UDC 517.946:566.5

ON THE ASYMPTOTICS OF SOLUTIONS OF A MIXED PROBLEM

T. I. Zelenyak

In 1935, Bochner and Neumann, in [1], considered a broad class of equations of the form

\[ \sum_0^N \frac{\partial^i}{\partial t^i} A_i u = 0 \tag{1} \]

(\(A_i\) are self-adjoint operators in a Hilbert space \(H\), \(t\) is a parameter) and gave a condition of almost periodicity in \(t\) for the solutions of these equations. Under natural assumptions on the operators \(A_i\), this condition is the compactness of the trajectory \(u(t) \in H\) for \(-\infty < t < +\infty\). For a number of problems arising in mathematical physics, such compactness can be established for any domain and for any solution of the corresponding equation when homogeneous boundary conditions are satisfied. Equations of the form (1) include also the equation of S. L. Sobolev

\[ \frac{\partial^2}{\partial t^2} \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \frac{\partial^2 u}{\partial z^2} = 0, \tag{2} \]

considered under the initial and boundary conditions

\[ u|_{t=0} = u_0(x,y,z); \qquad u_t|_{t=0} = u_1(x,y,z); \tag{3} \]

\[ u|_{\Gamma} = 0. \tag{4} \]

Equation (2) is equivalent to the linearized system arising in the study of small oscillations of a rotating fluid [2]. In [2] the existence and uniqueness of solutions of problem (2)—(4) were proved under certain smoothness assumptions on \(u_0, u_1\), prescribed in a bounded domain \(\Omega\) with boundary \(\Gamma\) in the space \(x,y,z\). Other problems for equation (2) were also considered. The solutions of these problems depend continuously on the initial data and on the boundary \(\Gamma\) [3, 4]. Thus, in [2] a mixed problem, well-posed in the classical sense, was indicated for an equation not solved with respect to the highest derivative in time. S. L. Sobolev posed the problem of studying the asymptotics of the solutions of equation (2) under conditions (3), (4).

It is not difficult to prove that the solution of problem (2)—(4) admits the estimate \(\|u\|_{\overset{\circ}{W}{}^{1}_{2}(\Omega)} \le C\), where \(C\) depends only on \(u_0, u_1\). From the results of R. A. Aleksandryan it follows that it is impossible to prove an analogous estimate in \(W^2_2(\Omega)\).

In [5] a system of generalized eigenfunctions was constructed for the operator arising when the variable \(t\) is separated in the planar case \(\left(\dfrac{\partial u}{\partial y}=0\right)\). These functions are solutions of the problem

\[ -\lambda^2\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial z^2}\right)+\frac{\partial^2 u}{\partial z^2}=0; \tag{5} \]

\[ u|_\gamma=0. \tag{6} \]

In [5] it was established that, although in the case where \(\gamma\) is a circle all solutions of the corresponding mixed problem are almost periodic, one can specify domains, arbitrarily little different from a disk and with analytic boundary, in which the solutions are not almost periodic.

In [7] the almost periodicity of solutions of problem (2)—(4) was established in the case where \(\Omega\) is a sphere, and an example of a domain \(\Omega\) (a cone) was constructed in which there are no almost periodic solutions of equation (2) vanishing on the boundary. Thus, an estimate \(\|u\|_{W_2^1(\Omega)}\le C\), where

\[ \frac{dc}{dt}=0, \]

which ensures compactness of the solution in \(W_2^1(\Omega)\) and thereby, by [1], almost periodicity of the solution, is impossible.

The construction of almost periodic solutions evidently leads to the study of a certain spectral problem. The resolvents of the operators arising here, or, equivalently, the Dirichlet problem for the string equation (5), (6), were studied in [8—12].

In the present paper the planar case of S. L. Sobolev’s problem (2)—(4) is considered. We shall prove that if, in a given convex domain, not all solutions of this problem are almost periodic, then there exist solutions decreasing at infinity whose first derivatives with respect to \(x,z\) are a linear combination of functions of the form \(\vartheta(x,z)e^{i\mu(x,z)t}\), while the second derivatives can grow no faster than the first power of \(t\). The functions \(\mu(x,z)\) depend continuously on the boundary. The construction of solutions of the indicated form makes it possible to construct invariant subspaces \(H_k\) \((k=1,2,\ldots)\) of the operator associated with problem (5), (6), the resolvent, the spectral function, and a simple representation for this operator in \(H_k\).

In [13—16] mixed problems were studied for equations generalizing (2). In [16] it was established that, for some such equations, the arising spectral problem is associated with operators of elliptic type, with the parameter entering the coefficients of the higher-order terms. We shall consider equations generalizing (1), and shall give examples of non-almost-periodic solutions of the mixed problem for some such equations.

§ 1. ALMOST PERIODIC SOLUTIONS OF S. L. SOBOLEV’S EQUATION WITH ALMOST PERIODS DEPENDING ON THE SPATIAL VARIABLES

We shall study the asymptotics of solutions of the equation

\[ \frac{\partial^2}{\partial t^2}\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)+\frac{\partial^2 u}{\partial y^2}=0, \tag{1} \]

satisfying the boundary condition

\[ u|_\Gamma=0, \tag{2} \]

where \(\Gamma\) is the boundary of the convex domain \(\Omega\) in the \(x,y\) plane. The existence, uniqueness, and continuous dependence on the boundary of the solutions of equation (1) under condition (2) and the initial conditions

\[ u\big|_{t=0}=u_0(x,y);\qquad \frac{\partial u}{\partial t}\bigg|_{t=0}=u_1(x,y) \tag{3} \]

have been studied in [2—4].

Let us construct solutions of equation (1) of the form

\[ u(x,y,t)=v(x,y)e^{i\mu(x,y)t}. \tag{4} \]

To determine the functions \(v\) and \(\mu\), we obtain the system of equations:

\[ -\mu^2\left(\frac{\partial^2 v}{\partial x^2}+ \frac{\partial^2 v}{\partial y^2}\right)+ \frac{\partial^2 v}{\partial y^2} -4\mu\left(\frac{\partial v}{\partial x}\frac{\partial\mu}{\partial x}+ \frac{\partial v}{\partial y}\frac{\partial\mu}{\partial y}\right)- \]

\[ -2v\mu\left(\frac{\partial^2\mu}{\partial x^2}+ \frac{\partial^2\mu}{\partial y^2}\right) -2v\left[\left(\frac{\partial\mu}{\partial x}\right)^2+ \left(\frac{\partial\mu}{\partial y}\right)^2\right]=0; \tag{5} \]

\[ -2\mu^2\left(\frac{\partial v}{\partial x}\frac{\partial\mu}{\partial x}+ \frac{\partial v}{\partial y}\frac{\partial\mu}{\partial y}\right) +2\frac{\partial\mu}{\partial y}\frac{\partial v}{\partial y}+ \]

\[ +v\left[\frac{\partial^2\mu}{\partial y^2} -\mu^2\left(\frac{\partial^2\mu}{\partial x^2}+ \frac{\partial^2\mu}{\partial y^2}\right)\right] -4v\mu\left[\left(\frac{\partial\mu}{\partial x}\right)^2+ \left(\frac{\partial\mu}{\partial y}\right)^2\right]=0; \tag{6} \]

\[ \mu^2\left[\left(\frac{\partial\mu}{\partial x}\right)^2+ \left(\frac{\partial\mu}{\partial y}\right)^2\right] = \left(\frac{\partial\mu}{\partial y}\right)^2 . \tag{7} \]

In the system (5)—(7), instead of the function \(\mu\) we introduce \(\tau\):

\[ \mu=\sqrt{\frac{\tau^2}{1+\tau^2}}. \tag{8} \]

After obvious transformations, for the functions \(v\) and \(\tau\) we obtain the system of equations:

\[ -\tau^2\frac{\partial^2 v}{\partial x^2} +\frac{\partial^2 v}{\partial y^2} -4\tau\frac{\partial\tau}{\partial x}\frac{\partial v}{\partial x} -2\left(\frac{\partial\tau}{\partial x}\right)^2 v -2\tau\frac{\partial^2\tau}{\partial x^2}v=0; \tag{9} \]

\[ -2\tau^2\frac{\partial v}{\partial x}\frac{\partial\tau}{\partial x} +2\frac{\partial v}{\partial y}\frac{\partial\tau}{\partial y} +v\left(\frac{\partial^2\tau}{\partial y^2} -\tau^2\frac{\partial^2\tau}{\partial x^2}\right) -4\tau\left(\frac{\partial\tau}{\partial x}\right)^2v=0; \tag{10} \]

\[ \tau^2\left(\frac{\partial\tau}{\partial x}\right)^2 = \left(\frac{\partial\tau}{\partial y}\right)^2 . \tag{11} \]

It is easy to see that the system (8)—(10) is obtained in constructing solutions of the form \(w=ve^{i\tau t}\) for the equation

\[ \frac{\partial^2}{\partial t^2}\frac{\partial^2 w}{\partial x^2} +\frac{\partial^2 w}{\partial y^2}=0. \tag{12} \]

Thus, as in the case of constant \(\mu\), to each solution \(u=ve^{i\mu t}\) of equation (1) there corresponds a solution of equation (12) \(w=ve^{\pm i\tau t}\), where

\[ \tau=\sqrt{\frac{\mu^2}{1-\mu^2}}. \]

1. Differential Equations

We shall construct all solutions of the system (9)—(11). Suppose that in a neighborhood of some point the equality

\[ \tau \frac{\partial \tau}{\partial x}=\frac{\partial \tau}{\partial y}. \tag{13} \]

holds.

Equation (10) in this case is equivalent to the following:

\[ \frac{\partial v}{\partial y} = \tau \frac{\partial v}{\partial x} + \frac{\partial \tau}{\partial x}v, \tag{14} \]

and it is easy to see that equation (9) is a consequence of (13), (14). Similarly, in the case of the equation

\[ \tau \frac{\partial \tau}{\partial x} = -\frac{\partial \tau}{\partial y} \tag{15} \]

we have

\[ \tau \frac{\partial v}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial \tau}{\partial x}v =0, \tag{16} \]

and equation (9), similarly to the preceding case, follows from (15), (16). Thus, in order to construct all solutions of the required kind it suffices to solve the system (13), (14). It is easy to see that in a neighborhood of the point \((x_0,y_0)\), where \(\dfrac{\partial \tau}{\partial x}\ne 0\), the solution of the Hopf equation (13) is found from the functional equation

\[ x+\tau y=f(\tau) \tag{17} \]

and is a function whose level lines are straight lines. To find \(v\) in this case, let us write the system defining the characteristics

\[ \frac{dx}{-\tau}=\frac{dy}{1}=\frac{dv}{-\dfrac{\partial \tau}{\partial x}v}. \tag{18} \]

Since

\[ \frac{d\tau}{dy} = \frac{\partial \tau}{\partial x}\frac{dx}{dy} + \frac{\partial \tau}{\partial y} = -\tau \frac{\partial \tau}{\partial x} + \frac{\partial \tau}{\partial y} =0, \]

we have \(x+\tau y=C_1\). From (17) we obtain \(\dfrac{\partial \tau}{\partial x}=-\dfrac{1}{y-f'}\) and \(\dfrac{dv}{dy}=-\dfrac{v}{y-f'(\tau)}\), whence \(v\Big/\dfrac{\partial \tau}{\partial x}=C_2\). The general solution of equation (14), therefore, has the form \(v=\dfrac{\partial \tau}{\partial x}F(\tau)\), if \(\dfrac{\partial \tau}{\partial x}\ne 0\). For constant \(\tau\) we obtain \(v=f(x+\tau y)\). Thus the following has been proved.

Lemma 1. In a neighborhood of the point \((x_0,y_0)\), where \(\dfrac{\partial \tau}{\partial y}\ne 0\), any solution of the system (5), (6), (7) can be represented in the form

\[ v=\frac{\partial \tau}{\partial x}F(\tau), \qquad \mu=\pm \sqrt{\frac{\tau^2}{1+\tau^2}}, \qquad \tau^2\left(\frac{\partial \tau}{\partial x}\right)^2 = \left(\frac{\partial \tau}{\partial y}\right)^2. \]

From Lemma 1 it follows that if \(\dfrac{\partial \mu}{\partial x}\ne 0\), then the solutions of equation (1) having the form (4) can be written as follows:

\[ u(x,y,t)=\frac{\partial \tau}{\partial x}F(\tau)e^{\pm i\sqrt{\frac{\tau^2}{1+\tau^2}}\,t} =\frac{\partial}{\partial x}\int_{\tau_0}^{\tau} F(\alpha)e^{\pm i\sqrt{\frac{\alpha^2}{1+\alpha^2}}\,t}\,d\alpha, \]

where \(\tau_0\) is a constant; \(F\) is an arbitrary function, and

\[ \tau^2\left(\frac{\partial \tau}{\partial x}\right)^2 = \left(\frac{\partial \tau}{\partial y}\right)^2. \]

In the case \(\dfrac{\partial \mu}{\partial x}\equiv 0\) we have

\[ u(x,y,t)=f(x\pm \tau y)e^{i\mu t},\qquad \mu^2=\frac{\tau^2}{1+\tau^2}. \]

It is easy to verify that the function

\[ u(x,y,t)=\int_{\tau_0}^{\tau(x,y)} F(\alpha)e^{\pm i\sqrt{\frac{\alpha^2}{1+\alpha^2}}\,t}\,d\alpha \tag{19} \]

also satisfies equation (1), if \(\tau\) is a solution of the Hopf equation.

From Lemma 1 the following follows.

Theorem 1. Every solution of the equation

\[ \sum_{0}^{N}\frac{\partial^k}{\partial t^k}A_k u=0 \tag{20} \]

\[ \left( \text{where the operator } A_k u=a_{k,1}\frac{\partial^2 u}{\partial x^2} +a_{k,2}\frac{\partial^2 u}{\partial y^2} \right), \]

whose derivatives with respect to \(x,y\) have the form (4), in a neighborhood of the point \(x_0,y_0\), where \(\dfrac{\partial \mu}{\partial x}\ne 0\), can be represented as follows:

\[ u(x,y,t)=\int_{\tau_0}^{\tau(x,y)} f(\alpha)e^{il(\alpha)t}\,d\alpha+\psi(t), \tag{21} \]

where

\[ P(l)+\alpha^2 Q(l)=0;\qquad P(\lambda)=\sum a_{k,1}(i\lambda)^k,\qquad Q(\lambda)=\sum a_{k,2}(i\lambda)^k, \]

\(\tau\) is a solution of the Hopf equation, and \(\tau_0\) is some constant. Here one must exclude the case \(A_k=C_kA\), where \(C_k\) are constants.

In the next paragraph we shall construct all solutions of equation (20) having the form

\[ u(x,y,t)=\int_{\tau_1(x,y)}^{\tau_2(x,y)} f(\alpha)e^{il(\alpha)t}\,d\alpha, \tag{22} \]

where \(\left.\tau_1\right|_{\Gamma}=\left.\tau_2\right|_{\Gamma}\), and, consequently, \(u\) satisfies the condition \(\left.u\right|_{\Gamma}=0\). For this purpose it is sufficient in the domain \(\Omega\) to study the following boundary-value problem for the Hopf equations:

\[ (-1)^k\tau_k\frac{\partial \tau_k}{\partial x} = \frac{\partial \tau_k}{\partial y}, \qquad \left.\tau_1\right|_{\Gamma}=\left.\tau_2\right|_{\Gamma}. \tag{23} \]

The asymptotics of the function \(u\) in (22) is easily studied. Let us note that, in the case of the \(n\)-dimensional equation of S. L. Sobolev,

\[ \frac{\partial^{2}}{\partial t^{2}} \left( \frac{\partial^{2}u}{\partial x_{1}^{2}}+\ldots+ \frac{\partial^{2}u}{\partial x_{n}^{2}} \right) + \frac{\partial^{2}u}{\partial x_{n}^{2}}=0 \]

one can construct solutions of the form \(u=v(x_{1},\ldots,x_{n})e^{i\mu t}\), where \(\mu\) is determined from the equation

\[ \sum_{1}^{n} a_i(\mu)x_i+f(\mu)=0;\qquad a_n^2=\mu^2\sum_{1}^{n}a_i^2;\qquad f,a_1,\ldots,a_{n-1} \]

are arbitrary, and

\[ v(x_1,\ldots,x_n)=\frac{\partial\mu}{\partial x_1}\, F\left( x_2-\frac{a_2}{a_1}x_1,\ldots, x_n-\frac{\mu^2-1}{\mu^2}\frac{a_n}{a_1}x_1 \right), \]

but satisfying boundary conditions analogous to the two-dimensional case, for \(n=3\), is apparently impossible.

§ 2. CONSTRUCTION OF SOLUTIONS OF A SPECIAL BOUNDARY-VALUE PROBLEM FOR THE HOPF EQUATIONS

With respect to the domain \(\Omega\), assume the following: the boundary of this domain is given by the equation \(r=\rho(\varphi)\), \(\rho(\varphi)\) has three continuous derivatives and satisfies the condition
\(\rho^{2}+2\dot{\rho}^{2}-\rho\ddot{\rho}>q>0\), where \(q\) is some constant; \(\varphi\) is the polar angle of a point on the contour. We shall agree to regard increasing \(\varphi\) as corresponding to motion of the radius vector counterclockwise.

Let (23) hold in \(\Omega\); \(\tau_1,\tau_2\) are functions continuous in \(\Omega\). If

\[ \tau_1|_{\Gamma}=\tau_2|_{\Gamma}=\lambda(\varphi), \]

then \(\tau_k\) is determined from the equations

\[ x+(-1)^k\tau_k y=x(\varphi)+(-1)^k\lambda(\varphi)y(\varphi); \tag{24} \]

\[ \tau_k=\lambda(\varphi), \tag{25} \]

where \(x(\varphi)=\rho(\varphi)\cos\varphi\), \(y(\varphi)=\rho(\varphi)\sin\varphi\). From (24), (25) it follows that, at the points \(\varphi_n\) lying on the contour and forming the vertices of a broken line whose tangent of the angle between the \(y\)-axis and its \(n\)-th segment is equal to \((-1)^{n+1}\lambda(\varphi_0)\), the equality \(\lambda(\varphi_n)=\lambda(\varphi_0)\) holds.

We shall call the broken line indicated above a cycle with \(2N\) segments if its \(2N+1\)-st vertex coincides with \(\varphi_0\). If the set of points \(\varphi\) where \(\lambda(\varphi)=\lambda(\varphi_0)\) has positive measure, then the corresponding function (22) is identically equal to zero on a set of positive planar measure.

Thus, from John’s work [9] it follows that, in order to construct nontrivial smooth solutions of the form (22), it suffices to consider only those functions \(\lambda_n(\varphi)\) which correspond to cycles with \(2n\) segments, with the function \(\lambda_n(\varphi)\) depending continuously on \(\varphi\). On the other hand, if \(\lambda_n(\varphi)\) associates with each point \(\varphi\) the indicated cycle and

\[ \frac{d\lambda_n}{d\varphi}=0 \]

on some set of points \(\varphi\) of positive measure, then, obviously, there exist functions \(f\) and \(g\) such that

\[ f(x+\lambda_n y)-g(x-\lambda_n y)\ne 0 \]

and

\[ u=[f(x+\lambda_n y)-g(x-\lambda_n y)]\, e^{\pm i\sqrt{\frac{\lambda_n^2}{1+\lambda_n^2}}\,t} \]

is a periodic solution of problem (1), (2). From what has been said above it follows that, in order to construct solutions of problem (1), (2) whose derivatives are almost-periodic functions, for fixed \(x,y\), of the form

\[ v_1 e^{i\mu_1(x,y)t}-v_2 e^{i\mu_2(x,y)t}, \tag{26} \]

it is sufficient to construct all continuous functions \(\lambda_n(\varphi)\) which generate cycles.

We shall construct functional equations for determining \(\lambda_n(\varphi)\), prove the differentiability of these functions, and thereby construct all possible smooth solutions of problem (1), (2) whose derivatives have the form (26). In essence, the functions \(\lambda_n(\varphi)\) determining cycles were already used in the work of R. A. Aleksandryan [5] for constructing generalized eigenfunctions of the operator associated with problem (1), (2). In that work an example was constructed of a function \(\lambda_n(\varphi)\) such that \(\lambda_n(\varphi)\ne \mathrm{const}\), and hence the conclusion was drawn that non-almost-periodic solutions of problem (1), (2) exist. Instead of the topological mapping of the boundary into itself used in [5, 8–12], it is more convenient for us, in constructing differentiable solutions, to use functional equations.

By virtue of the assumption on the convexity of \(\Gamma\), each straight line \(x-\operatorname{ctg}\alpha\,y=x(\varphi)-\operatorname{ctg}\alpha\,y(\varphi)\) intersects the contour in no more than two points. Let \(T(\varphi,\alpha)\) be the point, different from \(\varphi\), lying on the contour \(\Gamma\) and on our straight line. If the straight line is tangent to the contour, then \(T(\varphi,\alpha)\) coincides with \(\varphi\). Let \(f_1(\varphi,\alpha)\) be the polar angle of the point \(T(\varphi,\alpha)\), defined as follows: \(f_1(\varphi,\alpha)\) is a continuous function of its arguments for \(-\infty<\varphi<+\infty\), \(-\infty<\alpha<+\infty\), ranges from \(-\infty\) to \(+\infty\), and satisfies the condition

\[ f_1(\varphi+2\pi,\alpha)=f_1(\varphi,\alpha)-2\pi;\qquad f_1(\varphi,\alpha+\pi)=f_1(\varphi,\alpha)+2\pi. \tag{27} \]

We shall prove the existence of such a function.

Let \(\Phi(\varphi,\alpha)\) be the polar angle corresponding to the point \(T(\varphi,\alpha)\), with \(0\leq \Phi(\varphi,\alpha)<2\pi\) for \(\alpha_0\leq\alpha<\alpha_0+\pi\), \(0\leq\varphi<2\pi\), and let \(\Phi(0,\alpha_0)=0\).

As \(\alpha\) increases, the point \(T(\varphi,\alpha)\) moves continuously along the contour and, when \(\alpha\) changes by \(\pi\), traverses the whole contour once, just as when \(\varphi\) changes by \(2\pi\). By positivity of the curvature, the function \(\Phi(\varphi,\alpha)\) has a discontinuity only on a monotone curve \(\alpha=\alpha(\varphi)\) in the \(\varphi,\alpha\) plane; moreover \(\Phi(\varphi,\alpha)\to\Phi(\varphi_0,\alpha_0)\) if \(\alpha\geq \alpha(\varphi)\), \(\varphi\to\varphi_0\), \(\alpha\to\alpha_0\), \(\alpha_0=\alpha(\varphi_0)\), and \(\Phi(\varphi,\alpha)\to\Phi(\varphi_0,\alpha_0)+2\pi\) if \(\alpha<\alpha(\varphi)\), \(\varphi\to\varphi_0\), \(\alpha\to\alpha_0\). At the other points of the rectangle under consideration the function \(\Phi\) is continuous by continuity of the mapping \(T(\varphi,\alpha)\) as a function of the parameters \(\varphi,\alpha\).

Consider the function

\[ \Phi_1(\varphi,\alpha)= \begin{cases} \Phi(\varphi,\alpha), & \alpha\geq\alpha(\varphi),\\ \Phi(\varphi,\alpha)-2\pi, & \alpha<\alpha(\varphi), \end{cases} \]

in the rectangle \(\alpha_0\leq\alpha<\alpha_0+\pi,\ 0\leq\varphi<2\pi\). \(\Phi_1(\varphi,\alpha)\) is continuous, and
\(\Phi_1(0,\alpha)=\Phi_1(2\pi,\alpha)+2\pi\),
\(\Phi_1(\varphi,\alpha_0)=\Phi_1(\varphi,\alpha_0+\pi)-2\pi\).
Therefore the function \(f_1(\varphi,\alpha)=\Phi_1(\varphi,\alpha)+2(n-k)\pi\) for \(\varphi=\bar\varphi+2k\pi\), \(\alpha=\bar\alpha+n\pi\), \(0\leq\bar\varphi<2\pi\), \(\alpha_0\leq\bar\alpha<\alpha_0+\pi\), and integers \(k\) and \(n\), is continuous for all \(\varphi\) and \(\alpha\). The function \(f_1(\varphi,\alpha)\) differs from the principal value of the polar angle of the point \(T(\varphi,\alpha)\) by an integer multiple of \(2\pi\) for fixed \(\varphi\) and \(\alpha\), and satisfies the functional equation

\[ x(f_1)-\operatorname{ctg}\alpha\,y(f_1)=x(\varphi)-\operatorname{ctg}\alpha\,y(\varphi). \tag{28} \]

Lemma 2. \(f_1(\varphi,\alpha)\) is a twice continuously differentiable function of its arguments; moreover
\[ \frac{\partial f_1}{\partial \alpha}>q_1>0,\qquad \frac{\partial f_1}{\partial \varphi}<-\bar q_1<0, \]
where \(q_1,\bar q_1\) are certain constants.

For simplicity we shall carry out the proof only for \(\dfrac{\partial f_1}{\partial \alpha}\). From (28) it is easy to obtain
\[ \frac{\partial f_1}{\partial \alpha} = \frac{[x(\varphi)-x(f_1)]^2+[y(\varphi)-y(f_1)]^2} {\dot x(f_1)[y(\varphi)-y(f_1)]-[x(\varphi)-x(f_1)]\dot y(f_1)} . \tag{29} \]
Since \(f_1\) is continuous, \(\dfrac{\partial f_1}{\partial \alpha}\) is undefined only at those points where the denominator vanishes, i.e., for those \(\varphi\) and \(\alpha\) for which the tangent to the contour at the point \(\varphi\) coincides with the straight line (28). At such a point, without loss of generality, one may take \(f_1(\varphi,\alpha)=\varphi\). But at these points the numerator also vanishes. It is easy to compute that
\[ \frac{\partial f_1}{\partial \alpha}\bigg|_{f_1\to\varphi} \to 2\,\frac{\dot x^2+\dot y^2}{\ddot x\,\dot y-\dot x\,\ddot y}>0 \]
by virtue of the positivity of the curvature. Since
\[ \frac{\partial f_1}{\partial \alpha}\ne0 \quad \text{for } f_1=\varphi+2k\pi, \]
our assertion is proved. The proof for the remaining derivatives is carried out analogously. We construct the functions
\[ f_n(\varphi,\alpha)=f_1\bigl(f_{n-1},(-1)^{n+1}\alpha\bigr), \]
putting
\[ f_0(\varphi,\alpha)=\varphi \]
for \(n=1,2,\ldots\).

Lemma 3. \(f_n(\varphi,\alpha)\) is a twice continuously differentiable function,
\[ \frac{\partial f_{2n}}{\partial \varphi}>\bar q_{2n}>0; \]
the function
\[ f_{2n}(\varphi,\alpha)-\varphi+4n\alpha \]
is periodic with period \(2\pi\) in \(\varphi\) and with period \(\pi\) in \(\alpha\). For the function \(f_{2n+1}(\varphi,\alpha)\) the inequalities
\[ \frac{\partial f_{2n+1}}{\partial \varphi}<-\bar q_{2n+1}<0,\qquad \frac{\partial f_{2n+1}}{\partial \alpha}>q_{2n+1}>q_1, \]
hold, where \(q_n,\bar q_n\) are certain constants.

The proof of this lemma is easily carried out by induction.

Theorem 2. For every \(N\) and every point \(\varphi\) on the boundary \(\Gamma\), one can indicate angles \(\alpha_N^i(\varphi)\) \((i=0,\ldots,2N-1)\) such that the polygonal line issuing from the point \(\varphi\) at the angle \(\alpha_N^i(\varphi)\) forms a cycle with \(2N\) links. The \(\alpha_N^i(\varphi)\) are twice continuously differentiable periodic functions of \(\varphi\). Any continuous function \(\lambda(\varphi)\) generating a cycle is, for some \(N,k\), equal to \(\operatorname{ctg}\alpha_N^k(\varphi)\), where \(k=0,\ldots,2N-1\).

Proof. All cycles are evidently determined by functions
\[ \lambda(\varphi)=\operatorname{ctg}\alpha(\varphi), \]
where
\[ f_{2N}(\varphi,\alpha)=\varphi+2k\pi . \tag{30} \]
Since \(f_{2N}\) is a monotone function of \(\alpha\) for fixed \(\varphi\), equation (30) always has a unique solution \(\alpha_N^k(\varphi)\). Since
\[ \frac{\partial f_{2N}}{\partial \alpha}\ne0, \]
the assertion concerning differentiability of \(\alpha_N^k(\varphi)\) is evident. Let us show that \(\alpha_N^k(\varphi+2\pi)=\alpha_N^k(\varphi)\). Indeed,
\[ f_{2N}\bigl(\varphi+2\pi,\alpha_N^k(\varphi+2\pi)\bigr) = \varphi+2\pi+2k\pi, \]
or, by virtue of Lemma 3,
\[ f_{2N}\bigl(\varphi,\alpha_N^k(\varphi+2\pi)\bigr)=\varphi+2k\pi, \]
whence it follows, by uniqueness of the solution of equation (30), that
\[ \alpha_N^k(\varphi+2\pi)=\alpha_N^k(\varphi). \]

Next, for \(k=2lN+i\), where \(0\leq i\leq 2N-1\),
\[ f_{2N}(\varphi,\alpha_N^i(\varphi)+l\pi) =4Nl\pi+f_{2N}(\varphi,\alpha_N^i(\varphi)) =4Nl\pi+2i\pi+\varphi=2k\pi+\varphi, \]
whence it follows that \(\alpha_N^k(\varphi)=\alpha_N^i(\varphi)+l\pi\). Thus, different \(\lambda_N^k(\varphi)=\operatorname{ctg}\alpha_N^k(\varphi)\) can be determined only for \(0\leq k\leq 2N-1\). Further, \(\alpha_N^i(\varphi)=\alpha_N^i(\varphi_0)\) if \(\varphi=f_1(\varphi_0,\alpha_N^i(\varphi_0))\). Indeed, this follows from the equality \(f_{2N}[f_1(\varphi,\alpha),-\alpha]=f_1[f_{2N}(\varphi,\alpha),\alpha]\). The lemma is proved.

Lemma 4. Among the functions \(\lambda_N^i(\varphi)\) that generate cycles, there is an even number of distinct ones. If \(\lambda_k^i(\varphi_0)=\lambda_N^j(\varphi_0)\), then \(\lambda_k^i=\lambda_N^j\). For
\[ \alpha_N^i\ne \frac{k\pi}{2} \]
the estimate \(\infty>C_N>|\lambda_N^i|>d_N>0\) holds, where \(C_N\) and \(d_N\) are constants.

Proof. Let \(N=rk\). Then
\[ f_{2N}(\varphi,\alpha_k^i(\varphi)) =f_{2(N-k)}(f_{2k},\alpha_k^i) =f_0(\varphi,\alpha_k^i(\varphi))+2\pi ri. \]
But \(f_{2N}(\varphi_0,\alpha_k^i(\varphi_0))=\varphi_0+2\pi j\), whence it follows that \(j=ri\), and by Theorem 2 \(\lambda_k^i(\varphi)=\lambda_N^j(\varphi)\).

The case \(N=r_1k_1,\ k=r_2k_1\) is considered analogously. If \(N\) and \(k\) are relatively prime and \(N>k\), then \(N=rk+r_1\) and \(f_{2kr}(\varphi_0,\alpha_0)=\varphi_0+2\pi ri\),
\[ f_{2N}(\varphi_0,\alpha_0)=f_{2r_1}(\varphi_0,\alpha_0)+2\pi ri=\varphi_0+2\pi j, \]
whence \(f_{2r_1}(\varphi_0,\alpha_0)=\varphi_0+2\pi l_1\).

Considering \(r_3\) and \(k\) and applying the preceding arguments several times, we finally obtain
\[ f_2(\varphi_0,\alpha_0)=\varphi_0+2l_2\pi, \]
where \(l_2\) is a natural number. But this is possible only if
\[ \alpha_0=\frac{M\pi}{2},\quad M=0,\pm1,\pm2, \]
since a cycle with two links is closed only when the lines forming it are parallel to one of the axes and coincide. In this case
\[ f_{2k}\left(\varphi,\frac{M\pi}{2}\right)=\varphi+2l_3\pi \]
and, by the continuity of \(f_{2k}\), \(l_3=i\) and
\[ \alpha_N^i(\varphi)\equiv \frac{M\pi}{2}. \]
The lemma is proved.

From the proof of one of the lemmas given in [5], it follows that the envelope of the family of straight lines
\[ x(\varphi)-\lambda_N^k(\varphi)y(\varphi)=x-\lambda_N^k(\varphi)y \]
for fixed \(k\) and \(N\) cannot enter the interior of the domain \(\Omega\). We shall need the following.

Lemma 5. If \(x,y\in \overline{\Omega}\), then the simultaneous fulfillment of the equalities
\[ x-\lambda_N^i(\varphi)y=x(\varphi)-\lambda_N^i(\varphi)y(\varphi), \tag{31} \]
\[ -\dot{\lambda}_N^i(\varphi)y=\bigl(x(\varphi)-\lambda_N^i(\varphi)y(\varphi)\bigr)', \tag{32} \]
is possible only at those boundary points where
\[ \lambda_N^i(\varphi)=\frac{\dot{x}(\varphi)}{\dot{y}(\varphi)},\qquad \dot{\lambda}_N^i(\varphi)=0. \tag{33} \]

The straight lines (31) corresponding to distinct points \(\varphi\) and \(\psi\) cannot intersect inside the domain.

Let
\[ f_{2N}(\varphi,\alpha_N^i(\varphi))=\varphi+2\pi i \]
and let
\[ \alpha_N^i(\varphi_1)=\alpha_1,\qquad \alpha_N^i(\varphi_2)=\alpha_2, \]
while the straight lines
\[ x(\varphi_i)-\operatorname{ctg}\alpha_i\,y(\varphi_i) =x-\operatorname{ctg}\alpha_i\,y\qquad (i=1,2) \]
intersect inside the domain. The points \(\varphi_1\) and \(\varphi_2\) may be assumed sufficiently close (by virtue of the continuity of \(\lambda_N^i(\varphi)\)), so that
\[ \left.\frac{d\alpha_N^i}{d\varphi}\right|_{\varphi=\varphi_1}\geq 0 \quad\text{and}\quad \left.\frac{df_1(\varphi,\alpha_N^i(\varphi))}{d\varphi}\right|_{\varphi=\varphi_1}\geq 0. \]

Then

\[ \frac{d f_2\bigl(\varphi,\alpha_N^i(\varphi)\bigr)}{d\varphi} = \frac{\partial f_1\bigl(f_1,-\alpha_N^i\bigr)}{\partial f_1}\, \frac{d f_1}{d\varphi} - \frac{\partial f_1\bigl(f_1,-\alpha_N^i\bigr)}{\partial(-\alpha_N^i)}\, \frac{d\alpha_N^i}{d\varphi} \]

and, by the lemmas and our assumption, \(\dfrac{d f_2}{d\varphi}<0\). It is easy, by induction, to prove that then \(\dfrac{d f_{2N}}{d\varphi}<0\), which contradicts the identity

\[ \frac{d f_{2N}\bigl(\varphi,\alpha_N^i(\varphi)\bigr)}{d\varphi}\equiv 1 \]

and shows that the straight lines inside \(\overline{\Omega}\) cannot intersect. Further, the points \(x,y\) satisfying (31), (32) lie on the envelope of the pencil of straight lines (31) and cannot lie inside the domain, since in that case straight lines passing through sufficiently close points of the envelope would intersect inside the domain. If, however, a point of the envelope lies on the boundary, then \(x=x(\psi)\), \(y=y(\psi)\), and then \(\lambda_N^i(\varphi)=\lambda_N^i(\psi)\), \(\psi=f_1\bigl(\varphi,\alpha_N^i(\varphi)\bigr)\), or \(\varphi=\psi\). In the first case we have

\[ \left. \begin{aligned} x(f_1)-\lambda_N^i(\varphi)y(f_1)&=x(\varphi)-\lambda_N^i(\varphi)y(\varphi),\\ -\dot{\lambda}_N^i(\varphi)y(f_1)&=\bigl(x(\varphi)-\lambda_N^i(\varphi)y(\varphi)\bigr)' \end{aligned} \right\}. \tag{34} \]

From these two equalities it is easy to conclude that either \(\dfrac{d f_1}{d\varphi}=0\), and then from the identity \(\lambda_N^i(f_1)\equiv \lambda_N^i(\varphi)\) we obtain \(\dot{\lambda}_N^i(\varphi)=0\), while from (34)

\[ \lambda_N^i=\frac{\dot{x}(\varphi)}{\dot{y}(\varphi)}, \]

or

\[ \frac{\dot{x}(f_1)}{\dot{y}(f_1)}=\lambda_N^i(f_1). \]

In the case \(\psi=\varphi\) we have \(\dot{x}(\varphi)-\lambda_N^i\dot{y}(\varphi)=0\). Now it remains only to verify that if \(\dot{x}(\varphi_0)-\lambda_N^i(\varphi_0)\dot{y}(\varphi_0)=0\), then \(\dot{\alpha}_N^i(\varphi_0)=0\). Indeed, in this case the straight line (31) is tangent,

\[ f_1\bigl(\varphi_0,\alpha_N^i(\varphi_0)\bigr)=\varphi_0+2\pi l \]

and

\[ \frac{\alpha_N^i(\varphi)-\alpha_N^i\bigl(\varphi+[f_1-\varphi-2\pi l]\bigr)} {f_1-\varphi+2\pi l}=0, \]

whence our assertion follows. The lemma is proved.

Lemma 6. For fixed \(k\) and \(n\), through every point \(x,y\in\overline{\Omega}\) there passes one and only one straight line of the pencil (31).

Proof. Put \(\lambda(\varphi)=\lambda_N^i(\varphi)\) for some \(N\) and \(i\). Let \(M\) be the set of those points of \(\Omega\) through which pass straight lines of the pencil (31). \(M\) is obviously closed. Let \(p\in\Omega\) and \(p\notin M\). Then any straight line \(R\) passing through \(p\) contains points \(p_1,p_2\in M\) such that, if \(p^*\in(p_1,p_2)\), then \(p^*\notin M\). Consider the straight lines \(\Pi_1(\varphi_1,\alpha(\varphi_1))\), \(\Pi_2(\varphi_2,\alpha(\varphi_2))\) from the pencil (31), passing respectively through \(p_1\) and \(p_2\). None of the straight lines of the pencil can join a point of \((\varphi_1,\varphi_2)\) with points of the arc \(\bigl(f_1(\varphi_1,\alpha(\varphi_1))=\psi_1,\ f_2(\varphi_2,\alpha(\varphi_2))=\psi_2\bigr)\); otherwise, between \(p_1\) and \(p_2\) on the segment of the straight line \(R\) there would lie a point of \(M\), while by Lemma 5 straight lines of the pencil inside \(\Omega\) cannot intersect. Thus, if \(\varphi\in(\varphi_1,\varphi_2)\), then \(f_1(\varphi,\alpha(\varphi))\in(\psi_1,\psi_2)\) and, consequently, \(p_i\in\Gamma\), while \(\Pi_i\) are tangent. Among all the straight lines under consideration there is at least one \(R^*\) such that the corresponding \(\Pi_1\) and \(\Pi_2\) are not tangent.

at the points \(\varphi_1\) and \(\varphi_2\), which leads us to a contradiction. Indeed, otherwise we would have

\[ \frac{\dot x(\varphi)}{\dot y(\varphi)} = \frac{\dot x\bigl(f_1(\varphi,\alpha(\varphi))\bigr)} {\dot y\bigl(f_1(\varphi,\alpha(\varphi))\bigr)} \tag{35} \]

for a set of points \(\varphi\) of positive measure. But then, as in Lemma 5,

\[ \frac{d\alpha}{d\varphi}=0 \]

on a set of points \(\varphi\) of positive measure. Differentiating (35) with respect to \(\varphi\), we obtain a contradiction with the assumption of positive curvature, since

\[ \frac{\partial f_1}{\partial \varphi}<0. \]

The lemma is proved.

Remark. Analogously to the case \(n=1\), one can prove that the equality

\[ \frac{\dot x(\varphi)}{\dot y(\varphi)} = (-1)^n\, \frac{\dot x\bigl(f_n(\varphi,\alpha(\varphi))\bigr)} {\dot y\bigl(f_n(\varphi,\alpha(\varphi))\bigr)} \tag{36} \]

cannot hold for any \(n\) (if \(\alpha(\varphi)\ne k\pi/2\), \(k\) an integer) on a set of points \(\varphi\) of positive measure.

Consider the set of convex domains \(\Omega_\varepsilon\) such that \(\Gamma_\varepsilon\), the boundary of \(\Omega_\varepsilon\), is given by the equation

\[ \rho=\rho(\varphi,\varepsilon), \tag{37} \]

and the condition

\[ \rho(\varphi,\varepsilon)^2+2\dot\rho(\varphi,\varepsilon)^2-\rho(\varphi,\varepsilon)\ddot\rho(\varphi,\varepsilon)\ge q>0 \]

is satisfied with one and the same constant \(q\). Let \(\rho(\varphi,\varepsilon)\to \rho(\varphi,0)\) as \(\varepsilon\to0\) in the metric of \(C_1(\Gamma)\).

Lemma 7. The functions \(a_N^i(\varphi,\varepsilon)\), constructed for the domains \(\Omega_\varepsilon\), are continuous with respect to \(\varepsilon\) at the point \(\varepsilon=0\).

For simplicity, assume that the tangents to \(\Gamma_\varepsilon\) at the point \(\varphi=0\) are all parallel. We shall prove that the function \(f_1(\varphi,\alpha,\varepsilon)\) is continuous with respect to \(\varepsilon\). We use the definition of the function \(f_1(\varphi,\alpha,\varepsilon)\). Consider the point \(T(\varphi,\alpha,\varepsilon)\). As \(\varepsilon\to0\), evidently \(T(\varphi,\alpha,\varepsilon)\to T(\varphi,\alpha,0)\). Therefore the acute angle \(\psi_\varepsilon\), formed by the rays issuing from the origin of coordinates to the points \(T(\varphi,\alpha,\varepsilon)\) and \(T(\varphi,\alpha,0)\), vanishes as \(\varepsilon\to0\), i.e. \(\psi_\varepsilon\to0\). Since the tangents at the point \(\varphi=0\) are, by assumption, parallel, the number \(\alpha_0\) used in constructing \(\Phi(\varphi,\alpha,\varepsilon)\) can be chosen the same for all \(\varepsilon\). In the general case \(\alpha(\varepsilon)\to\alpha_0\) as \(\varepsilon\to0\). Further,

\[ |\Phi(\varphi,\alpha,\varepsilon)-\Phi(\varphi,\alpha,0)|=\psi_\varepsilon \quad \text{a) if } \alpha>\alpha(\varphi,\varepsilon),\ \alpha>\alpha(\varphi,0), \]

\[ \text{b) if } \alpha<\alpha(\varphi,\varepsilon),\ \alpha<\alpha(\varphi,0); \]

\[ |\Phi(\varphi,\alpha,\varepsilon)-\Phi(\varphi,\alpha,0)+2\pi|=\psi_\varepsilon \quad \text{c) if } \alpha>\alpha(\varphi,\varepsilon),\ \alpha<\alpha(\varphi,0); \]

\[ |\Phi(\varphi,\alpha,\varepsilon)-\Phi(\varphi,\alpha,0)-2\pi|=\psi_\varepsilon \quad \text{d) if } \alpha<\alpha(\varphi,\varepsilon),\ \alpha>\alpha(\varphi,0); \]

for

\[ \alpha_0\le \alpha<\alpha_0+\pi,\qquad 0\le\varphi<2\pi. \]

In all cases, obviously, we have

\[ |\Phi_1(\varphi,\alpha,\varepsilon)-\Phi_1(\varphi,\alpha,0)|=\psi_\varepsilon, \]

but

\[ f_1(\varphi,\alpha,\varepsilon)-f_1(\varphi,\alpha,0) = \Phi_1(\varphi,\alpha,\varepsilon)-\Phi_1(\varphi,\alpha,0) \]

for \(0\le\varphi<2\pi\), \(\alpha_0\le\alpha<\alpha_0+\pi\), and the difference \(f_1(\varphi,\alpha,\varepsilon)-f_1(\varphi,\alpha,0)\), by virtue of the properties of the func-

ci \(f_1(\varphi,\alpha,\varepsilon)\) is a function periodic with period \(2\pi\) in \(\varphi\) and with period \(\pi\) in \(\alpha\). The lemma is proved.

Theorem 3. Any solution of problem (20), (2), whose derivatives with respect to \(x\) and \(y\) have the form (26), can be constructed by means of functions \(\lambda^i_N(\varphi)\). In the case
\[ \frac{d\lambda^i_N}{d\varphi}\ne 0 \]
such a solution has the form
\[ u(x,y,t)=\int_{\mu^1_{N,i}}^{\mu^2_{N,i}} f(\alpha)e^{il(\alpha)t}\,d\alpha, \tag{38} \]
where
\[ \mu^k_{N,i}\big|_{\Gamma}=\lambda^i_N(\varphi)=\operatorname{ctg} a^i_N(\varphi); \qquad \mu^k_{N,i}\frac{\partial \mu^k_{N,i}}{\partial x} =(-1)^k\frac{\partial \mu^k_{N,i}}{\partial y}; \tag{39} \]
whereas if
\[ \frac{d\lambda^i_N}{d\varphi}=0 \]
on a set of points \(\varphi\) of positive measure, then the solution is the function
\[ u(x,y,t)=\bigl[f\bigl(R_1(x+\lambda^i_N y)\bigr)-f\bigl(R_2(x-\lambda^i_N y)\bigr)\bigr]e^{il(\lambda^i_N)t} \tag{40} \]
for certain \(R_1\) and \(R_2\) and an arbitrary \(f\,(u\ne 0)\). The functions \(\mu^k_{N,i}(x,y)\) depend continuously on the domain, are continuously differentiable in \(\overline{\Omega}\) and twice continuously differentiable in \(\Omega\).

Theorem 3 is a consequence of the lemmas we have proved. The functions \(\mu^k_{N,i}\) are obtained by eliminating \(\mu\) from the system
\[ x+(-1)^k\mu^k_{N,i}y=x(\varphi)+(-1)^k\lambda^i_N(\varphi)y(\varphi), \tag{41} \]
\[ \mu^k_{N,i}=\lambda^i_N(\varphi). \tag{42} \]

By Lemma 6 the function \(\mu\) is determined uniquely at each point of \(\overline{\Omega}\). The differentiability of the function \(\mu^k_{N,i}\) follows from Lemma 5. The existence of the functions \(R_1\) and \(R_2\) is easy to prove by specifying, on a naturally chosen segment of the boundary where
\[ \frac{d\lambda_{N,i}}{d\varphi}\equiv 0, \]
a nonconstant function \(R(\varphi)\) and extending it to the remaining points of the boundary so that
\[ R(\varphi)=R\bigl(f_1(\varphi,\alpha^i_N(\varphi))\bigr). \]

The choice of this function is, obviously, not unique. The continuous dependence of \(\mu^k_{N,i}\) on the boundary follows from Lemma 7. In the case of equation (2), the asymptotics of solutions of the form (38) is easily determined: they decay as \(t\to +\infty\), their derivatives with respect to \(x,y\) are almost periodic for fixed \(x,y\), and the second derivatives with respect to \(x,y\) grow no faster than the first power of \(t\). In the case of an arbitrary equation of the form (20), the asymptotics is determined by the properties of the function \(l(\alpha)\).

Let us study in more detail the set of initial data for which solutions of the form (38) are constructed:
\[ u(x,y,0)=\int_{\mu^1_{N,i}}^{\mu^2_{N,i}} f(\alpha)\,d\alpha =F(\mu^2_{N,i})-F(\mu^1_{N,i}) =f_1(x-\mu^2_{N,i}y)-f_2(x+\mu^1_{N,i}y). \]

This equality follows in an obvious way from (41), (42).

Thus, the set of initial data for which we know how to construct solutions whose derivatives, for fixed \(x, y\), are almost-periodic in \(t\), has one and the same form in the case
\[ \frac{d\lambda_N^i}{d\varphi}\ne 0 \]
and in the case of constant \(\lambda_N^i\). Equation (1) can be rewritten in the form
\[ \frac{\partial^2 u}{\partial t^2} = -\Delta^{-1}\frac{\partial^2}{\partial y^2}u = Au, \tag{43} \]
where
\[ \Delta^{-1}\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)=u,\qquad u|_\Gamma=0. \]

Theorem 4. \(A\) is a self-adjoint bounded operator in the Hilbert space \(\overset{\circ}{W}{}^{1}_{2}(\Omega)\). The invariant subspaces \(H_{n,i}\) of this operator are formed by functions
\[ f(x+\mu^1_{n,i}y)+g(x-\mu^2_{n,i}y) \]
for all such \(f\) and \(g\) that \(f+g|_\Gamma=0\). Among the \(H_{n,i}\) there is a countable number of distinct ones. If \(H_{n,i}\ne H_{N,j}\), then \(H_{n,i}\perp H_{N,j}\).

The proof of this theorem is based on Lemmas 1–7 and is given in [18].

Let us construct the resolvent and the spectral function for \(A\) in
\[ H=\sum_{n,i}\oplus H_{n,i}. \]
The function
\[ u_{n,i}(x,y)=f(x+\mu^1_{n,i}y)+g(x-\mu^2_{n,i}y), \]
obviously, can be represented as a finite sum of terms with nonintersecting supports, each of which is either an eigenfunction of the operator \(A\) and has the form \(f(x+\lambda y)+g(x-\lambda y)\), where \(\lambda\) is constant, \(f+g|_\Gamma=0\), or has the form
\[ \int_{\mu^1_{N,i}}^{\mu^2_{N,i}} g(\alpha)\,d\alpha. \tag{44} \]

It is proved in [18] that functions of the form (44) are orthogonal to the eigenfunctions of the operator \(A\). Thus, in order to construct easily the resolvent and the spectral function, it is enough to give a representation for \(A\) on functions of the form (44). But
\[ \frac{\partial^2}{\partial t^2} \int_{\mu^1_{N,i}}^{\mu^2_{N,i}} f(\alpha)e^{\,i\sqrt{\frac{-\alpha^2}{1+\alpha^2}}\,t}\,d\alpha = A\int_{\mu^1_{N,i}}^{\mu^2_{N,i}} f(\alpha)e^{\,i\sqrt{\frac{-\alpha^2}{1+\alpha^2}}\,t}\,d\alpha, \]
whence, for \(t=0\), we have
\[ A\int_{\mu^1_{N,i}}^{\mu^2_{N,i}} f(\alpha)\,d\alpha = -\int_{\mu^1_{N,i}}^{\mu^2_{N,i}} f(\alpha)\frac{\alpha^2}{1+\alpha^2}\,d\alpha, \]
\[ R_\lambda\int_{\mu^1_{N,i}}^{\mu^2_{N,i}} f(\alpha)\,d\alpha = \int_{\mu^1_{N,i}}^{\mu^2_{N,i}} \frac{1}{\lambda-\dfrac{\alpha^2}{1+\alpha^2}}\,f(\alpha)\,d\alpha, \]

\[ E_\lambda \int_{\mu_{N,i}^{1}}^{\mu_{N,i}^{2}} f(\alpha)\,d\alpha = \int_{\mu_{N,i}^{1}}^{\mu_{N,i}^{2}} f(\alpha)\, \frac{ \left|\lambda-\dfrac{\alpha^2}{1+\alpha^2}\right| + \left(\lambda-\dfrac{\alpha^2}{1+\alpha^2}\right) }{ 2\left|\lambda-\dfrac{\alpha^2}{1+\alpha^2}\right| }\,d\alpha . \]

For the eigenfunctions of the operator \(A\), the representation of \(R_\lambda\) and \(E_\lambda\) is evident. Thus, the solutions constructed by us completely characterize the operator \(A\) in \(H\). The continuous dependence of \(\mu_{n,i}^{k}\) on the domain makes it possible to conclude that, for each element \(H_{n,i}(0)\), with sufficiently small \(\varepsilon\), there exists an element close to it in \(H_{n,i}(\varepsilon)\) (here the notation is analogous to that adopted in Lemma 7). The question of the coincidence of \(H\) with \(\overset{0}{W}{}_{2}^{1}(\Omega)\) has not been resolved.

§ 3. SOME REMARKS CONCERNING EQUATIONS GENERALIZING (1), AND EXAMPLES

We first construct an example of a domain for which there exist not almost periodic solutions of problem (1), (2).

Let
\[ \mu_1|_{\Gamma}=\mu_2|_{\Gamma}=\lambda(\varphi); \qquad \frac{d\lambda}{d\varphi}\ne 0,\quad (-1)^{i+1}\mu_i\frac{\partial\mu_i}{\partial x} = \frac{\partial\mu_i}{\partial y} \quad (i=1,2). \]

Introduce the variables
\[ \xi=x+\mu_1 y;\qquad \eta=x-\mu_2 y. \tag{45} \]

As was already shown in § 2, the straight lines
\[ x+\lambda(\varphi)y=x(\varphi)+\lambda(\varphi)y(\varphi), \]
\[ x+\lambda(\psi)y=x(\psi)+\lambda(\psi)y(\psi) \]
do not intersect inside the domain \(\Omega\) when \(\varphi\ne\psi\). Suppose that the relations (45) establish a one-to-one correspondence between the points of the domain \(\Omega\) and the points of some domain \(D\) with boundary \(\Gamma_1\) in the \(\xi,\eta\)-plane. Obviously,
\[ \mu_1(x,y)=\bar{\mu}_1(\xi),\quad \mu_2(x,y)=\bar{\mu}_2(\eta),\quad \left.\bar{\mu}_1(\xi)-\bar{\mu}_2(\eta)\right|_{\Gamma_1}=0. \]

Indeed,
\[ \frac{\partial\mu_1}{\partial\eta} = \frac{ -\mu_1 \begin{vmatrix} 1+\dfrac{\partial\mu_1}{\partial x}y & \dfrac{\partial\mu_1}{\partial x} \\[6pt] 1+\dfrac{\partial\mu_1}{\partial x}y & \dfrac{\partial\mu_1}{\partial x} \end{vmatrix} }{ \left(1+\dfrac{\partial\mu_1}{\partial x}y\right) \left(1-\dfrac{\partial\mu_2}{\partial x}y\right) (\mu_1+\mu_2) } =0. \]

We shall not prove that the Jacobian of the transformation is different from zero. In constructing the example this condition will be verified directly. We see that to each pair of functions \(\mu_1,\mu_2\) there corresponds a solution of the equation
\[ \frac{\partial^2 v}{\partial \xi\,\partial \eta}=0, \qquad v|_{\Gamma_1}=0. \]
Starting from such solutions, we shall construct nonconstant \(\mu_1,\mu_2\). Consider the function
\[ v_\varepsilon(\xi,\eta)=\varepsilon(\xi^2+\eta^2-1); \]
obviously,
\[ \frac{\partial^2 v_\varepsilon}{\partial \xi\,\partial \eta}=0, \qquad v_\varepsilon|_{\xi^2+\eta^2=1}=0. \]
Put
\[ \mu_1(\xi)=\varepsilon \xi^2+1;\qquad \mu_2(\eta)=\varepsilon(1-\eta^2)+1;\qquad v_\varepsilon=\mu_1-\mu_2; \]
\[ \mu_1(\xi)+\mu_2(\eta)\ne 0 \quad \text{for } \xi^2+\eta^2\le 1. \]
Introduce the variables \(x,y\):
\[ \xi=x+\mu_1(\xi)y;\qquad \eta=x-\mu_2(\eta)y. \]

For \(\xi=\cos\varphi,\ \eta=\sin\varphi\), i.e., on the boundary of the circle, we obtain

\[ x=\frac{\sin\varphi+\cos\varphi}{2},\qquad y=\frac{\cos\varphi-\sin\varphi}{2(1+\varepsilon\cos^2\varphi)}. \tag{46} \]

\(\varphi\) in the \(x,y\) plane is no longer, obviously, a polar angle. We define the functions \(\mu_i(x,y)\) from the equations

\[ x+\mu_1 y=\cos\varphi,\qquad x-\mu_2 y=\sin\varphi, \]

\[ \mu_1=\varepsilon\cos^2\varphi+1,\qquad \mu_2=1+\varepsilon(1-\sin^2\varphi). \]

Hence

\[ \mu_1(x,y)= \frac{1-2\varepsilon xy-\sqrt{1-4\varepsilon xy-4\varepsilon y^2}} {2\varepsilon y^2}; \]

\[ \mu_2= \frac{2\varepsilon xy-1+\sqrt{1-4\varepsilon yx+4\varepsilon(1+\varepsilon)y^2}} {2\varepsilon y^2}. \tag{47} \]

To prove that the \(\mu_i\) satisfy the conditions of Theorem 3 in \(\Omega\), it is enough, by virtue of their construction, to verify that they are defined at every interior point of the domain \(\Omega_\varepsilon\), whose boundary is given by equations (46). Since

\[ \left.1-4\varepsilon xy-4\varepsilon y^2\right|_{\Gamma_\varepsilon} = \frac{(1+\varepsilon\sin\varphi\cos\varphi)^2} {(1+\varepsilon\cos^2\varphi)^2}>0 \quad (|\varepsilon|<1), \]

the curve \(1-4\varepsilon xy-4\varepsilon y^2=0\), which for sufficiently large \(N\) contains the points

\[ \left(\frac{1-4\varepsilon N^2}{4\varepsilon N},\,N\right), \]

lies entirely outside \(\Omega_\varepsilon\). The differentiability of the \(\mu_i\) inside the domain and the proof that they satisfy equation (39) are verified directly. Thus, we have constructed an example of a nontrivial solution having the form (38). Obviously, as \(\varepsilon\to0\), \(\mu_i\to1\). \(\Gamma_\varepsilon\) depends continuously on \(\varepsilon\), and for \(\varepsilon=0\) we obtain a circle. Thus, in any neighborhood of the circle one can indicate a contour bounding a domain \(\Omega\) in which problem (1), (2) has solutions of the form (38). As we have already pointed out, an example of a domain, arbitrarily close to a disk, in which there exist solutions that are not almost periodic, was first constructed in [5].

A particular equation generalizing (1) was considered in Theorem 3 (equation (20)). Of special interest are the equations

\[ \sum \frac{\partial^i}{\partial t^i} \left( A_i u+b_i(x,y)\frac{\partial u}{\partial x} +c_i(x,y)\frac{\partial u}{\partial y} +d_i(x,y)u \right)=0 \tag{48} \]

(\(A_i\) are the same as in (20)), since equations of this kind include the equation

\[ \frac{\partial^2}{\partial t^2} \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} - n^2u \right) + \frac{\partial^2 u}{\partial y^2}=0, \tag{49} \]

which arises in the theory of small oscillations of a rotating fluid in the case when the domain is a cylinder with generators parallel to the \(z\)-axis. In [17], for equation (48), solutions of the form

\[ u(x,y,t)=\int_{\mu_1}^{\mu_2} f(x,y,\alpha)e^{\lambda(\alpha)t}\,d\alpha \tag{50} \]

were obtained.

However, these solutions, generally speaking, are not classical, and it is unclear whether almost-periodic smooth solutions can be constructed with the aid of such solutions. For individual equations such solutions can be constructed. Let us note, however, that the equations for \(f(x,y,\alpha)\) are very cumbersome, and the functions \(f\) defined in [19] were obtained under certain simplifying assumptions. Let us write out these equations. A function \(u\) having the form (50) satisfies equation (48) if \(f,\mu_i\) satisfy the following equation:

\[ \begin{aligned} Nu={}&\sum_{i=1}^{2}(-1)^i \Bigg\{ \left[ P\bigl(\lambda(\mu_i)\bigr)\frac{\partial^2\mu_i}{\partial x^2} + Q\bigl(\lambda(\mu_i)\bigr)\frac{\partial^2\mu_i}{\partial y^2} +\right.\\ &\left. +P'(\lambda)\lambda'(\mu_i)\left(\frac{\partial\mu_i}{\partial x}\right)^2 + Q'(\lambda)\lambda'(\mu_i)\left(\frac{\partial\mu_i}{\partial y}\right)^2 \right] f(x,y,\mu_i)e^{\lambda(\mu_i)t} +\\ &+ \left[ P\bigl(\lambda(\mu_i)\bigr)\left(\frac{\partial\mu_i}{\partial x}\right)^2 + Q\bigl(\lambda(\mu_i)\bigr)\left(\frac{\partial\mu_i}{\partial y}\right)^2 \right]\lambda'(\mu_i) f(x,y,\mu_i)t e^{\lambda(\mu_i)t} +\\ &+ \left[ P\bigl(\lambda(\mu_i)\bigr)\left(\frac{\partial\mu_i}{\partial x}\right)^2 + Q\bigl(\lambda(\mu_i)\bigr)\left(\frac{\partial\mu_i}{\partial y}\right)^2 \right] \frac{\partial f(x,y,\mu_i)}{\partial\mu_i}e^{\lambda(\mu_i)t} +\\ &+ \left[ 2\frac{\partial f}{\partial x}P(\lambda)\frac{\partial\mu_i}{\partial x} + 2\frac{\partial f}{\partial y}Q(\lambda)\frac{\partial\mu_i}{\partial y} + \frac{\partial\mu_i}{\partial x}B(\lambda)f +\right.\\ &\left. +\frac{\partial\mu_i}{\partial y}C(\lambda)f \right] e^{\lambda(\mu_i)t} \Bigg|_{\alpha=\mu_i} \Bigg\} + \int_{\mu_1}^{\mu_2} \left[ P(\lambda)\frac{\partial^2 f}{\partial x^2} + Q(\lambda)\frac{\partial^2 f}{\partial y^2} +\right.\\ &\left. + B(\lambda)\frac{\partial f}{\partial x} + C(\lambda)\frac{\partial f}{\partial y} + D(\lambda)f \right] e^{\lambda(\alpha)t}\,d\alpha =0. \end{aligned} \]

Here

\[ \sum_{0}^{n} b_i\lambda^i=B(\lambda),\qquad \sum_{0}^{n} C_i\lambda^i=C(\lambda),\qquad \sum_{0}^{n} d_i\lambda^i=D(\lambda),\qquad P(\lambda)=\sum_{0}^{n} a_{i,1}\lambda^i, \]

\[ Q(\lambda)=\sum_{0}^{n} a_{i,2}\lambda^i, \]

and \(\lambda(\alpha)\) is a solution of the equation \(P(\lambda)+\alpha^2Q(\lambda)=0\).

To satisfy the equation \(Nu=0\), it is sufficient to set

\[ 2\frac{\partial f}{\partial x}P(\lambda)\frac{\partial\mu_i}{\partial x} + 2\frac{\partial f}{\partial y}Q(\lambda)\frac{\partial\mu_i}{\partial y} + \frac{\partial\mu_i}{\partial x}B(\lambda)f + \frac{\partial\mu_i}{\partial y}C(\lambda)f \Bigg|_{\alpha=\mu_i} =0, \]

\[ P\bigl(\lambda(\alpha)\bigr)\frac{\partial^2 f}{\partial x^2} + Q\bigl(\lambda(\alpha)\bigr)\frac{\partial^2 f}{\partial y^2} + B\frac{\partial f}{\partial x} + C\frac{\partial f}{\partial y} + Df=0, \tag{51} \]

\[ \mu_i\left(\frac{\partial\mu_i}{\partial x}\right) = (-1)^{i+1}\frac{\partial\mu_i}{\partial y}. \tag{52} \]

It is easy to see that these equations are equivalent to the following:

\[ -\frac{\partial\mu_1}{\partial x} \left\{ -\alpha\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{B+\alpha C}{2Q\alpha}f \right\}_{\alpha=\mu_1} =0, \tag{53} \]

\[ -\frac{\partial\mu_2}{\partial x} \left\{ \alpha\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} - \frac{B-\alpha C}{2Q\alpha}f \right\}_{\alpha=\mu_2} =0. \tag{54} \]

To satisfy the boundary conditions it is sufficient to set \(\mu_1|_\Gamma=\mu_2|_\Gamma\); if we assume \(\dfrac{\partial \mu_i}{\partial x}\ne 0\), then equations (51), (53), (54), for \(\mu_i\) satisfying (52), are equivalent to the problem with data on the characteristics for equation (51)

\[ -\alpha \frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} +\frac{B+\alpha C}{2Q\alpha}\,f\bigg|_{x+\alpha y=\bar A_j(\alpha)}=0, \tag{55} \]

\[ \alpha \frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} -\frac{B-\alpha C}{2Q\alpha}\,f\bigg|_{x-\alpha y=\bar B_i(\alpha)}=0. \tag{56} \]

The functions \(\bar A_i,\bar B_i\) are obtained as follows. If \(\dfrac{\partial \mu_i}{\partial x}=0\) on a set of positive measure, then (53), (54) are satisfied automatically on this set. If, however, in some part \(\Omega\) the derivative \(\dfrac{\partial \mu_i}{\partial x}\ne 0\), then from (41), (42) it follows that \(x+\mu_1 y=\bar A_j(\mu_1)\). The domain \(\Omega\) can be divided into parts whose boundaries are straight lines from our pencil, in each of which either \(\dfrac{\partial \mu_i}{\partial x}=0\), or \(\dfrac{\partial \mu_i}{\partial x}=0\) only on the boundary. The index \(j\) denotes the number of such a strip. It is not difficult to prove the solvability and differentiability with respect to \(\alpha\) of the solutions of problem (55), (56) for equation (51). In particular, for equation (48), in the case where \(\Gamma\) is given by equations (46), and \(\mu_1,\mu_2\) are defined by formulas (47), we obtain the solution

\[ u=\int_{\mu_1}^{\mu_2} J_0\!\left(n\sqrt{-F_1F_2}\right)\varphi(\alpha)e^{\,i\sqrt{\frac{\alpha^2}{1+\alpha^2}}\,t}\,d\alpha, \]

where

\[ F_1(x,y,\alpha)= \begin{cases} x+\alpha y+\sqrt{\alpha-1}, & x-y>0,\\ x+\alpha y-\sqrt{\alpha-1}, & x+y<0; \end{cases} \]

\[ F_2(x,y,\alpha)= \begin{cases} x-\alpha y+\sqrt{2-\alpha}, & 2y-x>0,\\ x-\alpha y-\sqrt{2-\alpha}, & 2y-x<0. \end{cases} \]

In this case the constructed function \(f(x,y,\alpha)\) is discontinuous on the straight lines \(x=2y,\ y=-x\), which are characteristics of equations (39).

In [19] examples of equations are also constructed for which \(f(x,y,\alpha)\), under such a construction, turns out to be continuous. This is possible, for example, for the equation

\[ \frac{\partial^2}{\partial t^2}\left( \frac{\partial^2u}{\partial x^2} +\frac{2}{x}\frac{\partial u}{\partial x} +\frac{\partial^2u}{\partial y^2} \right) +\frac{\partial^2u}{\partial y^2}=0, \]

which is obtained from the equation

\[ \frac{\partial^2}{\partial t^2}\left( \frac{\partial^2u}{\partial x_1^2} +\frac{\partial^2u}{\partial x_2^2} +\frac{\partial^2u}{\partial x_3^2} +\frac{\partial^2u}{\partial x_4^2} \right) +\frac{\partial^2u}{\partial x_4^2}=0. \]

when considering axisymmetric solutions. In this case \(f(x,y,\alpha)=\dfrac{\varphi(\alpha)}{x}\). For an equation not reducible to (1),

\[ \frac{\partial^2}{\partial t^2}\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}-(x^2+y^2)u\right)+\frac{\partial^2 u}{\partial y^2}-x^2u=0 \]

for \(\mu_i\) determined by formulas (47), we have

\[ f(x,y,\alpha)=\varphi(\alpha)J_0\left(\frac{1}{2\alpha}\sqrt{[(x-\alpha y)^2-(\sqrt{2-\alpha})^2][(x+\alpha y)^2-(\sqrt{\alpha-1})^2]}\right). \]

These examples show that in some cases from equations (51), (53), (54) one can determine smooth solutions having the form (50). The results obtained for equation (20) can be transferred to certain systems. In particular, an obvious generalization is for systems of the form

\[ \frac{\partial^2}{\partial t^2}E\Delta u+Mu=0, \]

where \(E\) is the identity matrix; \(M\) is a symmetric matrix whose coefficients are second-order differential operators containing only second derivatives with respect to \(x\) and \(y\), and \(u\) is a vector-function. When considering equations of the form

\[ \frac{\partial^2}{\partial t^2}M_1u+\frac{\partial}{\partial t}M_2u+M_3u=0, \]

where \(M_1\) are second-order differential operators, the spectrum of the operators arising here may coincide with those values \(\lambda\) for which the equation

\[ \lambda^2M_1u+\lambda M_2u+M_3u=0 \]

is elliptic. In this case as well, the spectrum may fill an entire curve continuously [16].

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Received by the editors
September 7, 1965

Institute of Mathematics, Siberian Branch, Academy of Sciences of the USSR