UDC 517.946.4

MATHEMATICS

M. E. LERNER

ON MAXIMUM PRINCIPLES FOR EQUATIONS OF MIXED ELLIPTIC-HYPERBOLIC TYPE OF THE SECOND KIND

(Presented by Academician I. N. Vekua, 19 XII 1967)

Consider the equation

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

in a domain \(D\), bounded by: 1) a simple Jordan arc \(\sigma\), situated in the upper half-plane and resting on the axis \(y=0\) at the points \(O(0;0)\), \(A(a,0)\), \(a>0\); 2) the characteristics \(OC\) and \(AC\). Let \(D_1\) and \(D_2\) be respectively the subdomains of ellipticity and hyperbolicity of the equation \((K_m)\), \(OA\) its line of transition; \(0<m<2\); \(M,N,F\in C^{(0)}(\overline D)\); \(M,N\in C^{(1)}(\overline D_2)\); \(F\leq 0\) in \(D_1\). Particular cases of \((K_m)\) are the equations studied in a mixed domain

\[ u_{xx}+\operatorname{sgn} y\cdot |y|^m u_{yy}=0, \qquad 0<m<1 \quad (^{1}); \quad (\Sigma_m) \]

\[ u_{xx}+yu_{yy}+au_y=0, \qquad a<1 \quad (^{2-5}); \quad (S_1^\alpha) \]

\[ u_{xx}+\operatorname{sgn} y\cdot |y|^m u_{yy} +a|y|^{m-1}u_y=0, \qquad 0<m<2,\quad m/2<\alpha<1 \quad (^{6}). \quad (S_m^\alpha) \]

We shall say that for equation \((K_m)\) in the domain \(D\) (in the domain \(D_2\)), in a certain class \([K]\) of its solutions, maximum principle I (maximum principle II) holds in the following case: let \(u\in [K]\) and \(u|_{OC}\equiv 0\); then \(\max_{\overline D}|u|\) \((\max |u|)\) is attained on \(\sigma\cup AC\) (on the characteristic \(AC\)).

§ 1. Consider the equation

\[ E[u]\equiv u_{\xi\eta}+a(\xi;\eta)u_\xi+b(\xi;\eta)u_\eta+c(\xi;\eta)u=0 \quad (E) \]

in the characteristic triangle \(O_0C_0A_0\) (the domain \(\Delta\), \(O_0(0;0)\), \(C_0(0;-\xi_0)\), \(A_0(\xi_0;-\xi_0)\), \(\xi_0>0\)); \(a,a_\xi,b,c\in C^{(0)}(\Delta\cup O_0C_0)\).

We shall say that in the domain \(\Delta\) the coefficients of equation \((E)\) satisfy conditions \((L)\), if: 1) \(a>0\) on \(O_0C_0\); 2)

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

in \(\Delta\), and conditions \((M)\), if in \(\Delta\cup O_0C_0\): 1) \(a>Q\); 2)

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

Here \(PQ\) is a segment of an arbitrary characteristic \(\eta=\mathrm{const}\),

\[ P\in O_0C_0,\quad Q\in\Delta;\quad \beta=\exp\left\{\int_0^\xi b\,d\xi\right\};\quad h=a_\xi+ab-c=h_1+h_2,\quad h_1\geq 0\ \text{in }\Delta; \]

\[ \gamma=\gamma_1+\gamma_2,\quad \gamma_1=-\beta h_1,\quad \gamma_2=-\beta h_2;\quad c=c_1+c_2,\quad c_1\geq 0. \]

We shall call the function \(u(\xi;\eta)\) a solution of equation \((E)\) of class \([W_0]\), if: \(E[u]\equiv 0\) in \(\Delta\), \(u\in C^{(2)}(\Delta)\), \(u\in C^{(0)}(\overline\Delta)\), \(u\in C^{(1)}(\Delta\cup O_0C_0)\).

Lemma 1 (the characteristic principle of the absolute extremum for hyperbolic equations). Let \(u(\xi;\eta)\) be a solution of equation (E) of class \([W_0]\), \(u \ne 0\) in \(\Delta\), and \(u|_{O_0C_0}=0\). Then \(\max_{\Delta}|u|\) cannot be attained in \(\Delta\), but is attained on the characteristic \(\overline{A_0C_0}\).

1. Suppose the contrary. Let \(\max_{\Delta}|u|\) be attained at some point \(Q \in \Delta\). We write equation (E) in the form
   \[ (\beta u_\eta)_\xi + (\alpha\beta u)_\xi + \gamma u = 0 \]
   and integrate this equation over the segment \(PQ\) \((P \in O_0C_0,\ \eta_P=\eta_Q)\). We obtain
   \[ \beta(Q)u_\eta(Q)= \left[-a(P)u(Q)-\int_{PQ}[u(Q)-u]\gamma_2\,d\xi -u(Q)\int_{PQ}\beta C_2\,d\xi\right]+ \]
   \[ +\int_{PQ}[u(Q)-u]\gamma_1\,d\xi -u(Q)\int_{PQ}\beta C_1\,d\xi; \tag{1} \]
   \[ \beta(Q)u_\eta(Q)=-\beta(Q)a(Q)u(Q)-\int_{PQ}\gamma u\,d\xi. \tag{2} \]
   By virtue of conditions (L) and (M), respectively, from (1) and (2) it follows that
   \[ u(Q)u_\eta(Q)<0, \tag{3} \]
   which is impossible.

2. Suppose that \(\max_{\Delta}|u|\) is not attained on \(\overline{A_0C_0}\). Then, by the preceding part, it is attained only on the open segment \(O_0A_0\) at some point \(Q\). Consequently, in \(\Delta\), in some neighborhood of the point \(Q\), there exists a point \(Q'\) at which \(u\ne 0\) and
   \[ |u(Q')|>\max_{A_0C_0}|u|. \tag{4} \]
   Draw through the point \(Q\) a segment \(O'_0A'_0 \parallel O_0A_0\) \((O'_0 \in O_0C_0,\ A'_0 \in C_0A_0)\), and denote by \(\Delta'\) the open triangular domain \(O'_0C_0A'_0\). \(\max_{\Delta'}|u|\) cannot be attained at a point belonging to \(\Delta'\cup O'_0A'_0\), since, by the preceding part, at such a point one would have \(u\cdot u_\eta<0\). Hence \(\max_{\Delta'}|u|\) is attained only on \(\overline{A'_0C_0}\), and
   \[ |u(Q')|\le \max_{A'_0C_0}|u|, \]
   which contradicts (4) and completes the proof of the lemma.

It can be shown that \(\max_{\Delta}|u|\) is attained only on the characteristic \(A_0C_0\), if
\[ |u(A)|<\max_{A_0C_0}|u|. \]

Denote by \((E_m^{\lambda\mu})\) equation (E) in which
\[ a=-\sigma/2(\xi+\eta)+f(\xi;\eta)/|\xi+\eta|^{\sigma+\lambda},\quad b=-\sigma/2(\xi+\eta)+g(\xi;\eta)/|\xi+\eta|^{\sigma+\lambda}, \]
\[ c=r(\xi;\eta)/|\xi+\eta|^\mu,\quad \sigma=m/(2-m),\quad 0<m<1,\quad \lambda<2(1-m)/(2-m), \]
\[ \mu<(4-3m)/2(2-m);\quad f,\ (\xi+\eta)f_\xi,\ g,\ r\in C^{(0)}(\overline{\Delta}). \]

Lemma 2. For equation \((E_m^{\lambda\mu})\), the assertion of Lemma 1 is valid if the length of the segment \(O_0A_0\) is sufficiently small.

The proof follows from the fact that conditions (L) are satisfied if the length of the segment \(O_0A_0\) is sufficiently small.

We note that, under the change of variables
\[ \xi=x-\frac{2}{2-m}(-y)^{(2-m)/2},\quad \eta=-x-\frac{2}{2-m}(-y)^{(2-m)/2}, \]
equation \((K_m)\) is transformed into an equation of the form (E), and the triangle \(OAC\) into a triangle of type \(\Delta\).

§ 2. We shall call a function \(u(x;y)\) a solution of equation \((K_m)\) of class \([W]\) if:
\[ K_m[u]\equiv 0 \quad \text{in } D_1\cup D_2,\quad u\in C^{(2)}(D_1\cup D_2),\quad u\in C^{(0)}(\overline D), \]
\[ u\in C^{(1)}(\overline D\setminus \overline{OA}). \]

Theorem 1. For equation \((K_m)\) in the domain \(D\) (in the domain \(D_2\)), in the class \([W]\) of its solutions there holds the maximum principle I (the maximum principle—

II), if the length of the transition line is sufficiently small and \(0<m<1\), \(\beta<1-m/2\), \(\alpha<1-m\), \(\gamma<1-3m/4\).

Proof. Let \(u(x;y)\) be an arbitrary solution of equation \((K_m)\) of class \([W]\), \(u|_{\overline{OC}}\equiv 0\) and \(u\not\equiv 0\) in \(D\). Then, by the sufficient smallness of the transition line, by Lemma 2, \(\max_{\overline{D}_2}|u|\) is attained on \(\overline{AC}\), and maximum principle II is proved.

Let \(\max_{\overline{D}}|u|\) be attained in \(\overline{D}_2\). Consequently, it is attained on \(\overline{AC}\).

Let \(\max_{\overline{D}}|u|\) be attained in \(\overline{D}_1\). Then, by the known property of elliptic equations, it is attained on \(\bar{\sigma}\cup OA\), and, if on \(OA\), then, by the preceding, also on \(AC\). Thus, \(\max_{\overline{D}}|u|\) is attained on \(\bar{\sigma}\cup AC\), and the theorem is completely proved.

Theorem 2. For equation \((K_m)\) in the domain \(D\) (in the domain \(D_2\)), in the class \([W]\) of its solutions, maximum principle I (maximum principle (II)) is valid if its coefficients are such that the coefficients of the corresponding equation of the form (E) satisfy in \(\Delta\) the conditions (I) or (M).

The proof is analogous to Theorem 1 and follows from Lemma 1.

We note that in Theorem 2 the requirement of continuity on the characteristic \(AC\) of the coefficients of equation \((K_m)\) and of the first partial derivatives of the functions \(M(x;y)\) and \(N(x;y)\) is superfluous.

Corollary of Theorem 2. For an arbitrary length of the transition line, for equation \((\Sigma_m)\) with \(0<m<2\) and for equation \((S_m^\alpha)\) with \(0<m<2\), \(\alpha<m/2\), in the class \([W]\) maximum principles I and II are valid.

§ 3. Suppose a one-to-one correspondence has been established between all points of the characteristic \(\overline{AC}\) and some set of points \(E\subset\sigma\); \(Q_1\) and \(Q_2\) are arbitrary corresponding points \((Q_1\in E,\ Q_2\in\overline{AC})\), \(E_1\equiv\bar{\sigma}\setminus E\), and at each point belonging to the set \(E\), the arc \(\sigma\) has a normal.

Problem \(\Phi\). Find a solution \(u(x;y)\) of equation \((K_m)\) of class \([W]\) from the data:
\[ \text{1. } u|_{E_1}=\varphi_1.\qquad \text{2. } \partial u/\partial n|_E=\varphi_2.\qquad \text{3. } u|_{\overline{OC}}=\psi.\qquad \text{4. } u(Q_1)-u(Q_2)=g(Q_1). \]

Here \(\varphi_1,\varphi_2,\psi,g\) are functions continuous on the indicated parts of the boundary of the domain \(D\); \(n\) is the interior normal. Boundary condition 4 defines a “curvilinear jump of compaction” \((^{7,8})\).

Theorem 3. For equation \((K_m)\), under the conditions of Theorems 1 and 2, the solution of problem \(\Phi\) is unique.

Suppose the contrary. Let \(u(x;y)\) be a solution of problem \(\Phi\) for equation \((K_m)\), satisfying homogeneous boundary conditions, but \(u\not\equiv 0\) in \(D\). By homogeneous boundary condition 3, \(\max_{\overline{D}}|u|\) is attained on \(\bar{\sigma}\cup AC\). Taking into account homogeneous boundary conditions 1 and 2, we obtain that \(\max_{\overline{D}}|u|\) cannot be attained on \(\bar{\sigma}\) and, consequently, is attained on \(\overline{AC}\). But then, by homogeneous boundary condition 4, it is attained on \(\bar{\sigma}\), which, as has already been shown, is impossible. The contradiction obtained proves the theorem.

Corollary of Theorem 3. For equations \((\Sigma_m)\) and \((S_m^\alpha)\), under the conditions of the corollary to Theorem 2, the solution of problem \(\Phi\) is unique.

We note that for equation \((S_1^\alpha)\), when \(\alpha<0\), there exists and is unique a solution of the Dirichlet problem in the class of functions continuous in \(\overline{D}\) and having a continuous derivative with respect to \(y\) on the transition line \((^4)\).

Remark 1. We have considered maximum principles I and II with data on the characteristic \(\widehat{OA}\). These principles are formulated analogously with data on the characteristic \(AC\). For their proof, in equation \((E)\) one should interchange the roles of the variables \(\xi\) and \(\eta\). Maximum principles I and II with data on the characteristic \(AC\) hold for equation \((K_m)\) under the conditions of Theorem 1 and for equations \((\Sigma_m)\) and \((S_m^\alpha)\) under the conditions of Theorem 2.

The author expresses his gratitude to S. P. Pul'kin and the participants of his seminar for discussion of the results presented above.

Kuibyshev Polytechnic Institute
named after V. V. Kuibyshev

Kuibyshev State Pedagogical Institute
named after V. V. Kuibyshev

Received
13 XII 1967

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