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MATHEMATICS
V. P. MIKHAILOV
ON THE PRINCIPLE OF LIMITING AMPLITUDE
(Presented by Academician I. G. Petrovskii, 12 VI 1964)
Let \(S\) be a closed smooth star-shaped surface in three-dimensional space, and let \(D\) be a domain of three-dimensional space containing infinity, whose boundary is the surface \(S\). Denote by \(u(x,t)\), \(x=(x_1,x_2,x_3)\), the solution in \(D\) of the wave equation
\[ -\frac{\partial^2 u}{\partial t^2}+\Delta u=f(x)e^{i\omega t}, \tag{1} \]
satisfying the initial conditions
\[ u(x,t)\big|_{t=0}=\left.\frac{\partial u(x,t)}{\partial t}\right|_{t=0}=0 \tag{2} \]
and one of the boundary conditions on \(S\):
\[ u(x,t)\big|_S=0, \tag{3} \]
\[ \left.\frac{\partial u(x,t)}{\partial n}\right|_S=0, \tag{4} \]
\[ \left(\frac{\partial u(x,t)}{\partial n}+\sigma(x)u\right)\Big|_S=0, \tag{5} \]
where \(n\) is the normal to the surface \(S\) exterior with respect to \(D\), and \(\sigma(x)>0\) is a smooth function on the boundary \(S\). The function \(f(x)\) in (1) is assumed smooth and finite, with support in some domain \(\Omega\), which may be regarded as a ball of sufficiently large radius. In addition, we shall assume the function \(f(x)\) to be equal to zero on \(S\), together with its derivatives up to some order \(k\). \(\omega\) in (1) is a certain real number.
It is known that the solutions of the problems (1), (2), (3); (1), (2), (4); (1), (2), (5) exist and are unique.
Let \(\lambda\) be some complex number. Denote by \(w(x,\lambda)\) the solution in the domain \(D\) of the Helmholtz equation
\[ \Delta w-\lambda^2 w=f(x), \tag{6} \]
satisfying one of the conditions
\[ w\big|_S=0, \tag{7} \]
\[ \left.\frac{\partial w}{\partial n}\right|_S=0, \tag{8} \]
\[ \left(\frac{\partial w}{\partial n}+\sigma(x)w\right)\Big|_S=0 \tag{9} \]
on the boundary \(S\), and the Sommerfeld radiation condition \((^{1,2})\) at infinity
\[ \frac{\partial w}{\partial r}+\lambda w=o\left(\frac{e^{-|\operatorname{Re}\lambda|r}}{r}\right). \tag{10} \]
Lemma 1. For every complex \(\lambda\ne0\) there exists a unique solution \(w(x,\lambda)\) of any of the problems (6), (7), (10); (6), (8), (10); (6), (9), (10).
For the first and second boundary-value problems the proof of this lemma is given in \((^{1-3})\). As for the third boundary-value problem, the proof of uniqueness of the solution is carried out by a method analogous to that in \((^1)\), while the proof of existence will be outlined below.
The aim of this note is to prove the following theorem:
Theorem (Principle of limiting amplitude). There exist \(\alpha>0\) and \(C(\alpha)>0\) such that
\[ \left|u(x,t)-v(x)e^{i\omega t}\right|\le C(\alpha)e^{-\alpha t} \tag{11} \]
for \(t \geqslant 0\), where \(u(x,t)\) is the solution of one of the exterior boundary-value problems for the wave equation (1), (2), (3); (1), (2), (4), or (1), (2), (5). The function \(v(x)\), called the limiting amplitude, is the solution, respectively, of the problems (6), (7), (10); (6), (8), (10), or (6), (9), (10) for \(\lambda=i\omega\).
For the first boundary-value problem (1), (2), (3) this theorem was proved by C. Morawetz, P. Lax, and R. Phillips in \((^{4,5})\). The limiting-amplitude principle for the first boundary-value problem was also studied in spaces of arbitrary dimension in the papers \((^{6,7})\); in these papers, under the same restrictions on the boundary \(S\) of the domain as in \((^{4,5})\), inequality (11) is proved with the factor \(e^{-\alpha t}\) replaced by \(1/\sqrt{t}\). A number of papers have also been devoted to the justification of the limiting-amplitude principle for the Cauchy problem, i.e., for the case when there is no reflecting body: in paper \((^8)\) this problem was first posed and solved for the wave equation; in papers \((^{9,10})\) the question is studied of the asymptotic behavior as \(t\to\infty\) of the solution of the Cauchy problem for equation (1), in which, in place of the operator \(\Delta\), there stands a linear homogeneous elliptic operator \(L(\partial/\partial x)\) with constant coefficients of order \(2m\). Under certain restrictions on \(L(\partial/\partial x)\) (its sign is assumed chosen so that \(L(i\xi)<0\) for real \(\xi\), \(|\xi|=1\)) it is proved that \(\lim_{t\to\infty} u(x,t)e^{-i\omega t}=v(x)\), where \(v(x)\) is the solution of the corresponding elliptic equation \(L(\partial/\partial x)v+\omega^2v=f(x)\). Here \(v(x)\) turns out to be the same solution of this equation as the solution singled out by means of the so-called “limiting absorption” principle. For the equation \(\Delta u-u_{tt}+a(x)u=f(x)e^{i\omega t}\) with smooth and finite \(a(x)\), the question of the limiting amplitude for the Cauchy problem was studied in \((^{11})\).
In proving the theorem we shall, for definiteness, dwell on the boundary condition (5) of the third boundary-value problem. Applying the Laplace transform with respect to \(t\) to equation (1) and condition (5), and using the initial conditions (2), we obtain
\[ \Delta \widetilde u(x,\lambda)-\lambda^2\widetilde u(x,\lambda)=\frac{f(x)}{\lambda-i\omega}, \tag{12} \]
\[ \left(\frac{\partial \widetilde u(x,\lambda)}{\partial n}+\sigma(x)\widetilde u(x,\lambda)\right)_S=0, \tag{13} \]
where
\[ \widetilde u(x,\lambda)=\int_0^\infty u(x,t)e^{-\lambda t}\,dt, \]
and \(\operatorname{Re}\lambda>0\). From the results of I. N. Vekua \((^1)\) it follows that \(\widetilde u(x,\lambda)\) satisfies the Sommerfeld radiation condition (10). From Lemma 1 it follows that
\[ \widetilde u(x,\lambda)=\frac{w(x,\lambda)}{\lambda-i\omega}, \tag{14} \]
where \(w(x,\lambda)\) is the solution of problem (6), (9), (10).
Lemma 2. The function \(w(x,\lambda)\) is a meromorphic function of \(\lambda\), analytic for \(\operatorname{Re}\lambda>-2\alpha\).
Lemma 3. There exists a number \(C(\alpha)>0\) such that, for \(|\operatorname{Re}\lambda|\leqslant\alpha\), \(|\operatorname{Im}\lambda|\geqslant 1\),
\[ |w(x,\lambda)|\leqslant \frac{C(\alpha)}{|\lambda|}. \tag{15} \]
With the help of these lemmas the proof of the theorem is completed as follows. The function \(u(x,t)\) is recovered from the function \(\widetilde u(x,\lambda)\) by the Mellin formula
\[ u(x,t)=\frac{1}{2\pi i}\int_L \widetilde u(x,\lambda)e^{\lambda t}\,d\lambda, \]
where \(L\) is the straight line \(\operatorname{Re}\lambda=a\). By virtue of (14) and Lemmas 2 and 3 we have
\[ u(x,t)=w(x,i\omega)e^{i\omega t}+\frac{1}{2\pi i}\int_{\operatorname{Re}\lambda=-\alpha} e^{\lambda t}\frac{w(x,\lambda)}{\lambda-i\omega}\,d\lambda =v(x)e^{i\omega t}+z(x,t), \]
since \(w(x,i\omega)=v(x)\). By virtue of Lemma 3,
\[ |z(x,t)| \leq C(\alpha)e^{-\alpha t} \int_{\operatorname{Re}\lambda=-\alpha} \frac{|d\lambda|}{|\lambda|\,|\lambda-i\omega|} = C_1(\alpha)e^{-\alpha t}, \]
which is what had to be established.
Let us dwell briefly on the proofs of the lemmas.
With the aid of Green’s formula and the radiation condition (10), we have
\[ w(x,\lambda)=\iiint_{D\cap\Omega} f(y)U(x-y,\lambda)\,dy- \tag{16} \]
\[ -\iint_S w(y,\lambda)\left[ \sigma(y)U(x-y,\lambda)+ \frac{\partial U(x-y,\lambda)}{\partial n_y} \right]dy =F(x,\lambda)-\theta(x,\lambda), \]
where
\[ U(x-y,\lambda)=\frac{e^{-\lambda|x-y|}}{4\pi|x-y|} \]
is the fundamental solution of the Helmholtz equation.
Passing in (16) to the limit as \(x\to X\in S\), we obtain, by the properties of the double-layer potential, the Fredholm integral equation
\[ w(X,\lambda)=2F(X,\lambda)-2\iint_S w(y,\lambda)\left[ \sigma(y)U(X-y,\lambda)+ \frac{\partial U(X-y,\lambda)}{\partial n_y} \right]dy \]
for the unknown function \(w(X,\lambda)\). It can be shown that this equation is always solvable.
Below we shall need the following property of the function \(F(x,\lambda)\).
Lemma 4. If \(\nabla^r f(x)\big|_S=0,\ r=0,\ldots,k\), then for every \(\mu>0\) there is a \(C_1(\mu)>0\) such that the entire function \(F(x,\lambda)\) of \(\lambda\) (see (16)) in the region \(|\operatorname{Im}\lambda|\geq 1,\ |\operatorname{Re}\lambda|\leq \mu\) satisfies the inequality
\[ |F(x,\lambda)|+|\nabla_x F(x,\lambda)|\leq C_1(\mu)/|\lambda|^k . \]
Using Lemma 4, from (16) we obtain that the function \(\theta(x,\lambda)\), being a solution of the homogeneous Helmholtz equation (6), satisfies condition (10) and the boundary condition
\[ \left.(\partial\theta/\partial n+\sigma(x)\theta)\right|_S =\varphi(x,\lambda)\big|_S, \tag{17} \]
where \(\varphi(x,\lambda)\) is a sufficiently smooth function in \(x\) and an entire function in \(\lambda\), satisfying, for \(|\operatorname{Im}\lambda|\geq 1,\ |\operatorname{Re}\lambda|\leq\alpha\), the inequality
\[ |\varphi(x,\lambda)|\leq C_2(\alpha)/|\lambda|^k . \tag{18} \]
For the function \(\theta(x,\lambda)\) the following basic lemma is valid.
Lemma 5. If \(\theta(x,\lambda)\) is a solution of equation (6) with \(f\equiv0\), satisfies the boundary condition (17) and condition (10), then for \(p>3\) the inequality
\[ \|\theta(x,\lambda)\|_{L_p(D)} \leq C_3(p)\|\varphi(x,\lambda)\|_{L_p(S)} \tag{19} \]
holds, with a constant \(C_3(p)\) independent of \(\lambda\).
Taking into account that in the domain \(D\), \(\lambda^2\theta=\Delta\theta\), and also the known embedding theorems and (16), we obtain
\[ \|w(x,\lambda)\|_{L_p(S)} \leq C_4|\lambda|^2\|\varphi(x,\lambda)\|_{L_p(S)}. \]
If now in Lemma 4 and inequality (18) we put \(k=4\) and use equality (16) once again, we obtain estimate (15) of Lemma 3. Since the function \(w(x,\lambda)\), \(x\in S\), is meromorphic in \(\lambda\), as follows from Fredholm theory, the function \(w(x,\lambda)\) for \(x\in D\) will also be meromorphic in \(\lambda\). Therefore Lemma 2 follows from Lemma 3.
Moscow State University
named after M. V. Lomonosov
Received
6 VI 1964
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