L. I. Kovalenko

A GENERALIZED SOLUTION OF THE TRICOMI PROBLEM

(Presented by Academician L. V. Kantorovich, December 4, 1964)

The question of the existence of a solution of the Tricomi problem for second-order mixed-type equations with lower-order terms was considered in papers \((^1)\) (for the general Lavrent’ev—Bitsadze equation) and \((^2)\). In \((^2)\) the existence of only a weak solution was proved, and its uniqueness was not proved. In the present note, by the method of finite differences, the existence of a generalized solution of the Tricomi problem for a mixed-type equation with lower-order terms is proved. A generalized solution is understood in the sense of paper \((^3)\). In \((^3)\) the uniqueness of such a solution was proved. For the first time, finite differences were used in proving the existence of solutions of boundary-value problems for a mixed-type equation in paper \((^4)\) (for the Lavrent’ev—Bitsadze equation).

Consider the equation

\[ K(y)u_{xx}^{\prime\prime}+u_{yy}^{\prime\prime}+\tilde a(x,y)u_x^{\prime}+\tilde b(x,y)u_y^{\prime}+\tilde c(x,y)u=\tilde f(x,y), \tag{1} \]

where \(K(y)=|y|^\alpha q(y)\operatorname{sgn} y,\ \alpha>0,\ q(y)>0,\ q(\pm0)>0\). By the substitution of the unknown function
\[ v=u\exp\left\{\int_0^y \bar b(x,t)\,dt/2\right\} \]
equation (1) is reduced to the form

\[ K(y)v_{xx}^{\prime\prime}+v_{yy}^{\prime\prime}+a(x,y)v_x^{\prime}+c(x,y)v=f(x,y). \tag{2} \]

In what follows we shall consider only such an equation.

Let \(D\) be a domain the same as in \((^3)\). For \(y<0\) it is bounded by the characteristics \(\Gamma_0\) and \(\Gamma_A\) of equation (2). Let, near the points \(O(0,0)\), \(A(x_1,0)\), the curve \(\sigma\) (the boundary of the domain \(D\) for \(y\ge 0\)) end in the arcs \(\sigma_0\) and \(\sigma_A\), respectively. The arc \(\sigma_0\) is given by the equation \(x=\lambda(y)\), \(\lambda(y)\in C^{(2)}\) for \(y>0\); \(\lambda(0)=\lambda'(0)=0\). In the case \(\lambda(y)\ne0\) for \(y>0\), the inequalities
\[ c_1K'\le \lambda'\sqrt K\le c_2K',\qquad c_1,c_2=\mathrm{const}>0 \]
hold. The arc \(\sigma_A\) is situated below the straight line \(y=c_3(x_1-x)\), \(c_3=\mathrm{const}>0\). Denote by \(B(x_B,y_B)\) the end of the curve \(\sigma_0\) for \(y>0\); \(B\in\sigma-\sigma_0\). We assume that in the rectangle \(\Delta_B\{0\le x\le x_B,\ 0\le y\le y_B\}\) there are no points of the curve \(\sigma\), except points of the arc \(\sigma_0\).

Suppose that the coefficients of equation (2) satisfy conditions I and II of paper \((^3)\) and, in addition, the following conditions.

Conditions III: 1) for any \(\varepsilon>0\), in the domain \(\varepsilon<y<Y_2\), \(K(y)\in C^{(1,\mu)}\), \(\mu=\mu(\varepsilon)\); there exist \(s_1,s_2=\mathrm{const}>0\) such that
\[ s_1|y|^{\alpha-1}\le K'\le s_2|y|^{\alpha-1} \]
for \(y\ne0\); 2) \(a\in C^{(2)}(D^++D^-+\Gamma_0)\); \(|y|^{2-\alpha/2}a_y'\to0\) as \(y\to-0\) in \(D^-\); \(a_x'\), \(a_{xx}^{\prime\prime}\) are bounded in \(D^++D^-\), and \(ya_y'\) and \(y^{2-\alpha/2}a_y'\) in \(D^+\); 3) \(c,f\in C^{(1)}(D^++D^-)\); \(c_x'\), \(f_x'\) are bounded in \(D^++D^-\), and \(yc_y'\) and \(yf_y'\) in \(D^+\); 4) \(f=0\) on \(\Gamma_0+\sigma_0\); 5) \(c+a_x'\le0\) in \(D^-+\Gamma_0\); 6) \(K'^2d+16K^2a_x'>0\) in \(D^-\); 7) for any sufficiently small \(\varepsilon,\varepsilon_1,\varepsilon_2>0\) there exists \(c_4=\mathrm{const}>0\) such that, for \(0<y\le\varepsilon,\ 0<\varepsilon_1\le x_1-x\le\varepsilon_2,\ (x,y)\in D^+\), one has
\[ |a|\le c_4\sqrt K; \]
8) in the case \(\lambda(y)\equiv0\), in a neighborhood of the point \(O\) in \(D^+\), the ratio \(a/x\) is bounded above.

Theorem 1. Under the assumptions made, there exists a generalized solution of the Tricomi problem \((^3)\) for equation (2) with boundary conditions

\[ v=0 \text{ on } \Gamma_0+\sigma_0,\qquad v=\varphi \text{ on } \sigma-\sigma_0, \]

where \(\varphi\) is a given continuous function, \(\varphi(B)=0\).

We note that the Tricomi problem for equation (2) with \(f\) and \(\varphi\) not necessarily equal to zero on \(\Gamma_0+\sigma_0\) reduces to the one considered, under certain assumptions on the behavior of \(f\) and \(\varphi\) on \(\Gamma_0+\sigma_0\).

To prove Theorem 1, introduce a mesh (see \((^3)\)) with sufficiently small step \(h\), and consider difference equations of the same kind as in \((^3)\) for the case \(b\equiv 0\); \(v_h\) is the desired mesh function, the solution of the system of equations

\[ Rv_h=f_h^* \quad \text{in } D_h,\qquad v_h=0 \quad \text{on } \sigma_{0h}+\Gamma_{0h},\qquad v_h=\varphi_D \quad \text{on } \sigma_h-\sigma_{0h}, \]

where \(f_h^*=f\) in \(D_h^+ + D_h^-\), \(f_h^*=0\) on \(\gamma_h\); \(\sigma_{0h}\) is the set of boundary nodes belonging to \(\Delta_B\).

All notation, except that introduced for the first time in this note, is as in \((^3)\). Let \(D_0\) be a domain such that \(\overline{D}_0 \subset \overline{D}-\sigma+\sigma_0\); \(\overline{D}_{0h}\) is the mesh domain for \(\overline{D}_0\), defined analogously to \(\overline{D}_h\) in \((^3)\).

From estimate (12) of \((^3)\) follows the uniform boundedness of \(v_h\) in \(\overline{D}_h\).

The most essential part of the proof of Theorem 1 is the proof of the uniform boundedness of the first divided differences of the function \(v_h\) in \(\overline{D}_{0h}\).

Using the fact that \(v_{0,n}=0\), we obtain that in \(\overline{D}_{0h}\)

\[ v_{k,n} = \sum_{j=1}^{k+1}\beta_j^{(k,n)}v_{k+1-j,0} - \sum_{\substack{p=0,1,\ldots,k\\ q=1,2,\ldots,n+k-p}} l_q l_{q+1}\delta_{p,q}^{(k,n)} f_{p,q}, \tag{3} \]

where \(\beta_j^{(k,n)}\), \(\delta_{p,q}^{(k,n)}\) are certain numbers depending on the coefficients of the difference system of equations for \(v_{k,n}\). They have the following properties:

\[ \beta_j^{(k,n)}>0,\qquad \sum_{j=1}^{k+1}\beta_j^{(k,n)}<1,\qquad l_1'<\bar c\left[1-\sum_{j=1}^{k+1}\beta_j^{(k,2)}\right], \]

\[ \sum_{\substack{p=0,1,\ldots,k\\ q=1,2,\ldots,n+k-p}} l_q l_{q+1}\left|\delta_{p,q}^{(k,n)}\right| < y_n(3Y_1-y_n), \tag{4} \]

where \(\bar c=2(Y_1+c_*^{-1})\), and \(c_*\) is the same constant as in (11) of \((^3)\).

Expressing \(v_{k,2}\) by formula (3) and using (5) of \((^3)\), we obtain, for \((x,0)\in\gamma_h\),

\[ \sum_{j=1}^{k+1}\beta_j^{(k)}v_h(x-jh,0) - v_h(x,0) + l_1'v_{hy}(x,0) = \sum_{\substack{p=0,1,\ldots,k\\ q=1,2,\ldots,2+k-p}} l_q l_{q+1}\delta_{p,q}^{(k)} f_{p,q}, \tag{5} \]

where \(k=x/h-1\), \(\beta_j^{(k)}=\beta_j^{(k,2)}\), \(\delta_{p,q}^{(k)}=\delta_{p,q}^{(k,2)}\).

Let

\[ v_{x,k,n}\equiv v_{hx}(x,-y_n),\qquad \text{where } x=kh+nh/2. \]

Putting \(v_h=0\) at the nodes of the characteristic

\[ x=-h+\int_y^0 \sqrt{-K(t)}\,dt, \]

we obtain that \(v_{x,-1,n}=0\). Then the differences \(v_{x,k,n}\) are expressed in terms of \(v_{x,m,0}\) by means of a formula analogous to (3). Similarly to relation (5), we obtain, for \((x,0)\in\gamma_h\), \((x+h,0)\in\gamma_h\),

\[ \sum_{j=1}^{k+1}\widetilde{\beta}_j^{(k)}v_{hx}(x-jh,0) - v_{hx}(x,0) + l_1'v_{hxy}(x,0) = \sum_{\substack{p=0,1,\ldots,k\\ q=1,2,\ldots,2+k-p}} l_q l_{q+1}\widetilde{\delta}_{p,q}^{(k)}\zeta_{p,q}, \tag{6} \]

where \(k=x/h-1\), \(\widetilde{\beta}_j^{(k)}=\widetilde{\beta}_j^{(k,2)}\), \(\widetilde{\delta}_{p,q}^{(k)}=\widetilde{\delta}_{p,q}^{(k,2)}\); \(\widetilde{\beta}_j^{(k,n)}\), \(\widetilde{\delta}_{p,q}^{(k,n)}\) are numbers analogous to \(\beta_j^{(k,n)}\), \(\delta_{p,q}^{(k,n)}\); \(\xi_{p,q}=f_{x,p,q}-c_{x,p,q}v_{p+1,q}-a_{xx}(ph+\theta_1h+qh/2,-y_q)v_{x,p,q}h/2\), \(0<\theta_1<1\).

For \(\widetilde{\beta}_j^{(k,n)}\), \(\widetilde{\delta}_{p,q}^{(k,n)}\) estimates analogous to (4) hold. We denote the left-hand side of (6) by \(R_0^x v_{hx}\).

For \(y>0\) the differences \(v_{hx}\) satisfy the following equations:

\[ K(v_{hx})_{xx}+(1+a_n')(v_{hx})_{yy}+[a(x+h,y_n')(v_{hx})_x+ \]
\[ +a(v_{hx})_x]/2+(c+a_x')v_{hx}=\zeta_h^+, \tag{7} \]

where

\[ \zeta_h^+=f_x-c_xv_h(x+h,y_n')-a_{xx}''(x+\theta_2h,y_n')v_{hx}h/2,\qquad 0<\theta_2<1. \tag{8} \]

In (7) and (8), as also in the subsequent formulas for \(y>0\), the values of all functions for which the arguments are not indicated are taken at the point \(x=kh\), \(y=y_n'\); \(f_x, c_x\), etc. are divided differences.

The uniform boundedness of the differences \(v_{hx}, v_{hy}\) at the nodes of any domain located, together with its boundary, in \(D^+\), is proved by means of S. N. Bernstein’s method (see \((^5,^6)\)).

By a method analogous to (7), the boundedness of the first differences on \(\sigma_{0h}\) for \(y\ge \varepsilon>0\) is proved. This makes it possible to prove the boundedness of \(v_{hx}\) and \(v_{hy}\) for \(y\ge \varepsilon>0\) in \(\overline{D}_{0h}^{+}\) up to \(\sigma_{0h}\).

Let \(\Delta^i\) be the domain \(\lambda(y)+ih<x<a_0\), \(il_1'<y<\varepsilon\) \((a_0<x_1,\ \varepsilon>0,\ i=0,1)\); let \(\overline{\Delta}_h^i\) be the mesh domain for \(\Delta^i\), defined analogously to \(\overline{D}_h\) in (3); \(G_h^i\) is the set of boundary nodes for \(\overline{\Delta}_h^i\). For proving the boundedness of the differences \(v_{hx}\) up to the line \(y=0\), the following is important.

Lemma 1. Suppose \(|ah|<2K\) in \(\Delta_h^0\); the function \(z\) is defined in \(\overline{\Delta}_h^0\). Suppose the estimates hold: \(\widetilde{\beta}_j^{(k)}>0\), \(\widetilde{\beta}_1^{(k)}+\widetilde{\beta}_2^{(k)}+\ldots+\widetilde{\beta}_{k+1}^{(k)}<1\). If

\[ R_x^x z\equiv Kz_{xx}+(1+a_n')z_{yy}+[a(x+h,y_n')z_x+az_x^-]/2\ge 0 \]

in \(\Delta_h^0\), \(R_0^x z\ge 0\) on \(\gamma_h^0\) (\(\gamma_h^0\) is the set consisting of the nodes with \(y=0\)), then \(\max z\) in \(\overline{\Delta}_h^0\) is attained on \(G_h^0-\gamma_h^0\).

Applying Lemma 1 to the function

\[ z_1=(a_0-x)^2 e^{M_1y^2}v_{hx}^2+M_2[v_h^2+v_h^2(x+h,y)]+M_3ye^y, \]

where \(M_i=\mathrm{const}>0\) \((i=1,2,3)\), \(x=kh\), \(y=y_n'\), we prove the boundedness of \(v_{hx}\) for \(0\le y\le \varepsilon\), \(\lambda(y)\le x\le x_1-\varepsilon_1\).

Here one uses the boundedness of \(v_{hx}\) on \(\sigma_{0h}\) for \(0\le y\le \varepsilon\), proved with the help of the barrier \(\omega_0=[x-\lambda(y)](c_5-e^{-c_6y})\) for \(y\ge 0\) and

\[ \omega_0=\left(x-\int_0^y\sqrt{-K(t)}\,dt\right)(c_5-e^{-c_6y}) \]

for \(y\le 0\), \(c_5,c_6=\mathrm{const}>0\).

The boundedness of \(v_{hx}\) in \(\overline{D}_{0h}^{-}\) follows from the fact that the differences \(v_{x,k,n}\) are expressed in terms of \(v_{x,m,0}\) by means of a formula analogous to (3). Moreover, in \(\overline{D}_{0h}^{-}\) the quantities \((v_{x,k,n}-v_{x,k,n+1})/a_n\) are uniformly bounded. To prove this it is enough to apply Lemma 1 from work (3) to the functions \(z^{(i)}=M_4(c_7-e^{-y})+(-1)^i v_{hx}\), where \(i=1,2\); \(M_4,c_7=\mathrm{const}>0\).

We proceed to the estimate of the differences \(v_{hy}\) for \(y\le \varepsilon\) \((\varepsilon>0)\) in \(\overline{D}_{0h}\). For \(y<0\), by \(v_{y,k,n}\equiv v_{hy}(x,-y_n)\) we denote the ratio \((v_{k+1,n-2}-v_{k,n})/(l_{n-1}+l_n)\), and by \(v_{l,k,n}\) the ratio \((v_{k,n-1}-v_{k,n})/l_n\) (the “oblique” difference). Using the fact that \(v_h\) satisfies system (3) from work (3) and that \(v_{l,0,m}=0\), we obtain:

\[ v_{l,k,n}=\sum_{i=0}^{k-1}\left[ \frac{a_{n+i}-\alpha_{k-i,n+i}}{1-a_{n+i}}\, \frac{h}{l_{n+i}}\,v_{x,k-i-1,n+i}\right. \]
\[ \left. -\frac{l_{n+i}+l_{n+i+1}}{2} (c_{k-i,n+i}v_{k-i,n+i}-f_{k-i,n+i}) \right]. \tag{9} \]

From (9), in view of the boundedness of \(v_h\) and \(v_{hx}\), there follows the boundedness of the “oblique” differences. The boundedness of the differences \(v_{hy}\) in \(\overline{D}_{0h}\) follows from the fact that they are expressed in terms of \(v_{hx}\) and \(v_{l,k,n}\).

In \(D_{0h}^{+}\), for \(0<y\leqslant \varepsilon\), we first prove the boundedness of \(\sqrt{y}\,v_{\bar h y}\). In doing so we use the fact that

\[ R_1 z_2 \equiv K z_{2xx}+(1+a_n')z_{2yy}+a(z_{2x}+z_{2\bar x})/2 \geqslant 0 \]

in \(\Delta_h^1\), for

\[ z_2=(a_0-x)^2 y_n v_{hy}^2+v_{hx}^2+(y_n')^2 v_{hx}^2(x,y_{n-1}) +M_5\bigl[v_h^2+v_h^2(x-h,y_n')+v_h^2(x+h,y_n')\bigr] +M_6[1-(y_n')^r], \]

\[ r=\min(\alpha,1/24),\qquad M_5,M_6=\mathrm{const}>0, \]

and, consequently, \(\max z_2\) in \(\overline{\Delta}_h^1\) is attained on the boundary \(G_h^1\). Similarly we prove the boundedness of \(v_{hy}\), taking

\[ z_3=(a_0-x)^2 v_{hy}^2+M_7(1-\sqrt{y_n'}) \]

and using the fact that in \(\Delta_h^0\)

\[ R_h''z_3\equiv R_1z_3-\nu_h z_{3y}\geqslant 0, \]

where

\[ \nu_h=(a_{n-1}'-a_n')/l_n'+(1+a_{n-1})K_{\bar y}/K(y_{n-1}). \]

Thus, in \(\overline{D}_{0h}\) the first differences of \(v_h\) are uniformly bounded.

Next, from the values of \(v_h\) we define bilinear functions \(v_h'\) and piecewise-constant ones (similarly to (8), Chap. I, § 6). On the basis of Arzelà’s theorem one can select a subsequence \(v_{h_m}'\) uniformly convergent in \(D_0\) to a certain continuous function \(v\). This function satisfies requirements 1)—5) of the generalized solution of the Tricomi problem \((^3)\). In proving this, the boundedness of the first differences of \(v_h\) and the boundedness of the quantities

\[ (v_{x,k,n}-v_{x,k,n+1})/a_n \]

are essentially used. The fact that \(v\) satisfies the integral identity (14) of \((^3)\) (\(b=0\)) is proved similarly to (8), Chap. III. The continuity of \(v\) at points of the curve \(\sigma-\sigma_0\) and the satisfaction of the condition \(v=\varphi\) on \(\sigma-\sigma_0\) are ensured by the regularity of the points of the curve \(\sigma-\sigma_0\) (see \((^9)\), p. 106; \((^{10})\), p. 254). For the point \(A\), the function

\[ \omega_A=c_8(x_1-x)-y-c_9y^2, \]

where \(c_8,c_9=\mathrm{const}>0\), may be used as a barrier. Thus, Theorem 1 is proved.

From the uniqueness of the generalized solution of the Dirichlet problem (understood in a sense analogous to \((^3)\)) in the domain situated in the half-plane \(y>0\), and the existence of a smooth solution of the Dirichlet problem (see \((^9)\)), it follows that \(v\) in \(D^{+}\) has continuous derivatives of the first and second order which enter equation (2), and satisfies (2) in the usual sense.

From Theorem 1 and the results of \((^3)\) it follows that, under the fulfillment of conditions I, II (see \((^3)\)) and III, there exists and is unique a generalized solution of the Tricomi problem for equation (2).

The author expresses gratitude to A. F. Filippov for posing the question and for consultations, and also to V. G. Karmanov for advice.

Moscow Institute of Physics and Technology

Received
16 XI 1964

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