Abstract Generated abstract
This paper studies discontinuities of the Green function for a mixed Dirichlet problem for the two-dimensional wave equation and relates them to short-wave diffraction asymptotics for the Helmholtz Green function. Using a double-layer potential representation, the authors develop an iterative decomposition of the boundary density and show that, under an illumination condition on the source point, the remainder can be made smooth to any prescribed finite order on a finite time interval. The resulting construction gives an effective description of the singularities: they occur at the direct travel time and at travel times along extremal reflected polygons satisfying the equal-angle reflection condition at the boundary. The method is indicated to extend to higher dimensions, more general boundary conditions, Maxwell equations, and diffraction by thin screens.
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MATHEMATICS
A. Ya. POVZNER and I. V. SUKHAREVSKII
ON DISCONTINUITIES OF THE GREEN FUNCTION OF A MIXED PROBLEM FOR THE WAVE EQUATION AND ON SOME DIFFRACTION PROBLEMS
(Presented by Academician S. L. Sobolev on 5 VI 1958)
Consider the following mixed problem. In a two-dimensional domain \(D\), bounded by a simple infinitely differentiable contour \(S\) (closed or else going off to infinity), it is required to find a solution of the equation
\[ \Delta u = u_{tt}, \]
satisfying the conditions*
\[ u(0,x)=0;\qquad u_t(0,x)=f(x);\qquad u|_S=0. \]
The solution of this problem can be represented in the form
\[ u(t,x)=u_0(t,x)+\int_D w(t,x,y) f(y)\,d\omega_y, \]
where \(u_0(t,x)\) is the solution of the Cauchy problem in the whole space with the same initial conditions (\(f(x)\) being continued by zero outside \(D\)). We shall be interested in the points \(t=t_k(x,y)\) of discontinuity of the function \(w\) and of its derivatives with respect to \(t\), and in the magnitudes of the corresponding jumps.
If one introduces the Green function
\[ v(x,a,k)=\frac{1}{2\pi}H_0^{(1)}(k|x-a|)+\gamma(x,a,k) \]
of the Dirichlet problem for the equation
\[ \Delta v+k^2v=0 \]
in the domain \(D\), then it is easy to establish that
\[ \gamma(x,a,k)=\int_0^{+\infty} e^{ikt} w(t,x,a)\,dt. \]
This points to a close formal connection between our problem and the fundamental problem of the theory of diffraction for short waves—the problem of finding the asymptotics of the function \(\gamma(x,a,k)\) for large \(k\): integration by parts in the integral representing \(\gamma\) expresses the desired asymptotics through the dis-
* \(x,y\) are the radius vectors of points in \(\overline{D}\); \(a\) is a point of \(D\) fixed in what follows. If \(s\in S\), then \(\xi\) is the arc coordinate of the point \(s=s(\xi)\).
discontinuities of the function \(w\). If we seek the function \(\gamma\) in the form of a double-layer potential
\[ \gamma(x,a,k)=\frac{1}{2\pi}\int_S g(s,k)\frac{\partial}{\partial n_s}H_0^{(1)}(k|x-s|)\,d\xi, \tag{1} \]
then we obtain for \(g(s,k)\) the integral equation
\[ g(x,k)-\int_S h(x,s,k)g(s,k)\,ds=h_0(x,a,k)\qquad (x\in S), \tag{2} \]
where
\[ h_0(x,a,k)=-\frac{1}{\pi}\int_0^\infty \frac{e^{ikt}\sigma(t-|x-a|)}{\sqrt{t^2-|x-a|^2}}\,dt; \]
\[ h(x,s,k)=\frac{ik}{\pi}\frac{\partial}{\partial n_s}\ln|x-s|\cdot \int_0^\infty \frac{e^{ikt}\,t\sigma(t-|x-s|)}{\sqrt{t^2-|x-s|^2}}\,dt \]
\[ (\sigma(t)=1\ \text{for }t>0;\ \sigma(t)=0\ \text{for }t<0;\ n_s\text{ is the unit vector of the exterior normal},\ \operatorname{Im}k>0). \]
Put
\[ g_0(x,k)=h_0(x,a,k), \]
\[ g_{\mu+1}(x,k)=\int_S h(x,s,k)g_\mu(s,k)\,ds \qquad (\mu=0,1,2,\ldots), \]
\[ \sum_{\mu=0}^{n}g_\mu(x,k)=g_n^{(1)}(x,k). \]
Represent \(g\) in the form
\[ g(x,k)=g_n^{(1)}(x,k)+g_n^{(2)}(x,k). \]
Then \(g_n^{(2)}\) satisfies the equation
\[ g_n^{(2)}(x,k)-\int_S h(x,s,k)g_n^{(2)}(s,k)\,ds = g_{n+1}(x,k), \tag{2′} \]
and the function
\[ \gamma(x,a,k)=\gamma_n^{(1)}(x,a,k)+\gamma_n^{(2)}(x,a,k), \]
where \(\gamma_n^{(j)}\) is a potential of type (1) with density \(g_n^{(j)}\). Further, representing \(\gamma_n^{(j)}\) in the form
\[ \gamma_n^{(j)}=\int_0^\infty e^{ikt}w_n^{(j)}(t,x,a)\,dt, \]
we obtain:
\[ w(t,x,a)=w_n^{(1)}(t,x,a)+w_n^{(2)}(t,x,a). \]
The principal result of the present note is the following theorem:
Theorem 1. If the point \(a\) is such that from it one cannot draw a single tangent to \(S\)* (the contour \(S\) is completely “illuminated” from the point \(a\)), then for any prescribed \(T>0\) and natural number \(l\) there exists such an \(n\) that
* This property is possessed, for example, by all interior points of convex domains.
\(w_n^{(2)}(t,x,a)\) will be continuous, together with all mixed derivatives with respect to \(t\) and the coordinates of the vector \(x\), up to order \(l\) inclusive in the domain \((0\le t\le T;\ x\in \overline D)\), and
\[ \frac{\partial^p w_n^{(2)}(0,x,a)}{\partial t^p}=0 \qquad (p=0,1,2,\ldots,l). \]
Taking into account that \(w_n^{(1)}\) is computed directly from \(g_n^{(1)}\), we thereby obtain an effective method for finding the discontinuities of the function \(w\).
Let us represent \(g_{n+1}(s,k)\) in the form
\[ g_{n+1}(s,k)=\int_0^\infty e^{ikt}w_{n+1}(t,s,a)\,dt \qquad (s\in S). \]
Suppose that, for sufficiently large \(n\), the function \(w_{n+1}(t,s,a)\) has, in the given interval \([0;T]\), an arbitrarily large number of mixed derivatives with respect to \(t\) and \(\xi\) equal to zero for \(t=0\). Since the function \(w_n^{(2)}\) is the solution of the mixed problem
\[ \Delta u=u_{tt};\qquad u(0,x)=u_t(0,x)=0;\qquad u(t,s)=w_{n+1}(t,s,a) \qquad (s\in S), \]
it follows, by virtue of a known existence theorem, that the function \(w_n^{(2)}(t,x,a)\), for sufficiently large \(n\), has the properties indicated in Theorem 1. In the present paper, in fact, it is proved that the above assumption concerning \(w_{n+1}\) does indeed hold.
From the explicit representation of the function \(w_n^{(1)}\), in particular, it follows:
Theorem 2. The discontinuities of the function \(w(t,x,a)\) and of its derivatives with respect to \(t\) are located at the point \(t=|x-a|\) and at the points
\[ t=|x-\bar s_1|+|\bar s_1-\bar s_2|+\cdots+|\bar s_m-a| \qquad (m=1,2,\ldots), \]
where \(\bar s_i\) are all points of the contour \(S\) having the property that adjacent sides of the polygons with endpoints \(x\) and \(a\) and with vertices at \(\bar s_1,\ldots,\bar s_m\) make equal angles with the normal to \(S\) at the common vertex.*
The proof of Theorem 1 is based on a number of auxiliary propositions, the principal ones of which are given below.
Denote by \(\alpha=\alpha(a,T)\) the lower bound of the lengths of the links of extremal polygons having one end on \(S\) and perimeter not exceeding \(T\). It can be shown that \(\alpha>0\).** Introduce functions
\[ u_1^{(m)}(s_1,s_2,\varepsilon),\qquad u_2^{(m)}(s_1,s_2,\delta),\qquad u_3^{(m)}(s_1,s_2,\delta,\varepsilon) \quad (s_i=s(\xi_i)), \]
having continuous mixed derivatives with respect to \(\xi_1,\xi_2\) up to order \(m\) inclusive and defined so that \(u_1^{(m)}=0\) for \(|s_1-s_2|\ge \varepsilon\), \(u_1^{(m)}=1\) for \(|s_1-s_2|\le \varepsilon/2\), and is positive at the remaining points; \(u_2^{(m)}=0\) for \(|s_1-s_2|\le \alpha-\delta\); \(u_2^{(m)}=1\) for \(|s_1-s_2|\ge \alpha-\delta/2\), and is positive at the remaining points \((\varepsilon<\alpha/2-\delta)\); \(u_3^{(m)}=1-u_1^{(m)}-u_2^{(m)}\). Introduce the iterated kernels
\[ h^p(s,a,k)= \int_S\cdots\int_S h(s,s_p,k)\cdots h(s_2,s_1,k)h_0(s_1,a,k)\,ds_1\cdots ds_p \]
and represent them in the form
\[ h^{(p)}=\sum_{j_0,\ldots,j_1=1}^{3} h_{j_p\cdots j_1}^{(p)}, \]
* Such polygons will be called extremal in what follows.
** We owe the proof of this proposition to A. V. Pogorelov.
where
\[ h_{j_p\cdots j_1}^{(p)}(s,a,k) = \int_S \cdots \int_S h(s,s_p,k)\, u_{j_p}^{(m)} \left(s,s_p,\delta,\frac{\varepsilon}{2^p}\right) \times \]
\[ \times\, h(s_p,s_{p-1},k)\, u_{j_{p-1}}^{(m)} \left(s_p,s_{p-1},\delta,\frac{\varepsilon}{2^{p-1}}\right) \cdots h_0(s_1,a,k)\,ds_1,\ldots,ds_p = \]
\[ = \int_0^\infty e^{ikt}\, \widetilde h_{j_p\cdots j_1}^{(p)}(t,s)\,dt . \]
Theorem 3. For each given \(l\) and \(n\) there exists an \(M\) such that, for \(m \ge M\), the functions
\(\sum \widetilde h_{j_{n+1}\cdots j_1}^{(n+1)}(t,s)\), where the summation is taken over all combinations \(j_{n+1},\ldots,j_1\) containing at least one index \(j\) equal to 3, have mixed derivatives with respect to \(t\) and \(\xi\) \((0 \le t \le T;\ s\in S)\) up to order \(l\) inclusive, equal to zero at \(t=0\).
Theorem 4. For any prescribed \(l\), one can indicate such \(n\) and \(M(n)\) that, if \(m \ge M(n)\), then the functions
\(\widetilde h_{j_{n+1}\cdots j_1}^{(n+1)}(t,s)\), in which none of the indices is equal to 3, have mixed derivatives with respect to \(t\) and \(\xi\) \((0 \le t \le T;\ s\in S)\) up to order \(l\) inclusive, equal to zero at \(t=0\).
Theorems analogous to Theorems 1 and 2 can be proved by the same method in the case of a space of any number of dimensions and under a boundary condition of the general form \(\partial u/\partial n + q(s)u=0\). Moreover, these same results can also be extended to the case of Maxwell’s equations, including problems connected with diffraction by infinitely thin screens of a prescribed form.
Kharkov State University
named after A. M. Gorky
Received
4 VI 1958