UDC 517.946.4

ON THE UNIQUENESS OF SOLUTIONS OF PROBLEMS WITH FRANKL AND TRICOMI CONDITIONS FOR THE GENERAL LAVRENT'EV–BICADZE EQUATION

M. E. LERNER, S. P. PUL'KIN

F. I. Frankl in [1, 2] posed the so-called [3] “shock” problems for the equation of S. A. Chaplygin

\[ K(y)u_{xx}+u_{yy}=0, \tag{*} \]

\[ yK(y)>0 \quad \text{for } y\ne 0,\qquad K(0)=0,\qquad K'(y)>0. \]

Denote by \(D_*\) the simply connected domain bounded by: a) the segment \(CR\) of the axis \(OY\) \((y_R=-y_C=1)\); b) a simple Jordan arc \(\sigma\), situated in the first quadrant and resting on the coordinate axes at the points \(R\) and \(B(b;0)\); c) the segment \(AB\) of the axis \(OX\) \((A(a,0),\ a<b)\); d) the characteristic \(AC\). Draw the characteristic \(OE,\ E\in AC\). Then the first “shock” problem of Frankl consists in finding a solution \(u(x;y)\) of equation (*), continuous in \(\overline{D}_*\) and satisfying the boundary conditions:

\[ 1)\quad u\big|_{RB}=\varphi_1;\qquad 2)\quad u\big|_{\overline{AB}}=\varphi_2;\qquad 3)\quad u_x\big|_{CR}=\varphi_3; \]

\[ 4)\quad u(0,y)-u(0,-y)=\varphi_4,\qquad 0\le y\le 1, \]

where \(\varphi_1,\varphi_2,\varphi_3,\varphi_4\) are given continuous functions.

A. V. Bitsadze, for equation (*) with \(K(y)=\operatorname{sgn}y,\ a=b\) and certain restrictions on the arc \(\sigma\), proved [4] the existence and uniqueness of a solution of this Frankl problem, which is continuously differentiable everywhere in \(D_*\), except, possibly, for points of the segment \(OE\), and twice continuously differentiable everywhere in \(D_*\), except, possibly, for points of the segments \(OE\) and \(OA\).

Yu. V. Devingtal' proved [5] the uniqueness and existence of a twice continuously differentiable solution everywhere in \(D_*\), except, possibly, for points of the segment \(OA\), of this same problem for equation (*) with \(K(y)=\operatorname{sgn}y\,|y|^m,\ m>1,\ a<b\).

In the present paper, for the general Lavrent'ev–Bitsadze equation in domains more general than \(D_*\), problems are posed in which the Frankl condition \(u_x|_{CR}=\varphi_3\) is replaced by the Tricomi condition \(u|_{AC}=\psi\), while on the remaining parts of the contour various conditions are prescribed. On the line of transition the sought function \(u(x,y)\) must satisfy the Tricomi “gluing” condition: \(u_y(x,0-0)=u_y(x,0+0)\). In addition, the “density jump” is generalized, which may be prescribed on a disconnected set, as well as on the entire “elliptic” curve.

§ 1. LEMMAS

Consider the general Lavrent'ev–Bitsadze equation

\[ Lu \equiv u_{xx}+\operatorname{sgn}y\,u_{yy}+M(x,y)u_x+N(x,y)u_y+F(x,y)=0 \tag{1} \]

in a simply connected domain \(D\) (Fig. 1), bounded by: 1) a simple Jordan curve \(\sigma\), situated in the upper half-plane and resting on the axis \(OX\) at the points \(L(l,0)\) and \(B(b,0)\), \(l<b\); 2) the segments \(LK\) belonging to the axis \(OX\) \((K(k,0),\ k<b,\ l\leq k\) or \(l>k)\) and \(AB\) \((A(a,0),\ a>k,\ a>l,\ a\geq b\) or \(a<b)\); 3) the segment \(AC\) of the characteristic \(x-y=a\) \((C(a+c,c),\ c<0)\); and 4) another \(KC\) simple Jordan line, situated in the lower half-plane, which either coincides completely with the characteristic \(CJ\) passing through the point \(C\) \((J(2c+a,0))\), or is situated between this characteristic and the characteristic \(KE\), \(E\in AC\).

Fig. 1

The arc \(KC\) is such that any characteristic can intersect it at only one point. Depending on the mutual position of the points \(L,K,A\), and \(B\), different types of domains are obtained (see Fig. 1). We shall measure the magnitude of the arc \(s\) along the boundary of the domain \(D\) from the point \(K\) clockwise. Let \(S\) be the length of \(KC\), \(\delta\) the length of \(LK\), and \(LR\) the segment of the arc \(\sigma\) whose length is equal to \(S\); \(D_1\) and \(D_2\) are the subdomains of ellipticity and hyperbolicity of equation (1). The coefficients of equation (1) are assumed continuous in \(\overline D\), and \(F(x,y)\leq 0\) in \(D_1\).

In the domain \(D_2\) equation (1) has the form

\[ u_{xx}-u_{yy}+M(x,y)u_x+N(x,y)u_y+F(x,y)u=0 . \tag{2} \]

We make the change of variables \(\xi=y+x,\ \eta=y-x\). Then equation (2) is transformed into the equation

\[ L_0u\equiv u_{\xi\eta}+a(\xi,\eta)u_\xi+b(\xi,\eta)u_\eta+c(\xi,\eta)u=0, \tag{3} \]

and the domain \(D_2\) (the triangle \(KCA\)) passes into the domain \(\Delta\) (the triangle \(K'C'A'\), Fig. 2).

We shall assume that the coefficients of equation (2) are such that the coefficients of equation (3) \(a(\xi,\eta),\ b(\xi,\eta),\ c(\xi,\eta),\ b_\eta(\xi,\eta)\) are continuous in \(\Delta\).

We shall say that the coefficients of equation (2) (of equation (3)) satisfy in the domain \(D_2\) (in the domain \(\Delta\)) conditions \((A)\), if: 1) \(b\leq 0\) on \(A'C'\), 2) \(b_\eta+ab-c\leq 0\) in \(\Delta\), 3) \(c\leq 0\) in \(\Delta\); and conditions \((B)\), if: 1) satisfy-

conditions \((A)\), 2) at least one of the following two conditions is fulfilled: i) \((A_1)\) holds with a strict inequality; ii) on each segment \(\xi=\mathrm{const}\) within the domain \(\Delta\), the measure of the set of points at which either \(b_\eta+ab-c=0\), or \(c=0\), is zero. Let us note that the classes of such equations are sufficiently broad [6].

Let \(P'Q'\) be a segment of the characteristic \(\xi=\mathrm{const}\) \((\eta_{P'}<\eta_{Q'})\), and let \(u(\xi,\eta)\) be a twice continuously differentiable solution of equation (3) in some domain containing the segment \(P'Q'\). Then, denoting

\[ \alpha=\exp\int a(\xi,\eta)\,d\eta,\qquad \beta=\alpha b,\qquad \gamma=-b_\eta+\alpha c, \tag{4} \]

Fig. 2

as in [7], we obtain

\[ \alpha u_\xi\big|_{P'}^{Q'} =-[u(Q')-u(P')]\beta(P') +\int_{P'Q'}[u(Q')-u]\gamma\,d\eta -u(Q')\int_{P'Q'}\alpha c\,d\eta . \tag{5} \]

Let \(PQ\) be a segment of the characteristic \(\xi=\mathrm{const}\), \(P\in \overline{A'C'}\), \(\eta_P<\eta_Q\), \(Q\in \overline{\Delta}\setminus \overline{A'C'}\).

Lemma 1. If the function \(u(\xi,\eta)\in C^2(\Delta)\), \(L_0u=0\) in \(\Delta\), \(u(\xi,\eta)\in C^1(\overline{\Delta})\),

\[ \max_{PQ}u(\xi,\eta)=u(Q)>0, \]

\(u|_{A'C'}=0\), and the coefficients of equation (2) satisfy conditions \((B)\) in \(D_2\), then \(U_\xi(Q)>0\).

Proof. Let \(Q\) be an interior point of the domain \(\Delta\). Take on \(PQ\) a point \(P'\), \(\eta_P<\eta_{P'}<\eta_Q\). Writing relation (5) for the segment \(P'Q\) and letting \(P'\) tend to \(P\), we obtain relation (5) for the segment \(PQ\). Suppose \(C'K'\) does not coincide with the characteristic \(C'J'\) (Fig. 2, б, в) and \(Q\) is an interior point of \(C'K'\cup K'A'\). Take on \(PQ\) points \(P'\) and \(Q'\), \(\eta_P<\eta_{P'}<\eta_{Q'}<\eta_Q\). Passing in equality (5) for the segment \(P'Q'\) to the limit, first as \(P'\to P\), and then as \(Q'\to Q\), we obtain relation (5) for \(PQ\). Finally, if \(C'K'\) (Fig. 2, а) is a segment of a characteristic and \(PQ\) is part of it, write relation (5) for the segment \(P'Q'\): \(P'(\xi_P+\varepsilon,-a+\varepsilon_1)\), \(Q'(\xi_P+\varepsilon,\eta_Q-\varepsilon_2)\), where \(\varepsilon,\varepsilon_1,\varepsilon_2\) are such positive numbers that the segment \(P'Q'\) is contained in the domain \(\Delta\), \(\eta_Q-\varepsilon_2>-a+\varepsilon_1\). Passing to the limit in this relation successively as \(\varepsilon\to0\), \(\varepsilon_1\to0\), and \(\varepsilon_2\to0\), we obtain relation (5) for the segment \(PQ\).

Thus, for any position of the point \(Q\) in \(\overline{\Delta}\) (and, in particular, for \(Q\equiv K'\)), relation (5) is valid. Hence, by the conditions of the lemma, \(\alpha u_\xi\big|_{P}^{Q}>0\) and \(u_\xi(Q)>0\).

Lemma 1′. The assertion of Lemma 1 is valid for any segment \(PQ\) belonging to the closure of any simply connected subdomain of the domain \(\Delta\), if in this subdomain the conditions of Lemma 1 are satisfied.

Lemma 2. If the function \(u(\xi,\eta)\in C^2(\Delta)\), \(L_0u=0\) in \(\Delta\), \(u(\xi,\eta)\in C(\overline{\Delta})\), \(u_\xi,u_\eta\in C(\overline{\Delta}\setminus K'A')\), \(u|_{A'C'}=0\), and the coefficients of equation (2) satisfy conditions \((B)\) in \(D_2\), then the maximum of \(u(\xi,\eta)\) in \(\overline{\Delta}\), if it is positive, is attained only on the segment \(K'A'\).

Proof. Suppose the contrary. Let the positive maximum of \(u(\xi,\eta)\) in \(\overline{\Delta}\) be attained at some point \(Q\notin K'A'\). Draw the segment \(PQ\), \(P\in A'C'\). Since \(u_\xi|_{A'C'}=0\), by Lemma \(1'\)

\[ u_\xi(Q)>0. \tag{6} \]

Let \(Q\) be an interior point of the domain \(\Delta\). Then at it one must have \(u_\xi=0\), which contradicts (6). Let \(Q\) be a point of \(K'C'\). Then at it one must have \(u_\xi\leqslant 0\), which contradicts (6) and completes the proof of the lemma.

As is seen from the proof of the lemma, its assertion remains valid also in the case when the coefficients of equation (3) are continuous at least in \(\overline{\Delta}\setminus K'A'\).

Denote by \(\Delta_1\) and \(\Delta_2\), respectively, the triangles \(K'E'C'\) and \(A'E'K'\) (see Fig. 2, б, в).

Lemma 3. If the function \(u(\xi,\eta)\in C(\overline{\Delta})\), \(u(\xi,\eta)\in C^2(\Delta_1\cup \Delta_2)\), \(L_0u=0\) in \(\Delta_1\cup \Delta_2\), \(u_\xi,u_\eta\in [(\Delta_1\setminus K'E')\cup(\Delta_2\setminus K'A')]\), \(u|_{A'C'}=0\), and the coefficients of equation (2) satisfy conditions \((B)\) in each of the domains \(\Delta_1\) and \(\Delta_2\), then the maximum of \(u(\xi,\eta)\) in \(\overline{\Delta}\), if it is positive, is attained only on the segment \(K'A'\).

The proof follows easily from Lemma \(1'\).

Following [7] (pp. 458—459), we prove Lemmas 4 and 5.

Lemma 4. If the function \(u(\xi,\eta)\) satisfies the conditions of Lemma 2 and the coefficients of equation (2) satisfy in \(D_2\) conditions \((A)\), then the maximum of \(u(\xi,\eta)\) in \(\overline{\Delta}\), if it is positive, is attained on \(K'A'\).

Lemma 5. If the function \(u(\xi,\eta)\) satisfies the conditions of Lemma 3 and the coefficients of equation (2) satisfy in \(\Delta_1\) and \(\Delta_2\) conditions \((A)\), then the maximum of \(u(\xi,\eta)\) in \(\overline{\Delta}\), if it is positive, is attained on \(K'A'\).

§ 2. FORMULATION OF PROBLEMS OF GROUP A. UNIQUENESS

In what follows, by a solution of equation (1) in the domain \(D\) we shall understand a function \(u(x,y)\) satisfying equation (1) in the domain \(D\) everywhere, with the possible exception of points of the straight line \(y=0\) and points of the segment \(KE\). We pose the following problems:

Problem \(A_1\). In the domain \(D\), find a solution \(u(x,y)\) of equation (1) satisfying the boundary conditions

\[ \begin{gathered} 1)\quad u|_{KL}=\varphi_1(x);\qquad 2)\quad u|_{RBA}=\varphi_2(s);\qquad 3)\quad u|_{AC}=\psi(s);\\[4pt] 4)\quad u(s+\delta)-u(-s)=g(s),\quad 0\leq s\leq S. \end{gathered} \]

Here, as also below, \(\varphi_1,\varphi_2,\psi,g\) are prescribed functions. Condition 4 defines a “curvilinear jump of compaction.” For \(K\equiv L\), condition 1 becomes the condition \(u(K)=u_0\), where \(u_0\) is a prescribed number.

Note that problem \(A_1\) is posed for any domain \(D\) described at the beginning of § 1, in particular also for all the domains shown in Fig. 1. For some special classes of domains \(D\) we pose the following problems:

Problem \(A_2\). In the domain \(D\), find a solution \(u(x,y)\) of equation (1) satisfying the boundary conditions

\[ \begin{aligned} &1)\quad \left.\frac{\partial u}{\partial \lambda}\right|_{LK}=\nu_1(x), \quad l\leq x\leq k; &&2)\quad \left.\frac{\partial u}{\partial \lambda}\right|_{AB\backslash A}=\nu_2(x), \quad a<x\leq b;\\ &3)\quad \left.\frac{\partial u}{\partial \lambda}\right|_{RB}=\nu_3(s); &&4)\quad \left.u\right|_{AC}=\psi(s);\\ &5)\quad u(s+\delta)-u(-s)=g(s), \quad 0\leq s\leq S, \end{aligned} \]

where \(\lambda\) is the vector directed inward to the domain \(D_1\).

Problem \(A_3\). In the domain \(D\), find a solution \(u(x,y)\) of equation (1) satisfying the boundary conditions

\[ 1)\quad \left.\alpha(x,y)u+\beta(x,y)\frac{\partial u}{\partial \lambda}\right|_{LK}=f_1(x), \quad l\leq x\leq k; \]

\[ 2)\quad \left.\alpha(x,y)u+\beta(x,y)\frac{\partial u}{\partial \lambda}\right|_{RBA}=f_2(x), \quad s_R\leq s\leq s_B\leq s_A,\quad a<b; \]

\[ 3)\quad \left.u\right|_{AC}=\psi(s); \qquad 4)\quad u(s+\delta)-u(-s)=g(s), \quad 0\leq s\leq S. \]

Here, as in what follows, \(\alpha(x,y)\) and \(\beta(x,y)\) are given functions.

Problem \(A\). Let the domain \(D\) (see Fig. 1, \(a\)) be such that \(l\leq k\) and \(a\leq b\). Suppose that a one-to-one correspondence is established between some infinite set \(E_1\subset LK\cup \sigma \cup AB\) and some set \(E_2\subset KC\), and that \(Q_1\) and \(Q_2\) are corresponding points \((Q_1\sim Q_2)\), \(Q_1\in E_1\) and \(Q_2\in E_2\), \(E_1' \equiv [LK\cup \sigma \cup AB]\backslash E_1\).

In the domain \(D\), find a solution \(u(x,y)\) of equation (1) satisfying the boundary conditions:

\[ 1)\quad \left.\alpha(x,y)u+\beta(x,y)\frac{\partial u}{\partial \lambda}\right|_{E_1'}=f(s); \qquad 2)\quad \left.u\right|_{AC}=\psi(s); \]

\[ 3)\quad u(Q_1)-u(Q_2)=g(Q_1). \]

Here condition 3) determines a “curvilinear jump of compaction” on an arbitrary set.

We shall say that a function \(u(x,y)\) belongs to the class of functions \(\Omega_1\) if it is continuous in \(\overline D\), satisfies equation (1) in \(D_1\cup D_2\), has \(u_x\) and \(u_y\) continuous everywhere in \(D_1\cup \overline{D}_2\), except possibly on the transition line, at the interior points of which \(u_y(x;0-0)=u_y(x,0+0)\).

We shall say that a function \(u(x,y)\) belongs to the class of functions \(\Omega_2\) if it possesses all the properties listed for functions of the class \(\Omega_1\), except, possibly, continuity on \(KE\) of the derivatives \(u_x\) and \(u_y\), which are continuous in the open triangular domain \(AEK\) up to the segment \(KE\).

Theorem 1. If the coefficients of equation (1) satisfy, in \(D_2\), conditions (A), then the solution of problem \(A_1\) is unique in the class \(\Omega_1\).

Proof. Suppose the contrary. Then there must exist a solution \(u(x,y)\) of problem \(A_1\), taking positive values and satisfying homogeneous boundary conditions.

Suppose that the positive maximum of \(u(x,y)\) in \(\overline D\) is attained in \(\overline{D}_1\) at some point \(Q\). By the ellipticity of equation (1) in \(D_1\) and by boundary conditions 1) and 2), the point \(Q\in D_1\cup LK\cup RBA\). Let \(Q\) be an interior point of the common part of the boundaries of the domains \(D_1\) and \(D_2\). Then at it, by the known

properties of elliptic equations, \(u_y(x,0+0)<0\). But since at the point \(Q\) the function \(u(x,y)\) attains its greatest positive value and in \(\overline{D}_2\) at this point \(u_y(x,0-0)>0\), this contradicts the “gluing” condition. Let \(Q\in LR\). Then, by virtue of boundary condition 4), on \(KC\) there is a point \(Q'\) such that \(u(Q')=u(Q)\). By virtue of boundary condition 3), by Lemma 4 on the common part of the boundaries of the domains \(D_1\) and \(D_2\) there is an interior point \(Q''\) such that \(u(Q'')=u(Q')=u(Q)\), which, as has already been shown, is impossible. Consequently, the positive maximum of \(u(x,y)\) in \(\overline{D}\) cannot be attained in \(\overline{D}_1\).

Assuming that the positive maximum of \(u(x,y)\) in \(\overline{D}\) is attained in \(\overline{D}_2\), we obtain that it is attained at an interior point of the common part of the boundaries of the domains \(D_1\) and \(D_2\). Consequently, the solution of problem \(A_1\) is unique in the class \(\Omega_1\).

Repeating the preceding arguments, with the aid of Lemmas 1–5 it is easy to prove that Theorems 2–4 are valid.

Theorem 2. If \(l<k\), \(a<b\), and 1) the coefficients of equation (1) satisfy conditions (A) in \(D_2\); 2) the vector \(\lambda\) forms an acute angle with the inward normal at every point of the indicated part of the boundary of the domain \(D\); 3) the arc \(\sigma\) is tangent to the \(OX\) axis and its tangent near the point \(L\) forms an obtuse angle with the \(OX\) axis, and near the point \(B\) an acute angle; 4) \(\nu_2(B)=\nu_3(B)\), then the solution of problem \(A_2\) is unique in the class \(\Omega_1\).

Theorem 3. If \(l<k\), \(a<b\), and 1) the coefficients of equation (1) satisfy conditions (A) in \(D_2\); 2) the vector \(\lambda\) and the arc \(\sigma\) satisfy the conditions of Theorem 2; 3) the functions \(\alpha(x,y)\) and \(\beta(x,y)\) at every point of the indicated parts of the boundary of the domain \(D\) satisfy the conditions \(\alpha\beta\leqslant 0\), \(\alpha^2+\beta^2\ne0\), \(\beta\ne0\), then the solution of problem \(A_3\) is unique in the class \(\Omega_1\).

Theorem 4. If \(l<k\), and 1) the coefficients of equation (1) in \(D_2\) satisfy conditions (A); 2) the set \(E_1\) is closed in \(\overline{LK}\cup\sigma\cup\overline{AB}\), and each of the arcs composing the set \(E_1\) belongs to the class \(A(1,\lambda)\) ([8], p. 10); 3) the vector \(\lambda\), the curve \(\sigma\), and the functions \(\alpha(x,y)\) and \(\beta(x,y)\) satisfy the conditions of Theorem 3, then the solution of problem \(A\) is unique in the class \(\Omega_1\).

Theorem 5. Suppose that the coefficients of equation (1) in each of the domains \(D'_2\) (the triangle \(KCE\)) and \(D''_2\) (the triangle \(KEA\)) satisfy conditions (A). Then in the class of functions \(\Omega_2\) there can exist no more than one solution of each of the problems with boundary conditions of the problems \(A_1\), \(A_2\), \(A_3\), and \(A\), if, respectively, all the conditions of Theorems 1–4 are fulfilled, except for conditions 1).

Remark 2. The problems \(A_2\), \(A_3\), and \(A\) can be posed in the class of functions for which at the points \(L\), \(K\), \(B\) the functions \(\nu_1(x)\), \(\nu_2(x)\), \(f_1(x)\), \(f(s)\) do not exist. To ensure uniqueness of the solutions of these and of the problems formulated below, it is sufficient to specify the values of the unknown function at these points. To ensure uniqueness of the solution of problems \(A_3\) and \(A\) in the absence of restrictions on the approach of the arc \(\sigma\) to the \(OX\) axis, it is sufficient to require that the function \(\beta(x,y)\) at the points \(L\) and \(B\) vanish with order not less than the order of the singularity of \(\partial u/\partial\lambda\).

§ 3. FORMULATION OF PROBLEMS OF GROUP B. UNIQUENESS

Let the domain \(D\) be such that \(k<l\) and \(a\geqslant b\). (Figs. 1, 2).

Problem \(B_1\). In the domain \(D\), find a solution \(u(x,y)\) of equation (1), satisfying the boundary conditions

\[ 1)\quad u\big|_{AC}=\psi(s);\qquad 2)\quad \left.\frac{\partial u}{\partial y}\right|_{\overline{KL}}=\nu_1(x); \]

\[ 3)\quad \left.\frac{\partial u}{\partial y}\right|_{\overline{BA}\setminus A}=\nu_2(x); \qquad 4)\quad \alpha(x,y)u+\beta(x,y)\left.\frac{\partial u}{\partial \lambda}\right|_{\overline{RB}\setminus B}=\varphi(s); \]

\[ 5)\quad u(s+\delta)-u(-s)=g(s),\qquad 0\leq s\leq S. \]

Problem \(B_2\). In the domain \(D\), find a solution \(u(x,y)\) of equation (1) satisfying conditions 1), 2) (or 3)), 4), and 5) of problem \(B_1\), and the condition: \(\left.u\right|_{AB}=\varphi_1(x)\) (or \(\left.u\right|_{KL}=\varphi_2(x)\)).

Problem \(B\). Suppose a one-to-one correspondence has been established between a certain infinite set \(E_1\subset \sigma\) and a certain set \(E_2\subset KC\), \(Q_1\sim Q_2\), \(Q_1\in E_1\), \(Q_2\in E_2\), \(E_1\equiv \sigma\setminus E_1'\).

In the domain \(D\), find a solution \(u(x,y)\) of equation (1) satisfying the boundary conditions

\[ 1)\quad \left.u\right|_{AC}=\psi;\qquad 2)\quad m(x,y)u+n(x,y)\left.\frac{\partial u}{\partial \gamma}\right|_{\overline{KL}\cup \overline{AB}\setminus A}=\nu(x); \]

\[ 3)\quad \alpha(x,y)u+\left.\beta(x,y)\frac{\partial u}{\partial \lambda}\right|_{E_1'}=\varphi(s); \qquad 4)\quad u(Q_1)-u(Q_2)=g(Q_1), \]

where the vector \(\lambda\) is directed into the domain \(D_1\); \(m(x,y)\) and \(n(x,y)\) are given functions.

Theorem 6. If 1) the coefficients of equation (1) satisfy, in \(D_2\), conditions (B); 2) the vector \(\lambda\) and the functions \(\alpha(x,y)\) and \(\beta(x,y)\) satisfy the conditions of Theorem 4, then in the class of functions \(\Omega_1\) there can exist no more than one solution of each of the problems \(B_1\) and \(B_2\).

We note that in Theorem 6, as well as in Theorem 7 formulated below, no restrictions are imposed on the approach of the arc to the \(OX\) axis.

Theorem 7. If 1) in the domain \(D_2\) the coefficients of equation (1) satisfy conditions (B); 2) the set \(E_1\) is closed in \(\sigma\) and each of the two sets constituting \(E_1'\) belongs to the class \(A(1,\lambda)\) ([8], p. 10); 3) the vector \(\lambda\) and the functions \(\alpha(x,y)\) and \(\beta(x,y)\) satisfy the conditions of Theorem 4; 4) the functions \(m(x,y)\) and \(n(x,y)\) satisfy on the \(OX\) axis the conditions \(mn\leq 0\) and \(m^2+n^2\neq 0\); 5) the vector \(\gamma\) forms with the \(OX\) axis at the point \(K\) an angle \(\omega\), \(\pi/4\leq \omega\leq \pi/2\), then the solution of problem \(B\) is unique in the class of functions \(\Omega_1\).

Proof. Suppose the contrary. Then there must exist a solution \(u(x,y)\) of problem \(B\), taking positive values and satisfying homogeneous boundary conditions.

Suppose that the positive maximum of \(u(x,y)\) in \(D\) is attained in \(\overline{D}_2\) at some point \(Q\). Then, by virtue of boundary condition 1), by Lemma 2 the point \(Q\) can belong only to \(\overline{KA}\setminus A\), and at it \(\partial u/\partial \gamma<0\), since at the corresponding point of the domain \(\Delta\), by Lemma \(1'\), we have \(u_\xi>0\) and at the same point \(u_\eta>0\). Therefore, by virtue of boundary condition 2), the point \(Q\) cannot belong to the segments \(\overline{KL}\) and \(\overline{BA}\). Consequently, the point \(Q\) must be an interior point of the common part of the boundaries of the domains \(D_1\) and \(D_2\), which is impossible by virtue of the “gluing” condition. Hence \(u(x,y)\) cannot attain a positive maximum in \(\overline{D}\) and \(\overline{D}_2\).

Suppose now that it is attained at a point \(Q\in \overline{D}_1\). Obviously, \(Q\in \overline{D}_1\). Let \(Q\in E_1'\). Then at it, by virtue of the known properties of elliptic equations and the properties of the functions \(\alpha(x,y)\) and \(\beta(x,y)\) and of the vector \(\lambda\), we obtain \(\alpha(x,y)u+\beta(x,y)\,\partial u/\partial \lambda\neq 0\), which contradicts boundary condition 3). Let \(Q\in E_1\). Then, by virtue of boundary condition 4), the positive maximum of \(u(x,y)\) in \(\overline{D}\) must be attained in \(\overline{D}_2\), which, as shown above, is impossible. The contradiction obtained completes the proof of the theorem.

Theorem 6 is proved similarly.

Remark 3. Let the domain \(D\) be such that \(K \equiv L\), and let the normal to the axis \(OX\) at the point \(K\) in a neighborhood of this point belong to the domain \(\overline D\) (for example, such a domain is the domain \(D_*\); see the introduction). Consider the problems \(B_{10}\), \(B_{20}\), and \(B_0\), respectively, with all the conditions of the problems \(B_1\), \(B_2\), and \(B\), except for the second boundary condition. The uniqueness of solutions of the problems \(B_{10}\), \(B_{20}\), and \(B_0\) holds if the conditions of Theorems 6 and 7 are satisfied and the derivative \(u_y\) is continuous at the point \(K\). The validity of this assertion follows easily from the arguments in the proof of Theorem 7 and from the vanishing at the point \(K\) of the first-order partial derivative with respect to \(y\) of the solution of each of these problems.

§ 4. FORMULATION OF PROBLEMS OF GROUP C. UNIQUENESS

Problem \(C_1\). Let the domain \(D\) be such that \(l<k\) and \(b<a\) (see Fig. 1, в). Suppose that a one-to-one correspondence is established between an infinite set \(E_1 \subset \overline{LK}\cup\sigma\) and some set \(E_2\subset KC\), with \(Q_1\sim Q_2\), \(Q_1\in E_1\), \(Q_2\in E_2\), \(E_1' \equiv (\overline{LK}\cup\sigma)\setminus E_1\). In the domain \(D\), find a solution \(u(x,y)\) of equation (1) satisfying the boundary conditions

\[ 1)\quad u|_{AC}=\psi(s);\qquad 2)\quad m(x,y)u+n(x,y)\frac{\partial u}{\partial \gamma}\bigg|_{\overline{BA}\setminus A}=\nu(x); \]

\[ 3)\quad \alpha(x,y)u+\beta(x,y)\frac{\partial u}{\partial \lambda}\bigg|_{E_1'}=\varphi(s);\qquad 4)\quad u(Q_1)-u(Q_2)=g(Q_1). \]

Problem \(C_2\). Let the domain \(D\) be such that \(k<l\) and \(a<b\) (Fig. 1, е). Suppose that a one-to-one correspondence is established between an infinite set \(E_1\subset \overline{\sigma}\cup AB\) and some set \(E_2\subset KC\), \(Q_1\sim Q_2\), \(Q_1\in E_1\), \(Q_2\in E_2\); \(E_1' \equiv (\overline{\sigma}\cup AB)\setminus E_1\).

In the domain \(D\), find a solution \(u(x,y)\) of equation (1) satisfying the boundary conditions

\[ 1)\quad u|_{AC}=\psi(s);\qquad 2)\quad m(x,y)u+n(x,y)\frac{\partial u}{\partial \gamma}\bigg|_{\overline{KL}}=\nu(x); \]

\[ 3)\quad \alpha(x,y)u+\beta(x,y)\frac{\partial u}{\partial \lambda}\bigg|_{E_1'}=\varphi(s);\qquad 4)\quad u(Q_1)-u(Q_2)=g(Q_1). \]

We easily prove Theorems 8 and 9.

Theorem 8. If \(l<k\) and \(b<a\), and 1) the coefficients of equation (1) satisfy in \(D_2\) (in \(D_2'\) and \(D_2''\)) the conditions \((A)\); 2) the set \(E_1\) is closed in \(\overline{\sigma}\cup AB\), and each of the arcs composing the set \(E_1'\) belongs to the class \(A(1;\lambda)\); 3) the vectors \(\lambda\) and \(\gamma\), the curve \(\sigma\), and the functions \(\alpha(x,y)\) and \(\beta(x,y)\) satisfy the conditions of Theorem 3; 4) the functions \(m(x,y)\) and \(n(x,y)\) satisfy the conditions of Theorem 7, then the solution of problem \(C_1\) is unique in each of the classes \(\Omega_1\) and \(\Omega_2\).

Theorem 9. If \(k<l\) and \(a<b\), and 1) the coefficients of equation (1) satisfy in \(D_2\) the conditions \((B)\), 2) the set \(E\) is closed in \(\overline{\sigma}\cup AB\), and each of the arcs composing the set \(E_1'\) belongs to the class \(A(1,\lambda)\), 3) the vectors \(\lambda\) and \(\gamma\), and the functions \(\alpha(x,y)\), \(\beta(x,y)\), \(m(x,y)\), and \(n(x,y)\) satisfy the conditions of Theorem 7, then the solution of problem \(C_2\) is unique in the class \(\Omega_1\).

Remark 4. If \(\sigma\) is a curve of class \(A(1,\lambda)\), then whatever the set \(E_1\) may be, the assertions of Theorems 4, 7, 8, and 9 are valid when all the conditions listed in them are fulfilled, except, perhaps, condition 2).

References

1. Frankl’ F. I. PMM, 20, no. 2, 1956.
2. Frankl’ F. I. PMM, 21, no. 1, 1957.
3. Linear Equations of Mathematical Physics, SMB. Publ. “Nauka,” 1965.
4. Bitsadze A. V. DAN SSSR, 109, no. 6, 1956.
5. Devingtal’ Yu. V. Izv. Vuzov, Mathematics, no. 2, 1958.
6. Lerner M. E. Volga Mathematical Collection, issue 3, 1965, p. 255.
7. Agmon S., Nirenberg L., Protter M. H. Comm. on Pure and Appl. Math., 6, no. 4, 1953.
8. Miranda K. Equations with Partial Derivatives of Elliptic Type. IL, Moscow, 1957.

Received by the editors
July 2, 1965

Kuibyshev Polytechnic
Institute

Kuibyshev State
Pedagogical Institute