, if \(z(x,y)\) attains its greatest positive value in \(\overline D_2\) at the point \((x_0,0)\), \(0<x_0<1\), then

\[ z_y(x_0,0)>0. \tag{11} \]

Proof of Theorem 3. Fix an arbitrary number \(\varepsilon>0\). According to the condition of the theorem, \(z(x,y)\in C(\overline D)\), and therefore it is continuous in the closed subdomain \(\overline D^{\,\varepsilon}\). Denote its greatest positive value in \(\overline D^{\,\varepsilon}\) by \(M_\varepsilon\). According to the first theorem, we have

\[ 0\le \max_{\overline D^{\,\varepsilon}_2} z(x,y) =\max_{\overline{AB}_\varepsilon} z(x,y) =z(x_0,0),\qquad 0\le x_0\le 1-\varepsilon, \]

where, naturally, \(z(x_0,0)\le M_\varepsilon\). By the known maximum principle for elliptic equations (see \((^3)\)) we have

\[ \max_{\overline D^{\,\varepsilon}_1} z(x,y) = \max_{\overline\sigma_0\cup \overline{AB}_\varepsilon\cup \overline\sigma_\varepsilon} z(x,y)\le M_\varepsilon, \]

where

\[ \sigma_0=\sigma\setminus\left\{\left.(x-1)^2+\frac{4}{(m+2)^2}y^{m+2}\le \varepsilon^2\right\}\right. . \]

Thus, we obtain

\[ \sup\left\{\max_{\overline D^{\,\varepsilon}_2} z(x,y),\ \max_{\overline D^{\,\varepsilon}_1} z(x,y)\right\} = \max_{\overline D^{\,\varepsilon}} z(x,y) = \max_{\overline\sigma_0\cup \overline{AB}_\varepsilon\cup \overline\sigma_\varepsilon} z(x,y). \tag{12} \]

On the other hand, \(z(x,y)\) is continuous in the closed domain \(\overline D_1\), twice continuously differentiable in \(D_1\), and satisfies condition (9). According to the maximum principle,

\[ \max_{\overline D_1} z(x,y)= \max_{\overline\sigma\cup \overline{AB}} z(x,y), \]

and, consequently,

\[ \max_{\overline\sigma_\varepsilon} z(x,y)\le \max_{\overline\sigma\cup \overline{AB}} z(x,y). \tag{13} \]

From (12) and (13) we obtain

\[ \max_{\overline{D}^{\varepsilon}} z(x,y)\leq \max_{\sigma\cup AB} z(x,y)=\max_{\overline{D}_1} z(x,y). \]

Suppose that \(\max_{\overline{D}_1} z(x,y)\) is attained on \(AB\) at some point \((x_0,0)\), \(0<x_0<1\). Then, according to Zaremba’s lemma \((^{2,4})\), at this point we shall have \(z_y(x_0,0)<0\), which contradicts inequality (11). Consequently, our supposition is false. Thus, finally we obtain

\[ M_\varepsilon=\max_{\sigma} z(x,y) \]

for arbitrary \(\varepsilon>0\), as was required to prove.

Proof of Theorem 2 for equation (8). Consider the function

\[ h(y)=K(\alpha-\beta e^{2y}), \tag{14} \]

where
\[ \beta>\sup\left\{(1+e^{2\overline{y}}),\ \left[1+2\left(\frac{2}{m+2}\right)^{m/(m+2)}e^{-1}\right]\right\},\quad \overline{y}=\max_{\overline{D}} |y|,\quad \alpha>1+\beta e^{2\overline{y}}. \]

By direct calculation we obtain

\[ h_\eta(y)=-\beta(-y)^{-m/2}e^{2y}K, \]

\[ -\left.h_\eta(y)\right|_{AC} =\beta\left(\frac{2}{m+2}\right)^{m/(m+2)} \exp\left[-2\left(\frac{m+2}{2}x\right)^{2/(m+2)}\right] x^{-m/(m+2)}K. \]

Hence, multiplying both sides by \(x^{m/(m+2)}\), we have

\[ -x^{m/(m+2)}[h_\eta(y)]_{AC}\geq K\geq \max_{\overline{AC}}\left|x^{m/(m+2)}\psi_\eta\right|, \]

i.e.

\[ -x^{m/(m+2)}[h_\eta(y)]_{AC}\geq \left|x^{m/(m+2)}\psi_\eta\right|_{AC}, \]

whence it follows that

\[ -x^{m/(m+2)}[h_\eta(y)]_{AC}\geq -x^{m/(m+2)}[\psi_\eta(x,y)]_{AC}, \tag{15} \]

\[ -x^{m/(m+2)}[h_\eta(y)]_{AC}\geq x^{m/(m+2)}[\psi_\eta(x,y)]_{AC}. \tag{16} \]

Further,

\[ -\{T[h]+c(x,y)h\} =\{4\beta e^{2y}-c(\alpha-\beta e^{2y})\}K \geq K\geq \max_{\overline{D}} |F(x,y)|. \tag{17} \]

In addition, the inequalities

\[ \left.h\right|_{\sigma}\geq \max_{\sigma}|\varphi(s)|, \tag{18} \]

\[ \left.h\right|_{AC}\geq \max_{\overline{AC}}|\psi(x,y)|. \tag{19} \]

are obvious.

Take the function \(z(x,y)=u(x,y)-h(y)\), where \(u(x,y)\) is a solution of problem (4), (5), (8). For it, according to (17) and (15), we have
\[ T[z]+cz=F(x,y)-\{T[h]+ch\}\geq F(x,y)+\max_{\overline{D}}|F(x,y)|\geq 0,\quad z_\eta(x,y)\big|_{AC}\geq 0. \]

From (18) and (19), respectively, the inequalities follow

\[ z(x,y)\big|_{\sigma}\leq 0,\qquad z(x,y)\big|_{AC}\leq 0. \tag{20} \]

Thus, the function \(z(x,y)\) satisfies all the requirements of Theorem 3. Therefore it must be that
\[ 0\leq \max_{\overline{D}} z(x,y)=\max_{\sigma} z(x,y), \]
whence, on the basis of inequality (20), we obtain

\[ z(x,y)=u(x,y)-h(y)\leq 0\qquad \text{everywhere in } \overline{D}. \tag{21} \]

Next consider \(z_1(x,y)=-u(x,y)-h(y)\), for which, analogously, we obtain
\(T[z_1]+cz_1\ge -F(x,y)+\max |F(x,y)|\ge 0\). On the basis of condition (16) we have \((z_1)_\eta|_{AC}\ge 0\), and from (18), (19) it follows that

\[ z_1(x,y)|_\sigma \le 0,\qquad z_1(x,y)|_{AC}\le 0. \tag{22} \]

According to Theorem 3 we may write

\[ z_1(x,y)=-u(x,y)-h(y)\le 0 \quad \text{everywhere in } \overline D. \tag{23} \]

Inequalities (22) and (23) give \(u(x,y)\le h(y)\), \(-h(y)\le u(x,y)\), whence we finally have \(|u(x,y)|\le h(y)\). Denoting
\(k=\max_{\overline D}(a-\beta e^{2y})\), we obtain estimate (6).

It is now easy to prove Theorem 2. Let there exist a solution of the Tricomi problem (1), (4), (5), namely \(u(x,y)\). Substituting it into equation (1), we obtain the identity \(T[u]\equiv f(x,y,u)\), which can be written in the form (5)
\(T[u]-f_u(x,y,\tilde u)=f(x,y,0)\), where \(\tilde u\) lies between zero and \(u(x,y)\). In the identity thus obtained, the coefficient of \(u(x,y)\) satisfies conditions (2), (3). Referring to the assertion proved above, we obtain the estimate \(|u|\le h\), which completes the proof of Theorem 2.

Suppose that the Tricomi problem (1), (4), (5) has two solutions \(u_1(x,y)\) and \(u_2(x,y)\). Then the difference \(v(x,y)=u_1(x,y)-u_2(x,y)\) vanishes on the curves \(\sigma\) and \(AC\), respectively. Consequently, \(v_\eta|_{AC}=0\). Moreover, we have the identity (see (6))
\[ T[v]\equiv T[u_1]-T[u_2]\equiv f(x,y,u_1)- \]
\[ -f(x,y,u_2)=(u_1-u_2)\int_0^1 \Phi_{u_1}(x,y,u_1,t)\,\frac{dt}{t}, \quad \text{where } \Phi(x,y,u_1,t)= \]
\[ =f(x,y,u_2+t(u_1-u_2)). \]
The coefficient of \(v=u_1-u_2\) satisfies the required conditions.

On the basis of Theorem 2 it follows that \(v(x,y)=0\) everywhere in \(\overline D\), i.e. \(u_1(x,y)\equiv u_2(x,y)\). The uniqueness theorem is proved.

CITED LITERATURE

\(^{1}\) A. Haar, C. R., 187, 23 (1928).
\(^{2}\) S. Agmon, L. Nirenberg, M. Protter, Comm. Pure and Appl. Math., 6, 455 (1953).
\(^{3}\) A. V. Bitsadze, Boundary-Value Problems for Elliptic Equations of Second Order, Moscow, 1966.
\(^{4}\) S. Zaremba, УМН, 1, 3–4, 125 (1946).
\(^{5}\) R. Courant, Partial Differential Equations, Moscow, 1964.
\(^{6}\) K. Miranda, Partial Differential Equations of Elliptic Type, Moscow, 1957.