Mathematics

N. N. VAKHANIYA

ON A BOUNDARY-VALUE PROBLEM WITH DATA ON THE ENTIRE BOUNDARY FOR A HYPERBOLIC SYSTEM EQUIVALENT TO THE EQUATION OF VIBRATION OF A STRING

(Presented by Academician S. L. Sobolev on 7 V 1957)

We consider the following problem, proposed by S. L. Sobolev: in the rectangle \(R:\ 0 \le x \le X,\ 0 \le t \le T\), with boundary \(\Gamma\) and ratio of side lengths \(T/X=\rho\), find a solution of the system

\[ \begin{aligned} \frac{\partial u_1}{\partial x} &= \frac{\partial u_2}{\partial t},\\ \frac{\partial u_1}{\partial t} &= \frac{\partial u_2}{\partial x} \end{aligned} \tag{1} \]

with the boundary condition

\[ a u_1\big|_{\Gamma}+b u_2\big|_{\Gamma}=f; \tag{2} \]

\(a\), \(b\), and \(f\) are prescribed on \(\Gamma\).

S. L. Sobolev, in the note \((^1)\), investigated this problem for a square \((\rho=1)\). S. L. Sobolev’s method can be extended to a rectangle with arbitrary rational \(\rho\). This is done in the first part of the present note.* In the second part the case of irrational \(\rho\) is considered.

1. \(\rho=m/n\), where \(m\) and \(n\) are relatively prime positive integers. Without loss of generality one may assume that \(T=m\), \(X=n\); \(a\), \(b\), and \(f\) are assumed to be continuous functions of the arc length \(s\), measured from the origin of the coordinates. Divide \(\Gamma\) into unit intervals \(I_1,I_2,\ldots,I_{2(m+n)}\), where \(I_k\) \((k=1,2,\ldots,2(m+n))\) is the interval \(k-1<s<k\). It is known \((^2)\) that the path of a light ray issuing from any point of any of the intervals \(I_k\) along one of the characteristic directions of system (1) (the lines \(x\pm t=\mathrm{const}\)), with elastic reflection at the boundary, is closed in the case of any rational \(\rho\). Further, one can prove that: a) into each interval \(I_k\) there falls one node (point of reflection) of the closed light path; thus the number of nodes is \(2(m+n)\); b) if the first node (the node in \(I_1\)) has the value of the parameter \(s\) equal to \(\xi\), then the \(k\)-th node has the value of \(s\) equal to \(k-1+\xi\) if \(k\) is odd, and \(k-\xi\) if \(k\) is even.

Taking these properties into account and repeating the arguments from \((^1)\), it is not difficult to show that the question of solvability of the problem posed reduces to the question of the existence of zeros in the interval \(0<\xi<1\) of the following determinant of order \(2(m+n)\):

\[ \Delta_{mn}(\xi)=(-1)^l\{\alpha_1\alpha_3\cdots \alpha_{2(m+n)-1}\beta_2\beta_4\cdots \beta_{2(m+n)} -\alpha_2\alpha_4\cdots \alpha_{2(m+n)}\beta_1\beta_3\cdots \beta_{2(m+n)-1}\}, \]

where

\[ \alpha_k=a(k-1+\xi)+b(k-1+\xi),\quad \beta_k=a(k-1+\xi)-b(k-1+\xi), \]

when \(k\) is odd \((k=1,3,\ldots,2(m+n)-1)\), and

\[ \alpha_k=a(k-\xi)+b(k-\xi), \]

* This case was also considered by Yu. Radvigin.

\(\beta_k=a(k-\xi)-b(k-\xi)\), when \(k\) is even \((k=2,4,\ldots,2(m+n))\); the parity of \(l\) depends on the parities of \(m\) and \(n\). \(\Delta_{11}(\xi)\) coincides with the determinant in \((1)\). If, instead of \(\Delta_{11}(\xi)\), one everywhere takes \(\Delta_{mn}(\xi)\), then all the conclusions from \((1)\) remain valid. In particular, the following theorem holds.

Theorem 1. If \(\Delta_{mn}(\xi)\ne 0\) for all \(\xi\) in the interval \(0<\xi<1\), then there exists a unique solution of the posed problem.

2. Let \(\rho\) be irrational. We shall assume that \(a^2-b^2\ne 0\) everywhere on \(\Gamma\). Then, obviously, we may suppose that everywhere on \(\Gamma\)

\[ a^2-b^2=1. \tag{3} \]

We first prove the uniqueness theorem.

Theorem 2. Let \(\rho\) be any irrational number; let \(u_1(x,t)\), \(u_2(x,t)\) be any (continuous) solution of system \((1)\). If \(f\equiv0\) on \(\Gamma\) and \(u_2(x,t)\) vanishes at some point \(M\) of the boundary \(\Gamma\), then \(u_1(x,t)\equiv0\) and \(u_2(x,t)\equiv0\) in \(R\).

Proof. The general solution of system \((1)\) has the form

\[ u_1(x,t)=\varphi(x+t)+\psi(x-t),\qquad u_2(x,t)=\varphi(x+t)-\psi(x-t), \]

where the function \(\varphi(\xi)\) is defined on the interval \(0\le \xi\le X+T\), and \(\psi(\xi)\) on the interval \(-T\le \xi\le X\). In terms of \(\varphi\) and \(\psi\), the boundary condition \((2)\) takes the form \((f\equiv0)\)

\[ (a+b)\varphi(x+t)\big|_{\Gamma}+(a-b)\psi(x-t)\big|_{\Gamma}=0. \tag{2'} \]

Draw through the point \(M\) one of the characteristics of system \((1)\), for example the line \(x+t=\mathrm{const}\). Denote by \(M_1\) the point of intersection of this line with \(\Gamma\). From \(M_1\) draw the characteristic \(x-t=\mathrm{const}\), and denote the point of intersection by \(M_2\). From \(M_2\) draw the characteristic \(x+t=\mathrm{const}\), denoting the point of intersection by \(M_3\). Drawing alternately the characteristics \(x-t=\mathrm{const}\) and \(x+t=\mathrm{const}\), and continuing this process indefinitely, we obtain a set of points \(M,M_1,M_2,\ldots\) everywhere dense\({}^{(2)}\) on \(\Gamma\). It is not difficult to see that, by virtue of \((2)\), \((2')\), and \((3)\), \(\varphi\) and \(\psi\) are zero on this set and, in view of continuity, are identically zero on \(\Gamma\). But this means that \(\varphi(\xi)\equiv0\) for all \(\xi\in[0,X+T]\) and \(\psi(\xi)\equiv0\) for all \(\xi\in[-T,X]\). Thus \(u_1(x,t)\equiv0\) and \(u_2(x,t)\equiv0\).

Remark. Taking \(a=1\), \(b=0\), we obtain the uniqueness theorem for the Dirichlet problem for the equation of vibrations of a string. Obviously, in this case the requirement that the function \(u_2\) vanish at a boundary point becomes superfluous. This requirement also becomes superfluous in some other cases, for example, in the case when \(a^2-b^2=0\) on any finite nonempty set of points on \(\Gamma\), but \(a^2+b^2\ne0\) on \(\Gamma\).

Passing to the question of existence of a solution, suppose\(^*\) that the irrational number \(\rho\) is not too rapidly approximated by rational numbers, more precisely

\[ \left|\rho-\frac{m}{n}\right|>\frac{A}{n^{K+1}}, \tag{4} \]

where \(A\) is a fixed positive constant; \(K\) is a fixed positive integer; \(m\) and \(n\) are arbitrary integers. We prove an existence theorem.

Theorem 3. Suppose: a) \(\rho\) satisfies inequality \((4)\) for some \(K\); \(a\), \(b\) satisfy condition \((3)\); b) on each side of the rectangle \(R\), \(f\in C^{K+4}\), \(\ln|a+b|\in C^{2K+6}\), and these functions and their even derivatives up to orders respectively \(2\left[\dfrac{K+4}{2}\right]\) and \(2K+6\) are continuous

\(^*\) The set of numbers \(\rho\) which, for an arbitrarily fixed \(K\), do not satisfy this requirement has Lebesgue measure zero. In particular, algebraic numbers of degree \(K+1\) satisfy this assumption.

at the four corner points of the rectangle. Under these assumptions there exists a twice continuously differentiable solution of the system (1), satisfying the boundary condition (2).

Proof. For brevity, let us call a pair of functions satisfying the system (1) hyperbolically conjugate functions. It is easy to verify the validity of the following elementary properties of hyperbolically conjugate functions: a) each member of a pair of hyperbolically conjugate functions is determined by the other up to an arbitrary constant summand; b) inside the rectangle \(R\) both hyperbolically conjugate functions have the same degree of smoothness, and if one of them is twice differentiable, then each of them satisfies the equation of vibration of a string; c) if \(v_1, v_2\) and \(w_1, w_2\) are two pairs of hyperbolically conjugate functions, then \(v_1 w_1+v_2 w_2\) and \(v_1 w_2+v_2 w_1\) are also hyperbolically conjugate functions. It is not difficult to see that these properties of hyperbolically conjugate functions make it possible to reduce the boundary-value problem (2) for the system (1) to the Dirichlet problem* for the equation of vibration of a string in the special particular case when the functions \(a\) and \(b\) are boundary values on \(\Gamma\) of functions hyperbolically conjugate in the domain \(R\). But it is clear that the problem of such a continuous continuation of the functions \(a\) and \(b\) into the interior of the rectangle is overdetermined and, generally speaking, has no solution. However, by multiplying the boundary condition (2) by a suitably chosen function, one can ensure that the new coefficients at \(u_1|_\Gamma\) and \(u_2|_\Gamma\) in relation (2) already have the required special property. To find such a “linking” function, the solution of the Dirichlet problem for the equation of vibration of a string will be needed. We give the exact proof.

Let \(\Theta(x,t)\) be the solution of the Dirichlet problem for the equation of vibration of a string with boundary condition
\[ \Theta|_\Gamma=\operatorname{ar\,th}\frac{b}{a}=\ln |a+b|. \]
Taking into account the result concerning the conditions for existence of a solution of the Dirichlet problem for the equation of vibration of a string \((^2)\), we may assert that, under our assumptions,** the function \(\Theta(x,t)\) does indeed exist and belongs to \(C^{K+4}\). Denote by \(\Lambda(x,t)\) the function hyperbolically conjugate to \(\Theta(x,t)\), and by \(\lambda\) the boundary value on \(\Gamma\) of the function \(\Lambda(x,t)\). It can be shown that on each side of the rectangle \(R\), \(\lambda\in C^{K+4}\) and, moreover, \(\lambda\) itself and all its even derivatives up to order
\[ 2\left[\frac{K+4}{2}\right] \]
are continuous at the four corner points of the rectangle \(R\). By assumption, the function \(f\) also has this property, so that the function \(e^\lambda f\) will likewise have this property. Therefore there exists a twice differentiable function \(F_1(x,t)\) which is a solution of the Dirichlet problem for the equation of vibration of a string with boundary function
\[ F_1|_\Gamma=e^\lambda f. \]
It is easy to see that, in view of condition (3), \(\operatorname{ch}\Theta|_\Gamma=a\) and \(\operatorname{sh}\Theta|_\Gamma=b\). Therefore, if \(u_1\) and \(u_2\) satisfy the system (1) and the boundary condition (2), then the function
\[ e^\Lambda \operatorname{ch}\Theta\, u_1+e^\Lambda \operatorname{sh}\Theta\, u_2, \]
which, as can be verified directly, satisfies the equation of vibration of a string, also assumes on \(\Gamma\) the boundary function \(e^\lambda f\). Hence, by the uniqueness of the solution of the Dirichlet problem for the equation of vibration of a string, we have
\[ e^\Lambda \operatorname{ch}\Theta\, u_1+e^\Lambda \operatorname{sh}\Theta\, u_2=F_1(x,t). \]

* The idea of reducing to the Dirichlet problem is borrowed from the theory of boundary-value problems for analytic functions \((^3)\).

** In the work \((^4)\) it is additionally required that at the four corner points of \(R\) the boundary function and all its even derivatives up to the corresponding order vanish. However, one can dispense with this restriction by adding to the sought solution a completely definite particular solution which, as is easily shown, can always be constructed.

Further, one can verify directly that the function \(e^\Lambda \operatorname{sh}\Theta\, u_1 + e^\Lambda \operatorname{ch}\Theta\, u_2\) is the hyperbolically conjugate function to \(F_1(x,t)\) (thus, the role of the “linking” function is played by the function \(e^\lambda\)). Thus, denoting by \(F_2(x,t)\) the function conjugate to \(F_1(x,t)\), we have

\[ e^\Lambda \operatorname{sh}\Theta\, u_1 + e^\Lambda \operatorname{ch}\Theta\, u_2 = F_2(x,t) + B, \]

where \(B\) is an arbitrary constant. From the last two equalities we find the solution \(u_1(x,t), u_2(x,t)\)

\[ u_1(x,t)=e^{-\Lambda}\{F_1\operatorname{ch}\Theta-F_2\operatorname{sh}\Theta-B\operatorname{sh}\Theta\}, \]

\[ u_2(x,t)=e^{-\Lambda}\{F_2\operatorname{ch}\Theta-F_1\operatorname{sh}\Theta+B\operatorname{ch}\Theta\}. \]

In accordance with the uniqueness theorem, the solution is determined up to an arbitrary constant \(B\), which can be found by prescribing \(u_2\) at some point of the boundary \((\operatorname{ch}\Theta)|_\Gamma=a\ne0\) everywhere by virtue of (3)).

Remark. If on each side of the rectangle the functions \(f\) and \(\ln |a+b|\) are trigonometric polynomials, then the problem is solvable for any irrational \(\rho\).

Remark. It is not difficult to see that the solution \(u_1,u_2\) depends continuously on the functions \(f\) and \(\ln |a+b|\) in the sense of the uniform metric, respectively of type \((K+4,2)\) and \((2K+6,2)\).

Tbilisi Mathematical Institute named after A. M. Razmadze
Academy of Sciences of the Georgian SSR

Received
6 V 1957

REFERENCES

1. S. L. Sobolev, DAN, 109, No. 4 (1956).
2. G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, Oxford, 1954, p. 376.
3. I. N. Vekua, Tr. Tbilissk. matem. inst., 11 (1942).
4. D. G. Bourgin, R. Duffin, Bull. Math. Soc., 45, No. 12 (1939).