Transmission eigenvalues for strictly concave domains (Q330848)

Under consideration is the transmission eigenvalue problem \[ \nabla c_{1}(x)\nabla u_{1} +\lambda n_{1}(x)u_{1}=0,\;\nabla c_{2}(x)\nabla u_{2} +\lambda n_{2}(x)u_{2}=0,\;x\in \Omega, \] \[ \;u_{1}|_{\Gamma}=u_{2}|_{\Gamma},\;\partial_{\nu}u_{1}|_{\Gamma}=\partial_{\nu}u_{2}|_{\Gamma}, \] where \(\Omega \subset {\mathbb R}^{n}\) is a bounded domain with a smooth boundary \(\Gamma\) and \(\nu\) is the outward unit normal to \(\Gamma\). The domain \(\Omega\) is assumed to be strictly concave (in a certain sense). Under natural conditions on the functions \(c_{i}, n_{i}\) (\(i=1,2\)) it is proven that there are no eigenvalues of this spectral problem in the domain \(\{\lambda:\mathrm{Re }\lambda \geq 0,\;|\mathrm{Im }\lambda|\geq c(1+\mathrm{Re }\lambda)^{1/2+\varepsilon}\}\) for some sufficiently small \(\varepsilon > 0\). Is shown that the counting function of this spectral problem is of the form \(N(r)\approx (\tau_{1}+\tau_{2})r^{n}\) with \(\tau_{j}=\frac{\omega_{n}}{(2\pi)^{n}}\int_{\Omega}\bigl(\frac{n_{j}}{c_{j}}\bigr)^{n/2}\,dx\) (\(\omega_{n}\) is the volume of the unit ball in \({\mathbb R}^{n}\)). The remainder term in this asymptotic formula is also described.