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UDC 517.947
MATHEMATICS
V. I. CHEKHLOV
A MIXED PROBLEM WITH DISCONTINUOUS BOUNDARY CONDITIONS FOR THE WAVE EQUATION
(Presented by Academician S. L. Sobolev on 26 III 1968)
The paper considers a mixed problem with discontinuous boundary conditions for the wave equation
\[ \frac{\partial^2 v}{\partial t^2} -\sum_{i=1}^{n}\frac{\partial^2 v}{\partial x_i^2} =f(t,x),\qquad t>0,\quad x_1>0,\quad x=(x_1,\ldots,x_n) \]
with initial conditions
\[ v(+0,x)=\varphi_0(x),\qquad \frac{\partial v}{\partial t}(+0,x)=\varphi_1(x),\quad x_1>0 \]
and boundary conditions
\[ v(t,+0,x')=g_0(t,x'),\quad x_2>0,\ t>0,\quad x'=(x_2,\ldots,x_n), \]
\[ \frac{\partial v}{\partial x_1}(t,+0,x')=g_1(t,x'),\quad x_2<0,\ t>0,\quad \text{where } x'=(x_2,\ldots,x_n). \]
Solving the auxiliary Cauchy problem and making the substitution \(v(x,t)=u(x,t)e^{at}\), \(a>0\), we reduce this problem to the form
\[ A(i\partial/\partial t,\ i\partial/\partial x)\equiv (\partial/\partial t+a)^2u-\Delta u=0,\qquad t>0,\ x_1>0; \tag{1} \]
\[ u(+0,x)=0,\qquad \frac{\partial u}{\partial t}(+0,x)=0,\quad x_1>0; \tag{2} \]
\[ u(t,+0,x')=h_0(t,x'),\qquad x_2>0,\ t>0, \]
\[ \frac{\partial u}{\partial x_1}(t,+0,x')=h_1(t,x'),\qquad x_2<0,\quad t>0. \tag{3} \]
Two questions are investigated: 1) the precise formulation of the boundary-value problem (1)—(3); 2) the smoothness of the solution.
The first question arises in connection with the fact that we cannot expect the existence of a classical solution of this problem for smooth \(h_0\) and \(h_1\) satisfying local compatibility conditions (of finite number) on the manifold of discontinuity.
In formulating the problem, the projection operators in \(L_2\), \(\Pi^+\) and \(\Pi^-\), described in (1), are used.
We introduce the necessary function spaces. Let \(u(t,x)\in L_2(R^{n+1})\). Its Fourier transform is
\[
Fu\equiv u(\tau,\xi)
=\int e^{it\tau+i(x\xi)}u(t,x)\,dt\,dx.
\]
To the operator \(A(i\partial/\partial t,\ i\partial/\partial x)\) from (1) there corresponds, in Fourier images, the operator of multiplication by
\[
A(\tau,\xi)=(-i\tau+a)^2+|\xi|^2=(\xi_1-\lambda)(\xi_1+\lambda).
\]
Here
\[
\lambda\equiv\lambda(\tau,\xi')
=\sqrt{(\tau+ia)^2-|\xi'|^2}
=\sqrt{(\tau+ia)^2-|\xi''|^2-\xi_2^2},
\quad
\xi''=(\xi_3,\ldots,\xi_n),
\]
with the condition
\[
\lambda(0,\xi')=i\sqrt{a^2+|\xi'|^2}.
\]
The function \(\lambda(s,\xi')\) is analytic in the complex plane \(s=\tau+i\sigma\) with a cut from the point \(s_1=|\xi'|-ia\) to the point \(s_2=-|\xi'|-ia\) in the lower half-plane. Let
\[
\varkappa(\tau,\xi'')=\lambda(\tau,0,\xi'').
\]
Then
\[
\lambda(\tau,\xi')=\sqrt{(\varkappa-\xi_2)(\varkappa+\xi_2)}.
\]
The function \(\lambda(\tau,z_2,\xi'')\) is analytic in the complex plane
planes \(z_2=\xi_2+i\eta_2\) with cuts from the points \(\pm\varkappa\) to \(\infty\), respectively, in the upper and lower half-planes. In the same plane with such cuts we consider the functions \(\eta_{\pm}(\tau,z_2,\xi'')=\sqrt{\varkappa\pm z_2}\), choosing the branches with the conditions \(\eta_+\to\sqrt{\xi_2}\) as \(\xi_2\to+\infty\), \(\eta_-\to i\sqrt{\xi_2}\) as \(\xi_2\to+\infty\). We have \(\lambda(\tau,\xi')=\sqrt{\varkappa+\xi_2}\sqrt{\varkappa-\xi_2}=\eta_+\eta_-\). Let us also note that if \(\lambda(\tau,\xi')=\operatorname{Re}\lambda+i\operatorname{Im}\lambda\), then \(\operatorname{Im}\lambda(\tau,\xi')\ge a\), \(\operatorname{Im}\lambda\to a\) as \(|\tau|\to\infty\), and \(\operatorname{Im}\lambda=O(\sqrt{a^2+|\xi'|^2})\) as \(|\xi'|\to\infty\).
Definition 1.
\[ V_s(R^{n+1})=\left\{u(t,x)\in L_2(R^{n+1}): |u|_{V_s(R^{n+1})}^2=\right. \]
\[ \left. =\int \frac{|A(\tau,\xi)|^2|\varkappa|^{1/2}(\operatorname{Im}\varkappa)^{1/2}}{|\eta_+|^2\operatorname{Im}\lambda} \left|\frac{A\varkappa\operatorname{Im}\varkappa}{\lambda\operatorname{Im}\lambda}\right|^{2(s-1)} |u(\tau,\xi)|^2\,d\tau d\xi<\infty \right\}, \]
\(s\ge 1\) an integer.
It can be proved that
\[ V_s(R^{n+1})\subset W_2^{(1,1,s,s)_{(x_1,x_2,x'',t)}}(R^{n+1})= \]
\[ =\left\{u(t,x)\in L_2(R^{n+1}):\int(1+|\xi_1|^2+|\xi_2|^2+|\xi''|^{2s}+|\tau|^{2s})|u(\tau,\xi)|^2\,d\tau d\xi<\infty\right\}. \]
Definition 2.
\[ X_s^0(R^n)=\left\{v(t,x)\in L_2(R^n):\right. \]
\[ \left. |v|_{X_s^0(R_n)}^2=\int |\eta_-|^2|\varkappa|^{3/2}(\operatorname{Im}\varkappa)^{1/2} |\varkappa\operatorname{Im}\varkappa|^{2(s-1)} |v(\tau,\xi')|^2\,d\tau d\xi'<\infty \right\}. \]
It can be proved that
\[ X_s^0(R^n)\subset W_2^{(1/2,\;s-1/4,\;s-1/4)_{(x_2,x'',t)}}(R^n). \]
Definition 3.
\[ X_s^1(R^n)=\left\{v(t,x')\in W_2^{(-1/2)}(R^n):\right. \]
\[ \left. |v|_{X_s^1(R^n)}^2=\int \frac{|\varkappa|^{5/2}(\operatorname{Im}\varkappa)^{1/2}}{|\eta_+|^2} |\varkappa\operatorname{Im}\varkappa|^{2(s-1)} |v(\tau,\xi')|^2\,d\tau d\xi'<\infty \right\}. \]
It can be proved that
\[ X_s^1(R^n)\subset W_2^{(-1/2),(s-3/4,\;s-3/4)}{}_{(x_2),(x'',t)}(R^n)= \]
\[ =\left\{v(t,x')\in W_2^{(-1/2)}(R^n):\int \frac{(1+\tau^2+|\xi''|^2)^{s-3/4}}{(1+\xi_2^2)^{1/2}} |v(\tau,\xi')|^2\,d\tau d\xi'<\infty \right\}. \]
Let us note that \(X_s^0(R^n)\) and \(X_s^1(R^n)\) are the natural spaces of traces of the functions \(u(t,x_1,x')\) and \(\dfrac{\partial u}{\partial x_1}(t,x_1,x')\) for \(x_1=\mathrm{const}\), where \(u(t,x)\in V_s(R^{n+1})\).
Let \(R_+^{n+1}=\{(t,x):x_1>0\}\).
Definition 4.
\[ V_s^+(R_+^{n+1})=\left\{u(t,x)\in L_2(R_+^{n+1}):\right. \]
\[ \exists u\ \text{and}\ \hat u(t,x)\in V_s(R^{n+1}):\ u(t,x)=\hat u(t,x),\ x_1>0; \]
\[ \left. |u|_{V_s^+(R_+^{n+1})}=\inf_{\hat u}|\hat u(t,x)|_{V_s(R^{n+1})} \right\}. \]
We now proceed to the formulation of the problem in terms of the spaces introduced. Let \(h_0(t,x')\), \(h_1(t,x')\) from (3) admit extensions \(\hat h_0(t,x')\) and \(\hat h_1(t,x')\) for all \(x_2\) and \(t\) in the classes \(X_s^0(R^n)\) and \(X_s^1(R^n)\), respectively. We shall assume that these functions are extended by zero for \(t<0\) (this imposes on them certain compatibility conditions with zero at \(t=0\)). Then the Fourier transforms \(\hat h_0(\tau,\xi')\) and \(\hat h_1(\tau,\xi')\) will have analytic continuations \(H_0(s,\xi')\) and \(H_1(s,\xi')\) in \(\operatorname{Im}s>0\), \(s=\tau+i\sigma\), and
\[ |H_0(s,\xi')|\le c(\xi'),\qquad |H_1(s,\xi')|\le c(\xi'). \tag{4} \]
Extend the functions from \(V_s^+(R_+^{n+1})\) by zero for \(x_1<0\), denoting these extensions by \(u^+(t,x)\). We denote by \(V_s^+\) the class of their Fourier transforms \(u^+(\tau,\xi)\). These functions admit a bounded analytic continuation into the half-plane \(\operatorname{Im} z_1>0\), where \(z_1=\xi_1+i\eta_1\). Let \(v_-(t,x')\in X_s^0(R^n)\), \(w_+(t,x')\in X_s^1(R^n)\), and be equal to zero for \(x_2<0\) and \(x_2>0\), respectively. Then their Fourier transforms \(v_-(\tau,\xi')\), \(w_+(\tau,\xi')\) admit analytic continuations \(V_-(\tau,z_2,\xi'')\), \(W_+(\tau,z_2,\xi'')\) into the complex half-planes \(\operatorname{Im} z_2<0\) and \(\operatorname{Im} z_2>0\), respectively, and
\[ \left|V_-(\tau,z_2,\xi'')\right|\le c(\tau,\xi''),\qquad \operatorname{Im} z_2<0; \tag{5} \]
\[ \left|W_+(\tau,z_2,\xi'')\right|\le c(\tau,\xi'')|z_2|^{1/2},\qquad \operatorname{Im} z_2>0. \tag{6} \]
For the classes of Fourier transforms of functions from \(X_s^0(R^n)\), \(X_s^1(R^n)\) we retain the same notation.
Formulation of the problem: find \(u^+(\tau,\xi)\), \(v_-(\tau,\xi')\), \(w_+(\tau,\xi')\) such that:
\(1^\circ.\) \(u^+(\tau,\xi)\in V_s^+\).
\(2^\circ.\) \(u^+(\tau,\xi)\) admits an analytic continuation into the complex half-plane \(\operatorname{Im}s>0\), \(s=\tau+i\sigma\), and
\(\left|u^+(s,\xi)\right|\le c(\xi)\), \(\operatorname{Im}s>0\).
\(3^\circ.\) \(\Pi_{\xi_1}^+A(\tau,\xi)u^+(\tau,\xi)=0\).
\(4^\circ.\)
\[
\frac{1}{2\pi}\int u^+(\tau,\xi)\,d\xi_1=\hat h_0(\tau,\xi')+v_-(\tau,\xi').
\]
\(5^\circ.\)
\[
\frac{1}{2\pi}\int \Pi_{\xi_1}^+(-i\xi_1)u^+(\tau,\xi_1,\xi')\,d\xi_1
=
\hat h_1(\tau,\xi')+w_+(\tau,\xi').
\]
\(6^\circ.\) \(v_-(\tau,\xi')\in X_s^0(R^n)\) and satisfies (5).
\(7^\circ.\) \(w_+(\tau,\xi')\in X_s^1(R^n)\) and satisfies (6).
\(8^\circ.\) \(v_-(\tau,\xi')\), \(w_+(\tau,\xi')\) admit analytic continuations \(V_-(s,\xi')\), \(W_+(s,\xi')\) into the complex half-plane \(\operatorname{Im}s>0\), with
\[
\left|V_-(s,\xi')\right|\le c(\xi'),\qquad
\left|W_+(s,\xi')\right|\le c(\xi').
\]
Definition. By a generalized solution of problem (1)—(3) we shall mean a function \(u(t,x)\in V_s^+(R_+^{n+1})\) satisfying \(1^\circ\)—\(5^\circ\).
The functions \(v_-(t,x')\), \(w_+(t,x')\) appear in the formulation of the problem because of the arbitrariness in the continuation of \(h_0(t,x')\), \(h_1(t,x')\) for \(x_2<0\), \(x_2>0\), respectively.
Theorem. The solution of problem \(1^\circ\)—\(8^\circ\) exists and is unique in
\(V_s^+\times X_s^0\times X_s^1\) for arbitrary
\(\hat h_0(t,x')\in X_s^0(R^n)\) and \(\hat h_1(t,x')\in X_s^1(R^n)\), and the estimate holds
\[ |u|_{V_s^+(R_+^{n+1})} \le c\left\{ \left\|\varkappa^{3/4}(\operatorname{Im}\varkappa)^{1/4}(\varkappa\,\operatorname{Im}\varkappa)^{s-1}\Pi_{\xi_2}^+\eta_-\,\hat h_0(\tau,\xi')\right\|_{L_2(R^n)} + \right. \]
\[ \left. + \left\|\varkappa^{3/4}(\operatorname{Im}\varkappa)^{1/4}(\varkappa\,\operatorname{Im}\varkappa)^{s-1}\Pi_{\xi_2}^-\frac{\hat h_1(\tau,\xi')}{\eta_+}\right\|_{L_2(R^n)} \right\}. \tag{7} \]
We note that the norms on the right do not depend on the continuations of \(h_0(t,x')\) and \(h_1(t,x')\) in the classes \(X_s^0(R^n)\) and \(X_s^1(R^n)\), but depend only on the functions themselves.
It is proved that the function \(u_+(t,x)\) from \(1^\circ\)—\(8^\circ\) has the form
\[ u^+(\tau,\xi) = \frac{ 2\left[\Pi_{\xi_2}^+\eta_-(\tau,\xi')\hat h_0(\tau,\xi') +i\Pi_{\xi_2}^-\hat h_1(\tau,\xi')/\eta_+(\tau,\xi')\right] }{ i\eta_-(\tau,\xi')\bigl(\xi_1+\lambda(\tau,\xi')\bigr) }. \tag{8} \]
Formula (8) shows that, as the domain of definition of the operator \(\mathfrak A\) described in items \(1^\circ\)—\(8^\circ\), one cannot take a space of the type \(W_2^s(R_+^{n+1})\) and the spaces of traces of functions from \(W_2^s(R^{n+1})\). The natural domain of definition of this operator is
\[
V_s^+(R_+^{n+1})\times X_s^0(R^n)\times X_s^1(R^n),
\]
\(s\ge 1\), with nonsmoothness in \(x_1\) and \(x_2\) that cannot be improved. Finally, using the finiteness of the domain of dependence for the solution \(u^+(t,x)\) from \(1^\circ\)—\(8^\circ\), one can prove that it will have smoothness of type \(V_s\) inside the characteristic “cone,” whose “axis” is the manifold of discontinuity of the boundary
conditions \(x_1 = 0,\ x_2 = 0\), and will have the smoothness corresponding to an ordinary mixed problem outside it.
As for the “interior smoothness” of the solution, it has been proved that in the region \(R_{+\delta_1}^{+} = \{(t,x): x_1 \geqslant 0 > 0\}\) the solution has the smoothness of the solution of an ordinary mixed problem. In particular, from the partial hypoellipticity of the hyperbolic equation with respect to \(x\) and the proved smoothness properties up to the boundary, it follows that if \(h_0(t,x')\) and \(h_1(t,x') \in C_0^\infty(R^n)\), then \(u^{+}(t,x) \in C^\infty(R^{n+1})\) (see (3), p. 146, Theorem 4.2.4.). This fact is a consequence of the discontinuity of the boundary conditions occurring on a “time-like” manifold which nowhere intersects the characteristic surfaces.
Moscow Institute of Physics and Technology
Received
11 III 1968
REFERENCES
- M. I. Vishik, G. I. Eskin, UMN, 23, no. 1 (133), 15 (1967).
- N. Wiener, R. Paley, Fourier Transforms in the Complex Domain, “Nauka,” 1964.
- L. Hörmander, Linear Partial Differential Operators, Moscow, 1965.