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Asymptotic behavior of the Green function of the first boundary value problem. (Russian. English summary) Zbl 1052.35501

The behavior, as \(\epsilon\to 0\), of the Green function \(G_\epsilon(x,y)\) of the first boundary value problem for the Helmholtz equation in three-dimensional space is investigated in the case where there is a \(2\pi\epsilon\)-periodic lattice of small solids whose diameters \(l=l(\epsilon)\) tend to zero too and, at the same time, the frequency increases like \(\epsilon^{-1}\). The domain in question is \(D_\epsilon:= \mathbb R^3 \bigcup_{m\in\mathbb Z^3}\overline{F_{l(\epsilon)}+\epsilon 2\pi m}\), where \(F_{l(\epsilon)}=l(\epsilon)F\), \(F\) a simply connected bounded domain, \(l(\epsilon)=o(\epsilon^{3/2})\); \(G=G_\epsilon(x,y)\) obeys the equation \(\Delta G+k(\epsilon)^2G=\delta(x-y)\) in \(D_\epsilon\), \(G(x,y)=0\) for \(x\in\partial D\) and \(G(x,y)\to 0\) as \(x\to\infty\) (for \(y\) fixed in \(D_\epsilon\)), \(k(\epsilon)=\alpha\epsilon^{-1}+i\beta (\alpha,\beta\) are positive constants, \(\alpha<\frac 12)\). On the other hand, let \(\phi_\epsilon\) be the (well-known) fundamental solution to \(\Delta\phi+(k(\epsilon)^2-q(\epsilon))\phi=\delta(x-y)\) in \(\mathbb R^3\), \(\phi(x,y)\to 0\) as \(x\to\infty\). Then it is proved that \(\|G_\epsilon-\phi_\epsilon\|\leq C(Q_x)\cdot\mu(\epsilon)\) in
\(L^2(Q_x\times\mathbb R^3)\), with \(\lim_{\epsilon\to 0}\mu(\epsilon)=0\), for any bounded domain \(Q_x\subset\mathbb R^3\). Here \(q(\epsilon)=|F_{l(\epsilon)}|/|Y_\epsilon|\), where \(|\cdot|\) denotes the volume; \(Y_\epsilon\) is the “period cell” \((-\epsilon\pi,+\epsilon\pi)^3\), i.e., \(|Y_\epsilon|=8\pi^3\epsilon^3\). The construction of the asymptotic representation is based on Fourier transformations weighted by eigenfunctions of a special operator in a reference cell.

MSC:

35B25 Singular perturbations in context of PDEs
35A08 Fundamental solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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