Full range scattering estimates and their application to cloaking
The authors study an acoustic scattering problem from an inhomogeneity surrounded by an absorbing layer. They prove uniform estimates with respect to the frequency for all frequencies. These estimates are also uniform with respect to the parameter inside the inhomogeneity. For an inhomogeneity with a diameter \(\epsilon\) surrounded by an absorbing layer with an index of refraction \(1+\frac{i}{\omega\epsilon\lambda}\) (\(0<\lambda<1\)) and a thickness \(\epsilon/2\), the scattered field \(v_{\epsilon}\) from an incident plane wave satisfies estimates for low and high frequencies. Precisely, scattering estimates for large frequencies \(\omega>\frac1\epsilon\) hold, namely \[ \forall \beta > \epsilon \;,\quad \| v_{\epsilon} \|_{0,B_{\beta}\setminus B_{\epsilon}}\lesssim \epsilon^{d-1} \beta. \] Scattering estimates for moderate to small frequencies \(0<\omega \leqslant \frac1\epsilon\) are given: \[ \forall \beta > \epsilon \;,\quad \| v_{\epsilon} \|_{0,B_{\beta}\setminus B_{\epsilon}}\lesssim C_{\omega,\epsilon} \epsilon^{2} \beta \quad (d=3)\;, \] \[ \forall \beta > \epsilon \;,\quad \| v_{\epsilon} \|_{0,B_{2\beta}\setminus B_{\beta}}\lesssim C_{\omega,\epsilon}\tfrac{|F(\beta \omega)|^2 }{|F(\epsilon \omega)|^2 } \beta^2 \quad (d=2) \;. \] Here, \(C_{\omega,\epsilon}=\max\{1, \frac{\lambda^2}{\omega^2\epsilon^2}\}\), \(d=2,3\) is the dimension of the ambient space, and \(F=H_{0}^{(1)}\) is the Hankel function of the first kind of order zero. The analysis depends on whether \(\omega\) is smaller or larger than the inverse of the diameter of the scattering inhomogeneity. For the large frequencies, the proof of estimates is based on a technique of Morawetz's multiplier together with the effect of the absorbing layer. The authors present several applications to cloaking. Uniform estimates are used to assess the effectivity of approximate cloaking schemes. The degree of cloaking can be estimated as a function of the frequency. For incident plane waves and a given scheme, it is not possible to obtain cloaking uniformly in frequency. This is due to low frequency probing. The authors present also an analysis of damped cloaking for the full wave equation.
scientific article; zbMATH DE number 6101993
scientific article