Surface potentials and the method of matching of asymptotic expansions in the Helmholtz resonator problem. (Russian) Zbl 0787.35024

Gadyl’shin, R. R.

Algebra Anal. 4, No. 2, 88-115 (1992).

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)

MSC:

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

Keywords:

Helmholtz resonator; matching asymptotic expansions; surface potentials