UDC 517.216.21

MATHEMATICS

V. P. MIKHAILOV

ON THE GENERALIZED TRICOMI PROBLEM

(Presented by Academician I. G. Petrovskii, 17 X 1966)

Let \(K(y)\) be a function continuously differentiable on the interval \(y_0 \leqslant y \leqslant y_1\), \(y_0 < 0\), \(y_1 > 0\), having the following properties: \(K(y)<0\) for \(y_0 \leqslant y<0\); \(K(y)>0\) for \(0<y\leqslant y_1\); \(K(0)=0\), \(K'(0)>0\).

Consider in the plane \(\{x,y\}\) a domain \(Q\) bounded by the closed curve \(\Gamma=\Gamma_0\cup\Gamma_1\cup\Gamma_2\). The curve \(\Gamma_0\): \(x=x(s)\), \(y=y(s)\), \(0\leqslant s\leqslant S\), \(x(0)=1\), \(x(S)=0\), \(y(0)=0\), \(y(S)=0\) (\(s\) is the arc length along this curve) is situated in the half-plane \(y\geqslant 0\) and has only two common points with the axis \(Ox\), \((0,0)\) and \((1,0)\); \(\max_{0\leqslant s\leqslant S} y(s)=y_1\). The functions \(x(s)\) and \(y(s)\) will be assumed differentiable on \([0,S]\), and the curve \(\Gamma_0\) to have no singular points. Moreover, suppose that \(\dot{x}(0)\dot{y}(0)<0\), and if \(\dot{y}(S)=0\), then \(\dot{x}(S)<0\). The curve \(\Gamma_1\) is situated in the half-plane \(y\leqslant 0\), passes through the origin, and has the equation \(x=\mu(y)\), \(y\in[y_0,0]\), where \(\mu(y)\) is continuous for \(y_0\leqslant y\leqslant 0\), differentiable for \(y_0\leqslant y<0\); the derivative of the function \(\mu(y)\) for \(y_0\leqslant y<0\) satisfies the inequality \(\mu'(y)\leqslant -\sqrt{-K(y)}\). The curve \(\Gamma_2\) is also situated in the half-plane \(y\leqslant 0\) and has the equation \(x=1-\int_y^0 \sqrt{-K(\eta)}\,d\eta\), i.e., passes through the point \((1,0)\). The curves \(\Gamma_1\) and \(\Gamma_2\) intersect at the point \((\nu,y_0)\), \(0<\nu<1\).

Consider in the domain \(Q\) Chaplygin’s equation

\[ T_\lambda(u)\equiv K(y)\partial^2 u/\partial x^2+\partial^2 u/\partial y^2+\alpha(x,y)u_x+\beta(x,y)u_y+ \]
\[ +\gamma(x,y)u-\lambda u=f(x,y), \tag{1} \]

where \(\alpha(x,y)\), \(\beta(x,y)\) are continuously differentiable, and \(\gamma(x,y)\) is a continuous function in \(\overline{Q}\), \(\lambda\) is a real parameter. The function \(f(x,y)\in L_2(Q)\). We note that the curve \(\Gamma_2\) is a characteristic of equation (1), while the curve \(\Gamma_1\) has characteristic directions only for those \(y\) for which \(\mu'(y)=-\sqrt{-K(y)}\).

By the generalized Tricomi problem (1)—(2) we mean the problem of finding in \(Q\) a solution of equation (1) satisfying the boundary conditions

\[ u\big|_{\Gamma_0\cup\Gamma_1}=0. \tag{2} \]

Denote by \(\mathring{W}_2^1(Q)\) the Hilbert space of functions given in \(Q\), obtained by completion in the norm \(W_2^1(Q)\) of the set of functions smooth in \(\overline{Q}\) and satisfying conditions (2).

Definition. A generalized solution from \(\mathring{W}_2^1(Q)\) of problem (1)—(2) is a function \(U(x,y)\in \mathring{W}_2^1(Q)\) such that the integral identity

\[ \iint_Q [K(y)u_x v_x+u_y v_y-(\alpha u_x+\beta u_y)v+\lambda uv]\,dx\,dy =-\iint_Q fv\,dx\,dy \tag{3} \]

is satisfied for every function \(v(x,y)\in \mathring{W}_2^1(Q)\).

Theorem 1. Under the assumptions made concerning the coefficients of equation (1) and the boundary \(\Gamma\) of the domain \(Q\), for every function \(f(x,y)\in \widetilde W_{2}^{-1}\) there exists a unique generalized solution of problem (1)—(2), provided only that \(\lambda\ge \lambda_0\), where \(\lambda_0\) is some real number.

Since \(L_2(Q)\subset \widetilde W_{2}^{-1}(Q)\), Theorem 1 implies, under its assumptions, the unique solvability of problem (1)—(2) for every function \(f(x,y)\in L_2(Q)\).

The principal role in the proof of this theorem is played by the following

Lemma 1. There exists a constant \(C>0\) such that for any function \(U\in W_2^2(Q)\cap \widetilde W_2^1(Q)\), for sufficiently large \(\lambda\), the inequality

\[ \iint_Q (u_{xx}^2+u_y^2+u^2)\,dx\,dy +\int_0^1 u^2\big|_{y=0}\,dx +\int_{\Gamma_2}\left(\frac{\partial u}{\partial s}\right)^2 ds +\int_{\Gamma_2}u^2\,ds+ \]
\[ +\int_{\Gamma_0}\bigl(\dot x^2(S)+A(S-s)\bigr) \left(\frac{\partial u}{\partial n}\right)^2 ds \le C\iint_Q (Ku_{xx}+u_{yy}-\lambda u)^2\,dx\,dy . \tag{4} \]

If the curve \(\Gamma_1\) has no characteristic points, then to the left-hand side of inequality (4) one may add the term \(\displaystyle \int_{\Gamma_1}\left(\frac{\partial u}{\partial n}\right)^2 ds\); if, on the curve \(\Gamma_1\), there is a set \(E\) of characteristic points with \(\operatorname{mes} E<\operatorname{mes}\Gamma_1\), then to the left-hand side of (4) one may add the term
\[ A_0(\sigma)\int_{\Gamma_1\setminus E_\sigma}\left(\frac{\partial u}{\partial n}\right)^2 ds, \]
where \(E_\sigma\), \(\sigma>0\), is an open set containing the set \(E\), the distance from whose boundary to the set \(E\) is equal to \(\sigma\), \(\operatorname{mes}E_\sigma<\operatorname{mes}\Gamma_1\), the constant \(A_0(\sigma)>0\), \(A_0(\sigma)\to 0\) as \(\sigma\to 0\); \(A>0\).

We outline the proof of this lemma. Let \(\delta>0\) be a sufficiently small number, whose smallness is determined only by the behavior of the function \(K(y)\) near \(y=0\) and by the behavior of the functions \(x(s)\) and \(y(s)\) near \(s=0\) and \(s=S\). Draw in the domain \(Q^+=Q\cap(y>0)\) a smooth curve \(\Lambda\), lying in the strip
\[ 0<y\le \max\bigl(y(\delta),\,y(S-\delta)\bigr) \]
and passing through the points \((x(\delta),y(\delta))\) and \((x(S-\delta),y(S-\delta))\). We may assume that the curve \(\Lambda\) adjoins the curve \(\Gamma_0\) at the points \((x(\delta),y(\delta))\) and \((x(S-\delta),y(S-\delta))\) so smoothly that the boundary of the domain \(Q_\delta\), lying between the curves \(\Lambda\) and \(\Gamma_0\), has a smooth inward normal at these points. Let
\[ Q_\delta'=Q^+\setminus Q_\delta,\qquad Q^-=Q\cap(y<0), \]
and let \(\Gamma_{0,\rho}\), \(0<\rho<S/2\), be the part of the curve \(\Gamma_0\) for which the parameter \(s\in[\rho,S-\rho]\).

Construct the functions \(a(x,y)\) and \(b(x,y)\) as follows: in the domain
\[ Q^-\cup Q_\delta' \]
\[ a(x,y)=\varepsilon^{5/4}+\varepsilon^\theta(1-x),\qquad 0<\theta<1/4,\qquad b(x,y)=\varepsilon(x/2-1) \]
for some sufficiently small \(\varepsilon>0\), while in the domain \(Q_\delta\), \(a\) and \(b\) are solutions of the equations
\[ \Delta^2 a=\Delta^2 b=0, \]
satisfying the boundary conditions: on the contour \(\Lambda\),
\[ a\big|_\Lambda=\varepsilon^{5/4}+\varepsilon^\theta(1-x)\big|_\Lambda,\quad b\big|_\Lambda=\varepsilon(x/2-1)\big|_\Lambda,\quad \partial a/\partial n\big|_\Lambda=-\varepsilon^\theta\cos(\mathbf n,\mathbf x)\big|_\Lambda, \]
\[ \partial b/\partial n\big|_\Lambda=\varepsilon\frac{\cos(\mathbf n,\mathbf x)}{2}\bigg|_\Lambda, \quad\text{on the contour }\Gamma_{0,\delta}\quad a\big|_{\Gamma_{0,\delta}}=-\dot y(s),\quad b\big|_{\Gamma_{0,\delta}}=\dot x(s). \]

On the portions of the curves \(\Gamma_{0,\delta}/\Gamma_{0,2\delta}\), the functions \(a(x,y)\) and \(b(x,y)\) are obtained by linear interpolation between the corresponding values of \(a(x,y)\) and \(b(x,y)\) at the points
\[ (x(\delta),y(\delta)),\quad (x(2\delta),y(2\delta)),\quad (x(S-\delta),y(S-\delta)),\quad (x(S-2\delta),y(S-2\delta)), \]
\[ \frac{\partial a}{\partial n}\bigg|_{\Gamma_{0,\delta}}=g(s),\qquad \frac{\partial b}{\partial n}\bigg|_{\Gamma_{0,\delta}}=g_1(1), \]
where \(g_1(s)\) and \(g(s)\) are arbitrary smooth functions on the interval \(\delta<s<S-\delta\), taking at the endpoints of this interval the values
\[ g(\delta)=-\varepsilon^\theta\cos(\mathbf n,\mathbf x)\big|_{\Gamma_0,\,(s=\delta)}, \]

\[ g(S-\delta)=-\varepsilon^\theta \cos(\mathbf n,\mathbf x)\big|_{\Gamma_0\,(s=S-\delta)},\qquad g_1(\delta)=g(\delta)\frac{\varepsilon^{1-\theta}}{2},\qquad g_1(S-\delta)=g(S-\delta)\frac{\varepsilon^{1-\theta}}{2}. \]

By \(c(x,y)\) we denote a function equal to a sufficiently large constant \(C_0>0\) in the domain \(Q^+\) and equal to
\[ \frac{C_0K'(0)}{4\sqrt[4]{-K(y)}}\int_y^0\frac{e^{n\eta}\,d\eta}{\sqrt[4]{-K^3(\eta)}} \]
for sufficiently large \(n>0\) in the domain \(Q^-\) (the latter integral is convergent, since \(K(y)\) is a smooth function and \(K'(0)\ne0\)).

Applying the Friedrichs \(a,b,c\) method, i.e., multiplying equation (1) for \(\alpha=\beta=\gamma=0\) by \(au_x+bu_y+cu\), with the \(a,b\), and \(c\) just constructed, and integrating the equality thereby obtained over the domain \(Q\), we obtain the required estimate (4). We note that first a sufficiently small \(\delta>0\) is fixed, then \(\varepsilon>0\) is chosen sufficiently small, then \(C_0>0\) is taken sufficiently large, and only after that is \(n>0\) chosen sufficiently large.

Lemma 1 immediately implies the following.

Lemma 2. If \(u(x,y)\in \widetilde W_2^2(Q)\cap \dot W_2^1(Q)\), then for sufficiently large positive \(\lambda\)
\[ \iint_Q (u_x^2+u_y^2+u^2)\,dx\,dy \le C\iint_Q (Ku_{xx}+u_{yy}-\lambda u)\,dx\,dy \]
with a constant \(C>0\) independent of the chosen function \(u\).

The proof of Theorem 1 is then carried out as follows. Denote by \(R_\lambda\) the extension of the operator \(T_\lambda\) (1), under the conditions (2), defined by the integral identity (3). The domain of definition of the operator \(R_\lambda\) is \(\widetilde W_2^1(Q)\), and its range is the Hilbert space \(\dot W_2^{-1}(Q)\). It is easy to see that the operator \(R_\lambda\) is bounded from \(\widetilde W_2^1(Q)\) into \(\dot W_2^{-1}(Q)\); therefore, by Lemma 2, for \(u\in \widetilde W_2^1(Q)\) and sufficiently large \(\lambda\) the inequality
\[ \|u\|_{\widetilde W_2^1(Q)}\le C\|R_\lambda u\|_{\dot W_2^{-1}(Q)} \tag{5} \]
holds, and if \(R_\lambda u\in L_2(Q)\), then also the inequality
\[ \|u\|_{\widetilde W_2^1(Q)}\le C\|R_\lambda u\|_{L_2(Q)}. \]

From the last inequality follows the uniqueness theorem, for large \(\lambda\), for the generalized solution of the problem (1)—(2). Since the set \(\{R_\lambda u\}\), \(u\in \widetilde W_2^1(Q)\), is obviously closed, in order to prove solvability, for sufficiently large \(\lambda\), of the problem (1)—(2), it is enough to establish that the orthogonal complement in \(\dot W_2^{-1}(Q)\) to \(\{R_\lambda u\}\), \(u\in \widetilde W_2^1(Q)\), is empty. The latter is also derived from (3) and (5).

Further, by known methods \((^3)\) one can obtain the proof of Theorem 2.

Theorem 2. The problem of finding a generalized solution to the problem (1)—(2) for arbitrary \(\lambda\) is Fredholm.

We note that a number of important results on the generalized Tricomi problem for certain special cases of equation (1) were obtained in the works \((^{1,2,4-6})\).

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
30 IX 1966

CITED LITERATURE

1. F. Tricomi, On linear equations of mixed type, 1947.
2. A. V. Bitsadze, Equations of Mixed Type, Publishing House of the Academy of Sciences of the USSR, 1959.
3. V. P. Mikhailov, Mat. sbornik, 63, no. 2 (1964).
4. Yu. M. Berezanskii, Expansion in Eigenfunctions of Self-Adjoint Operators, Kiev, 1965.
5. C. S. Morawetz, Comm. Pure and Appl. Math., 11, no. 3 (1958).
6. M. H. Protter, Duke Math. J., 21, 1 (1954).