In this article we are interested in the rigorous construction of geometric optics expansions for hyperbolic corner problems. More precisely we focus on the case where self-interacting phases occur. Those phases are proper to the high frequency asymptotics for the corner problem and correspond to rays that can display a homothetic pattern after a suitable number of reflections on the boundary. To construct the geometric optics expansions in that framework, it is necessary to solve a new amplitude equation in view of initializing the resolution of the WKB cascade.

In this article we are interested in energy estimates for initial boundary value problem when surface waves occur that is to say when the uniform Kreiss Lopatinskii condition fails in the elliptic region or in the mixed region. More precisely we construct rigorous geometric optics expansions for elliptic and mixed frequencies and we show using those expansions that the instability phenomenon is higher in the case of mixed frequencies even if the uniform Kreiss Lopatinskii condition does not fail on hyperbolic modes. As a consequence this result allow us to give a classification of weakly well posed initial boundary value problems according to the region where the uniform Kreiss Lopatinskii condition degenerates.

Consider an elliptic operator in divergence form with symmetric coefficients. If the diffusion coefficients are periodic, the Bloch theorem allows one to diagonalize the elliptic operator, which is key to the spectral properties of the elliptic operator and the usual starting point for the study of its long-time homogenization. When the coefficients are not periodic (say, quasi-periodic, almost periodic, or random with decaying correlations at infinity), the Bloch theorem does not hold and both the spectral properties and the longtime behavior of the associated operator are unclear. At low frequencies, we may however consider a formal Taylor expansion of Bloch waves (whether they exist or not) based on correctors in elliptic homogenization. The associated Taylor-Bloch waves diagonalize the elliptic operator up to an error term (an "eigendefect"), which we express with the help of a new family of extended correctors. We use the Taylor-Bloch waves with eigendefects to quantify the transport properties and homogenization error over large times for the wave equation in terms of the spatial growth of these extended correctors. On the one hand, this quantifies the validity of homogenization over large times (both for the standard homogenized equation and higher-order versions). On the other hand, this allows us to prove asymptotic ballistic transport of classical waves at low energies for almost periodic and random operators.

No abstract