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Surface potentials and the method of matching of asymptotic expansions in the Helmholtz resonator problem. (Russian) Zbl 0787.35024
This paper deals with ideal soft screen, ideal hard screen and asymptotics of quasiproper frequency. Because of the lack of space it follows a short presentation of the first subject.
Let \(\Omega\) be a simply connected and bounded domain in \(\mathbb{R}^ 3\), \(\Gamma_ 0=\partial \Omega \in C^ \infty\), \(\omega_ \varepsilon \subset \Gamma_ 0\) an open simply connected set, \(\partial \omega_ \varepsilon \in C^ \infty\), \(\omega_ \varepsilon \subset B(x_ 0,\varepsilon)\), \(x_ 0 \in \Gamma_ 0\), \(0 <\varepsilon \ll 1\), and \(\Gamma_ \varepsilon=\Gamma_ 0 \backslash \overline\omega \varepsilon\). Consider the problem: \[ (1)\quad (\Delta+k^ 2)u_ \varepsilon=0 \text{ on }\mathbb{R}^ 3 \backslash \overline \Gamma_ \varepsilon,k>0, \qquad (2) \quad u_ \varepsilon=p_ \varepsilon f \text{ on } \Gamma_ \varepsilon; \] \(f \in C^ \infty(\Gamma_ 0)\); \(p_ \varepsilon\) is the contraction operator on \(\Gamma_ \varepsilon\), \[ {\partial u_ \varepsilon \over \partial r}-iku_ \varepsilon =o(r^{-1}) \text{ for } r \to+\infty,\;r=| x |. \tag{3} \] If \(E(k,r)=(4\pi r)^{-1} \exp(ikr)\), \((A_ 0(k) \varphi)x=\int_{\Gamma_ 0} \varphi(y)E(k,| x-y |)ds_ y\), \(\varphi \in C^ \infty (\Gamma_ 0)\) and \(A_ \varepsilon(k)=p_ \varepsilon A_ 0(k)\), then the problem (1)–(3) is equivalent to the equation \(A_ \varepsilon(k) \varphi=p_ \varepsilon f\).
Consider the Hilbert space \(H_ s(\mathbb{R}^ 2)\) with the scalar product \((u,v)_{s,\mathbb{R}^ 2}=\int_{\mathbb{R}^ 2} (1+ | \xi |)^{2s}Fu\;\overline{Fv} d \xi\), \(F\) being the Fourier transform. Let \(\{V_ 1, V_ 2 \}\) be a covering of \(\Gamma_ 0\), \(\chi_ j:B(0,1) \to V_ j\), \(j=1,2,\) a diffeomorphism, \(J_ j\) the Jacobian of \(\chi_ j\), \(\{\chi^ 2_ j\}\) the partition of unity on \(\Gamma_ 0\). The completion of \(C^ \infty(\Gamma_ 0)\) with respect to the norm \(\| u \|_ s^{(0)}=(\sum_ j \| | J_ j |^{{1\over 2}}\chi_ ju \|^ 2_{s,\mathbb{R}^ 2})^{1\over 2}\), is the space denoted by \(H_ s (\Gamma_ 0)\); let \(H^ 0_ s(\Gamma_ \varepsilon)\) be the completion with respect to the norm of \(H_ s(\Gamma_ 0)\). Let \(H_ \varepsilon (\Gamma_ \varepsilon)\) be the space of distributions on \(\Gamma_ \varepsilon\), belonging to \(H_ s(\Gamma_ 0)\), with norm \(\| u\|_ s^{(\varepsilon)}=\inf\|\ell_ \varepsilon u\|_ s^{(0)}\), where \(\ell_ \varepsilon u\in H_ 0(\Gamma_ 0)\) are all extensions to \(\Gamma_ 0\) of the distributions on \(\Gamma_ \varepsilon\).
Then, the equation \(A_ \varepsilon(i)\varphi=f\) has a unique solution in \(H^ 0_{-{(1/2)}} (\Gamma_ \varepsilon)\) for all \(f\in H_{(1/2)} (\Gamma_ \varepsilon)\), and the operator \(A^{-1}_ \varepsilon(i)\) is uniformly bounded with respect to \(\varepsilon\). Moreover, under some assumptions, the operator \(A_ \varepsilon (k)\) is invertible from \(H_{(1/2)} (\Gamma_ \varepsilon)\) to \(H^ 0_{- (1/2)} (\Gamma_ \varepsilon)\) and the inequality \[ \| A^{-1}_ \varepsilon(k)p_ \varepsilon f \|^{(0)}_{-{1/2}} \leq {c \over | \tau_ \varepsilon-k |} \| p_ \varepsilon f \|^{(\varepsilon)}_{{1/2}} \] holds.
Reviewer: D.M.Bors (Iaşi)

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P25 Scattering theory for PDEs