Full Text
T. I. Zelenyak
ON A PROBLEM OF S. L. SOBOLEV
(Presented by Academician S. L. Sobolev on 25 VI 1962)
Consider the equation:
\[ \sum_{i=0}^{N}\frac{\partial^i}{\partial t^i} \left( a_{i,1}\frac{\partial^2 u}{\partial x^2} + a_{i,2}\frac{\partial^2 u}{\partial y^2} + b_i(x,y)\frac{\partial u}{\partial x} + c_i(x,y)\frac{\partial u}{\partial y} + d_i(x,y)u \right)=0, \tag{1} \]
where \(a_{i,1}a_{i,2}\) are constant real numbers; \(b_i,c_i,d_i\) are continuous functions of their arguments in a bounded, closed domain \(\Omega\) with thrice differentiable boundary \(\Gamma\), whose curvature is positive. In \((^2)\) the question of the almost-periodicity of solutions of the equation
\[ \frac{\partial^2}{\partial t^2} \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) + \frac{\partial^2 u}{\partial y^2} =0 \tag{2} \]
was studied under the condition
\[ u\big|_{\Gamma}=0. \tag{3} \]
This problem is connected with the study of the spectrum of the singular integral operator \(A\):
\[ A=\Delta^{-1}\frac{\partial^2}{\partial y^2} \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right)\Delta^{-1}u=u, \qquad \Delta^{-1}u\big|_{\Gamma}=0. \]
In \((^2)\) the dependence of the character of the spectrum \(A\) on the topological properties of \(\Gamma\) is shown. In \((^3)\) a certain class of solutions of equation (2), satisfying condition (3), is constructed. Using their properties, in \((^4)\) invariant subspaces of the operator \(A\) are constructed, in which a simple representation of this operator is given. Using this representation, it is easy to construct the spectral function and the resolvent for \(A\) in a certain infinite-dimensional Hilbert space \(H \subset \overset{0}{W}{}^{1}_{2}(\Omega)\). We shall construct solutions of equation (1), satisfying condition (3) and being a generalization of the solutions constructed in \((^3)\). These solutions are not identically zero if the operator \(A\), constructed for the domain \(\Omega\), does not have a purely point spectrum.
Let
\[ (-1)^{i+1}\mu_i\frac{\partial \mu_i}{\partial x} = \frac{\partial \mu_i}{\partial y}, \qquad \mu_1\big|_{\Gamma}=\mu_2\big|_{\Gamma}=\rho(\varphi). \tag{4} \]
The existence of such functions was established in \((^3)\).
Introduce the notation:
\[ P(\lambda)=\sum a_{i,1}\lambda^i, \qquad Q(\lambda)=\sum a_{i,2}\lambda^i, \qquad \sum b_i(x,y)\lambda^i=B(\lambda), \]
\[ \sum c_i\lambda^i=C(\lambda), \qquad \sum d_i\lambda^i=D(\lambda). \tag{5} \]
Let \(\lambda(\alpha)\) be a solution of the equation
\[ P(\lambda)+\alpha^2Q(\lambda)=0. \tag{6} \]
We shall construct solutions of equation (1) of the form
\[ u(x,y,t)=\int_{\mu_1}^{\mu_2} f(x,y,\alpha)e^{\lambda(\alpha)t}\,d\alpha . \tag{7} \]
The functions (7), obviously, satisfy condition (3). Using (4) and the equalities obtained from (4) by differentiation, for \(f(x,y,\alpha)\) one may obtain the equations:
\[ \frac{\partial\mu_2}{\partial x} \left\{ \alpha\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} -\frac{B-\alpha C}{2Q(\lambda)\alpha}f \right\}_{\alpha=\mu_2}=0; \tag{8} \]
\[ \frac{\partial\mu_1}{\partial x} \left\{ -\alpha\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} +\frac{B+\alpha C}{2Q(\lambda)\alpha}f \right\}_{\alpha=\mu_1}=0; \tag{9} \]
\[ -\alpha^2\frac{\partial^2 f}{\partial x^2} +\frac{\partial^2 f}{\partial y^2} +\frac{B}{Q}\frac{\partial f}{\partial x} +\frac{C}{Q}\frac{\partial f}{\partial y} +\frac{D}{Q}f=0. \tag{10} \]
If it is assumed that \(Q(\lambda(\alpha))\ne 0\), i.e., that \(P,Q\) have no common zeros, then the coefficients of equations (8), (9), (10) will be continuous functions of \(x\) and \(y\). As in (4), the domain \(\Omega\) can be divided into domains \(\Pi_{i,j}\) such that in each of them, for the functions
\[ F_i(x,y,\alpha)=x+\alpha y-A_i(\alpha),\qquad R_j(x,y,\alpha)=x-\alpha y-B_j(\alpha) \tag{11} \]
the identities are fulfilled
\[ F_i(x,y,\mu_1)\equiv 0,\qquad R_j(x,y,\mu_2)\equiv 0. \tag{12} \]
Among the \(\Pi_{i,j}\) there may be domains in which \(\partial\mu_1/\partial x\equiv 0\), or \(\partial\mu_2/\partial x\equiv 0\), or both of these identities hold simultaneously. Then the corresponding identities (12) do not occur, but equations (8), (9) are fulfilled by virtue of \(\partial\mu_i/\partial x\equiv 0\).
Thus, in order to satisfy equations (8), (9), (10), it suffices to find a solution \(f(x,y,\alpha)\) of equation (10) such that for \(x,y\in\Pi_{i,j}\) the conditions
\[ \left. \alpha\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} -\frac{B-\alpha C}{2Q(\lambda)\alpha}f \right|_{x-\alpha y-B_j(\alpha)=0}=0; \tag{13} \]
\[ \left. -\alpha\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} +\frac{B+\alpha C}{2Q(\lambda)\alpha}f \right|_{x+\alpha y-A_i(\alpha)=1}=0. \tag{14} \]
The existence of a solution of problem (13), (14) for equation (10), as well as its differentiability with respect to the parameter \(\alpha\), is easily proved by using the method of successive approximations. Multiplication of solutions of problem (13), (14) for equation (10) by a function depending only on \(\alpha\) again leads to a solution of this problem. Thus we have constructed solutions of equation (1), under condition (3), having the form (7). The first derivatives with respect to \(x\) and \(y\) of the functions (7) differ from functions almost periodic in \(t\), for fixed \(x,y\), by a term vanishing as \(t\to\infty\), if \(\operatorname{Re}\lambda(\alpha)\equiv 0\).
In the general case the functions (7) have discontinuities on the characteristics of equation (4) that are the boundaries of the domains \(\Pi_{i,j}\). In some cases \(f(x,y,\alpha)\) are continuous. Let us give a few examples. Let \(\Omega\) be a domain whose boundary equation \(\Gamma\) has the form
\[ x=\frac{\sin\varphi+\cos\varphi}{2},\qquad y=\frac{\cos\varphi-\sin\varphi}{2(1+\cos^2\varphi)}. \tag{15} \]
Then the functions
\[ \mu_1=\frac{1-2xy-\sqrt{1-4xy-4y^2}}{2y^2},\qquad \mu_2=\frac{|2xy-1+\sqrt{1-4xy+8y^2}|}{2y^2} \tag{16} \]
are defined in \(\Omega\), satisfy equations (4), and
\[ \mu_1|_{\Gamma}=\mu_2|_{\Gamma}=1+\cos^2\varphi . \]
Example 1. For the equation
\[ \frac{\partial^2}{\partial t^2}\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}-n^2u\right)+\frac{\partial^2u}{\partial y^2}=0, \tag{17} \]
\[ f(x,y,\alpha)=J_0\left(n\sqrt{-F_1F_2}\right), \]
where \(J_0\) is the Bessel function,
\[ F_1(x,y,\alpha)= \begin{cases} x+\alpha y+\sqrt{\alpha-1}, & x+y<0,\\ x+\alpha y-\sqrt{\alpha-1}, & x+y>0; \end{cases} \]
\[ F_2(x,y,\alpha)= \begin{cases} x-\alpha y+\sqrt{2-\alpha}, & 2y-x>0,\\ x-\alpha y-\sqrt{2-\alpha}, & 2y-x<0; \end{cases} \]
\(f(x,y,\alpha)\) is discontinuous on the straight lines \(x=2y,\ y=-x\).
Equation (17) is obtained from the equation considered in (1),
\[ \frac{\partial^2}{\partial t^2}\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right)+\frac{\partial^2u}{\partial y^2}=0, \tag{18} \]
and the solution of equation (18), vanishing on the boundary of a certain cylinder in the space \(x,y,z\), has the form:
\[ u(x,y,z,t)=\sin nz\int_{\mu_1}^{\mu_2}J_0\left(n\sqrt{-F_1F_2}\right)f(\alpha)\exp\left[\pm i\sqrt{\frac{\alpha^2}{1+\alpha^2}}\,t\right]\,d\alpha. \]
Example 2. The equation
\[ \frac{\partial^2}{\partial t^2}\left(\frac{\partial^2u}{\partial x_1^2}+\frac{\partial^2u}{\partial x_2^2}+\frac{\partial^2u}{\partial x_3^2}+\frac{\partial^2u}{\partial x_4^2}\right)+\frac{\partial^2u}{\partial x_4^2}=0 \tag{19} \]
admits solutions
\[ u=v(x,y,t);\qquad x=\sqrt{x_1^2+x_2^2+x_3^2};\qquad x_4=y; \]
\[ \frac{\partial^2}{\partial t^2}\left(\frac{\partial^2v}{\partial x^2}+\frac{2}{x}\frac{\partial v}{\partial x}+\frac{\partial^2v}{\partial y^2}\right)+\frac{\partial^2v}{\partial y^2}=0. \tag{20} \]
In the case of equation (20) we have:
\[ f(x,y,\alpha)=\frac{g(\alpha)}{x}, \]
and one can obtain a solution of equation (19), vanishing on the boundary of a certain torus in the space \(x_1,x_2,x_3,x_4\) and not continuously differentiable.
Example 3. For the equation
\[ \frac{\partial^2}{\partial t^2}\left[\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}-(x^2+y^2)u\right]+\frac{\partial^2u}{\partial y^2}-x^2u=0 \tag{21} \]
we have
\[ F_1(x,y,\alpha)=(x+\alpha y)^2-(\alpha-1), \]
\[ F_2(x,y,\alpha)=(x-\alpha y)^2-(2-\alpha), \]
\[ f(x,y,\alpha)=J_0\left(\frac{1}{2\alpha}\sqrt{F_1F_2}\right). \tag{22} \]
The solution of equation (21), constructed with the aid of the function (22), is continuously differentiable in \(\Omega\).
In conclusion we note that, by separation of variables, equation (18) can be reduced to equations having the form (1) in the case where \(\Omega\) is a domain of revolution about the \(y\)-axis in the space \(x,y,z\).
Institute of Mathematics with Computing Center
of the Siberian Branch of the Academy of Sciences of the USSR
Received
19 VI 1962
REFERENCES
- S. L. Sobolev, Izv. AN SSSR, ser. matem., 18, No. 1, 3 (1954).
- R. A. Aleksandryan, Dissertation, Moscow State University, 1949.
- T. I. Zelenyak, DAN, 139, No. 3 (1961).
- T. I. Zelenyak, Sibirsk. matem. zhurn., 3, No. 3 (1962).