UDC 517.946.4

MATHEMATICS

M. E. LERNER

ON A MAXIMUM PRINCIPLE FOR HYPERBOLIC EQUATIONS AND ITS APPLICATION TO EQUATIONS OF MIXED TYPE

(Presented by Academician I. N. Vekua on 22 IV 1967)

§ 1. Consider the equation

\[ \mathcal L[u]\equiv u_{\xi\eta}+a(\xi;\eta)u_\xi+b(\xi;\eta)u_\eta+c(\xi;\eta)u=0 \qquad(\mathcal L) \]

in the characteristic triangle \(O_0A_0C_0\) (the domain \(\Delta\), \(O_0(0;0)\), \(A_0(\xi_0;-\xi_0)\), \(C_0(0;-\xi_0)\), \(\xi_0>0\)), \(a,a_\xi,b,c\in C^{(0)}(\bar\Delta\setminus\overline{O_0A_0})\). Let

\[ h=a_\xi+ab-c=h_1+h_2,\qquad h_1\le 0\ \text{in }\Delta; \quad \beta=\exp\left\{\int_0^\xi b\,d\xi\right\}; \quad \alpha=\beta a; \]

\[ \gamma=-\beta h_1-\beta h_2=\gamma_1+\gamma_2;\qquad c=c_1+c_2,\qquad c_1\le 0\ \text{in }\Delta. \]

We shall say that the coefficients of \((\mathcal L)\) satisfy in \(\Delta\) conditions (C) if: 1) \(a<0\) on \(\overline{O_0C_0}\setminus O_0\); 2)

\[ -a(P)>\int_{PQ}\bigl[2|\gamma_2|+\beta|c_2|\bigr]\,d\xi \]

in \(\Delta\cup C_0A_0\), and conditions (D), if in \(\Delta\cup C_0A_0\): 1) \(a<0\), 2)

\[ -\beta(Q)a(Q)>\int_{PQ}|\gamma|\,d\xi, \]

where \(P\in \overline{O_0C_0}\setminus O_0\), \(PQ\) is a segment of the characteristic \(\eta=\text{const}\). We shall call a function \(u(\xi;\eta)\) a solution of equation \((\mathcal L)\) of class \([R]\) if \(\mathcal L[u]\equiv 0\) in \(\Delta\), \(u\in C^{(2)}(\Delta)\), \(u\in C^{(0)}(\bar\Delta)\), \(u\in C^{(1)}(\bar\Delta\setminus\overline{O_0A_0})\), \(u|_{\overline{O_0C_0}}=0\),

\[ \max_{\bar\Delta} u>0,\qquad \max_{\bar\Delta} u\ge -\min_{\bar\Delta} u. \]

Theorem 1 (maximum principle). Let the coefficients of \((\mathcal L)\) satisfy in \(\Delta\) conditions (C) or (D), and let \(u(\xi;\eta)\) be an arbitrary solution of \((\mathcal L)\) of class \([R]\).

Then \(\max_{\bar\Delta}u\) is attained only on \(\overline{O_0A_0}\).

Suppose the contrary. Let

\[ \max_{\bar\Delta}u=u(Q), \qquad Q\in \bar\Delta\setminus\overline{O_0A_0}. \]

Represent \((\mathcal L)\) in the form \({}^{(1)}\)

\[ (\beta u_\eta)_\xi+(\alpha u)_\xi+\gamma u=0 \]

and integrate this equation along the segment \(PQ\):

\[ \beta(Q)u_\eta(Q)= \left\{ -a(P)u(Q)+\int_{PQ}[u(Q)-u]\gamma_2\,d\xi -u(Q)\int_{PQ}\beta c_2\,d\xi \right\} +\int_{PQ}[u(Q)-u]\gamma_1\,d\xi -u(Q)\int_{PQ}\beta c_1\,d\xi; \tag{1} \]

\[ \beta(Q)u_\eta(Q)=-\beta(Q)a(Q)u(Q)-\int_{PQ}\gamma u\,d\xi. \tag{2} \]

By virtue of conditions (C) and (D), respectively, from (1) and (2) we obtain \(u_\eta(Q)>0\), which is impossible and proves the theorem.

Example 1. Denote by \((\mathcal L_m^{\lambda\mu})\) the equation \((\mathcal L)\) in which

\[ a=\frac{\sigma_1}{2(\xi+\eta)}+\frac{f(\xi;\eta)}{(-\xi-\eta)^{2\sigma_1+\lambda}}, \qquad b=\frac{\sigma_1}{2(\xi+\eta)}+\frac{g(\xi;\eta)}{(-\xi-\eta)^{2\sigma_1+\lambda}}, \]

\[ c=\frac{r(\xi;\eta)}{(-\xi-\eta)^{2\sigma_1+\mu}}; \]

\[ 0<m<2,\qquad \lambda<\frac{2-m}{2+m},\qquad \mu<\frac{4-m}{2(2+m)},\qquad \sigma_1=\frac{m}{2+m}, \]

\[ f,\;(\xi+\eta)f_\xi,\;g,\;r\in C^{(0)}(\overline{\Delta}). \]

Lemma 1. If the length of the segment \(O_0A_0\) is sufficiently small, then the maximum principle is valid for the equation \((\mathcal L_m^{\lambda\mu})\).

Indeed, for a sufficiently small length of \(O_0A_0\) the coefficients of \((\mathcal L_m^{\lambda\mu})\) satisfy conditions (C)

\[ a(P)=\frac{\sigma_1}{2\eta}\left[1-(-\eta)^\sigma \frac{2}{\sigma_1}f(0;\eta)\right], \qquad \sigma=\frac{2-m}{2+m}-\lambda>0, \]

\[ h=-\frac{m(4+m)}{4(2+m)}\frac{1}{(\xi+\eta)^2} \left[1-(-\xi-\eta)^\sigma \mathfrak W(\xi;\eta)\right], \]

\[ -a(P)-\int_{PQ}\beta |c|\,d\xi =-\frac{\sigma_1}{2\eta}\left[1-(-\eta)^\sigma\Omega(\xi;\eta)\right], \]

\[ \mathfrak W(\xi;\eta)= \frac{4(2+m)^2}{m(4+m)} \left[ \left(\lambda+\frac{3}{2}\sigma_1\right)f(\xi;\eta) -\frac{\sigma_1}{2}g(\xi;\eta) -(\xi+\eta)f_\xi(\xi;\eta)+ \right. \]

\[ \left. +(-\xi-\eta)^\sigma f(\xi;\eta)g(\xi;\eta) -(-\xi-\eta)^{1+\lambda-\mu}r(\xi;\eta) \right], \]

\[ \Omega(\xi;\eta)=\frac{2}{\sigma_1} \left[f(0;\eta)+(-\eta)^{3/2\,\sigma_1+\lambda}\omega(\xi;\eta)\right], \]

\[ \omega(\xi;\eta)= \int_0^\xi \frac{|r(t;\eta)|}{(-t-\eta)^{3/2\,\sigma_1+\mu}} \exp\left\{ \int_0^t \frac{g(\tau;\eta)}{(-\tau-\eta)^{2\sigma_1+\lambda}}\,d\tau \right\}\,dt. \]

Example 2.

\[ \mathfrak M_{\lambda\mu}[u]\equiv u_{\xi\eta} +\left(\frac{\lambda}{\xi+\eta}-\frac{\mu}{\xi-\eta}\right)u_\xi +\left(\frac{\lambda}{\xi+\eta}+\frac{\mu}{\xi-\eta}\right)u_\eta=0. \qquad (\mathfrak M_{\lambda\mu}) \]

Lemma 2. For the equation \((\mathfrak M_{\lambda\mu})\), for any \(0\le \lambda\le 1\), \(0\le \mu\le 1\), \(\lambda+\mu\ne 0\), the maximum principle is valid.

Indeed, in \(\Delta\) conditions (C) are satisfied

\[ a(P)=\frac{\lambda+\mu}{\eta}<0;\qquad h_1=-\frac{\lambda-\lambda^2}{(\xi+\eta)^2}\le 0;\qquad h_2=\frac{\mu-\mu^2}{(\xi-\eta)^2}\ge 0, \]

\[ -a(P)-2\int_0^\xi |\gamma_2|\,d\xi > -\frac{\lambda+\mu}{\eta} +\frac{2\mu}{\eta}(1-2^{\mu-1})>0. \]

It can be shown that for the equation \((\mathfrak M_{\lambda\mu})\) with \(\lambda=0\), \(0<\mu<1\) (the Euler–Poisson–Darboux equation) the well-known maximum principle for hyperbolic equations does not hold \(\bigl(({}^1),\) theorem 1\(\bigr)\).

§ 2. Consider the general generalized F. Tricomi equation

\[ T_m[u]\equiv \operatorname{sgn} y\cdot |y|^m u_{xx} +u_{yy} +\frac{M(x;y)}{|y|^\beta}u_x +\frac{N(x;y)}{|y|^\alpha}u_y +\frac{F(x;y)}{|y|^\gamma}u=0 \qquad (T_m) \]

in the domain \(D\), bounded by: 1) a simple Jordan arc \(\Gamma\), situated in the upper half-plane and resting on the axis \(y=0\) at the points \(O(0;0)\) and \(A(a;0)\), \(a>0\); 2) the characteristics \(OC\) and \(AC\). Let \(D_1\) and \(D_2\) be, respectively, the subdomains of ellipticity and hyperbolicity of the equation \((T_m)\); \(OA\) is the line of transition; \(M,N,F\in C^{(0)}(\overline{D_1})\); \(M,N,F\in C^{(0)}(\overline{D_2})\); \(F\le 0\) in \(D_1\); \(M,N\in C^{(1)}(\overline{D_2})\); \(N\ge 0\) in \(D_1\) only when \(\alpha>0\); \(m\ge 0\), \(\beta<1-m/2\), \(\alpha<1\), \(\gamma<1-m/4\). We shall call the function \(u(x;y)\) a regular solution of the equation \(T_m[u]\equiv G(x;y)\) \(\bigl(G(x;y)\in C^{(0)}(\overline D)\bigr)\), if \(T_m[u]\equiv G(x;y)\) in \(D_1\cup D_2\); \(u\in C^{(2)}(D_1\cup D_2)\); \(u\in C^{(0)}(\overline D)\); \(u\in C^{(1)}[D_1\cup(D_2\setminus \overline{OA})]\); \(u_y(x;0+0)=u_y(x;0-0)\) for \(0<x<a\), and these limits are finite—the gluing condition. A regular solution of the equation \((T_m)\) will be regarded as belonging to the class \([P_0]\) if

\[ u\big|_{\overline{OC}}\equiv 0,\qquad \max_{\overline D}u>0,\qquad \max_{\overline D}u\ge -\min_{\overline D}u. \]

Problem T (F. Tricomi). Find a regular solution \(u(x,y)\) of the equation \(T_m[u]=G(x;y)\) from the data: 1) \(u|_{\Gamma}=\varphi\), 2) \(u|_{\overline{OC}}=\psi\), where \(\varphi\) and \(\psi\) are continuous functions; \(\varphi(0)=\psi(0)\).

We shall say that for equation \((T_m)\) the principle of the (strict) maximum for problem T is valid if an arbitrary solution of class \([P_0]\) attains its maximum in \(\overline D\) (only) in the closure of the “elliptic” arc \(\Gamma\).

Theorem 2. If in equation \((T_m)\), \(0<m<2\), \(\alpha=\beta=\gamma=0\), and the length of the transition line is sufficiently small, then the principle of the strict maximum for problem T is valid for it.

Suppose the contrary. Let \(u(x;y)\in [P_0]\), \(\max_{\overline D}u=u(Q)\), but \(Q\in \overline D\setminus \overline\Gamma\). By the known property of elliptic equations and Lemma 1 the point \(Q\in OA\). Consequently, by the modified lemma of E. Hopf \((^{1}),\) Lemma 2, \(u_y(x_Q;0+0)<0\). But \(u_y(x_Q;0-0)\ge 0\), which contradicts the gluing condition and proves what is required.

Theorem 3. If in equation \((T_m)\), \(0<m<2\), and the length of the transition line is sufficiently small, then the maximum principle for problem T is valid for it.

Suppose the contrary. Let \(u(x;y)\in [P_0]\), \(\max_{D_1}u=u(Q)\), but \(Q\in \overline\Gamma\). Consequently, \(Q\in OA\). But then, by virtue of the lemma of K. I. Babenko \((^{2})\) (one can show that its assertion is also valid in the case \(\alpha>0\), but with \(N\ge 0\) in \(D_1\)), \(u_y(x_Q;0+0)<0\), which is impossible by the gluing condition.

Theorem 4. Let the coefficients of an equation of the form \((\mathcal L)\), corresponding to the equation \((T_m)\), satisfy in the domain \(\Delta\) the conditions (C) or (D). Then for equation \((T_m)\) the following are valid: 1) the principle of the strict maximum for problem T, if \(\alpha=\beta=\gamma=0\); 2) the maximum principle for problem T.

The proof is analogous to the proofs of Theorems 2 and 3.

Theorem 5. For the equation \(T_m[u]=G(x;y)\), the solution of problem T is unique if \(0<m<2\) and the length of the transition line is sufficiently small.

The proof follows easily from Theorems 2 and 3.

Theorem 6. If the coefficients of equation \((T_m)\) satisfy the conditions of Theorem 4, then for the equation \(T_m[u]=G(x;y)\) the solution of problem T is unique.

§ 3. Consider an equation of mixed type with two mutually perpendicular parabolicity lines of the first kind

\[ T_{mn}[u]\equiv \operatorname{sgn} y\cdot |y|^m U_{xx}+\operatorname{sgn} x\cdot |x|^n u_{yy}=0,\qquad (T_{mn}) \]

\[ m\ge 0,\qquad n\ge 0,\qquad m+n\ne 0 \]

in a domain \(D\), consisting of three subdomains \(D_1,D_2,D_3\). The domain \(D_1(x>0;\ y>0)\) is bounded by the coordinate axes and by a simple Jordan arc \(\Gamma\) resting on them at the points \(A(a;0)\) and \(B(0;b)\). The domains \(D_2(x>0;\ y<0)\) and \(D_3(x<0;\ y>0)\) are bounded respectively by two pairs of characteristics \(OC\) and \(AC\), \(OE\) and \(BE\). We shall call a function \(u(x;y)\) a regular solution of equation \((T_{mn})\) if \(T_{mn}[u]\equiv 0\) in \(D_1\cup D_2\cup D_3\), \(u\in C^{(2)}(D_1\cup D_2\cup D_3)\), \(u\in C^{(0)}(\overline D)\), \(u\in C^{(1)}[D_1\cup(\overline D_2\setminus \overline{OA})\cup(\overline D_3\setminus \overline{OB})]\),

\[ u_y(x;0+0)=u_y(x;0-0)\quad \text{for }0<x<a,\qquad u_x(0+0;y)=u_x(0-0;y) \]

for \(0<y<b\), and these limits are finite—the gluing conditions. We shall regard a regular solution \(u(x;y)\) of equation \((T_{mn})\) as belonging to the class \([P_{00}]\), if

\[ u\big|_{\overline{OC}\cup \overline{OE}}\equiv 0,\qquad \max_{\overline D}u>0,\qquad \max_{\overline D}u\ge -\min_{\overline D}u . \]

Problem T. Find a regular solution \(u(x;y)\) of equation \((T_{mn})\) from the data: 1) \(u|_{\Gamma}=\varphi\), 2) \(u|_{\overline{OC}}=\psi_1\), 3) \(u|_{\overline{OE}}=\psi_2\), where \(\varphi,\psi_1,\psi_2\) are continuous functions; \(\psi_1(0)=\psi_2(0)\).

Theorem 7 (the strong maximum principle for problem T). An arbitrary solution of the equation \((T_{mn})\) of class \([P_{00}]\) attains its maximum in \(\bar D\) only in the closure of the “elliptic” arc \(\Gamma\).

The proof is analogous to Theorem 2 and follows from the modified E. Hopf lemma \({}^{1}\) and Lemma 2.

Theorem 8. For the equation \((T_{mn})\), the solution of problem T is unique.

The proof follows from Theorem 7.

In conclusion we note that a somewhat different extremum principle for problem T in the case of the Lavrent’ev–Bitsadze and Tricomi equations belongs to A. V. Bitsadze \({}^{3}\).

The author expresses his gratitude to S. P. Pul’kin and the participants of his seminar, especially V. F. Volkodavov, for their discussion of the results presented above.

Kuibyshev Polytechnic Institute
named after V. V. Kuibyshev

Received
14 III 1967

REFERENCES

\({}^{1}\) S. Agmon, L. Nirenberg, M. H. Protter, Comm. Pure and Appl. Math., 6, No. 4, 455 (1953).
\({}^{2}\) K. I. Babenko, On the theory of equations of mixed type, Abstract of doctoral dissertation, Moscow, 1951.
\({}^{3}\) A. V. Bitsadze, Equations of mixed type, Itogi Nauki, Publishing House of the Academy of Sciences of the USSR, 1959, p. 84.