This year's lecture focused on the characterization of asymptotic profiles of the solutions of partial differential equations corresponding to various models.
These asymptotic or corrective profiles are associated with singularities, defects or interfaces... We will always assume a characteristic dimension noted E (> 0) for these defects. It's worth noting that, often for realistic applications, this parameter e is not " very small " but can be of the order of 0.1.
Furthermore, we will systematically consider situations where these defects of size E are present in periodic environments whose period is also of order E. Note that this is in fact the only interesting case, because if the period is much larger, we can consider the environment to be independent of e, whereas if the period is much smaller we need to resolve these oscillations before anything else, for example by applying or attempting to apply homogenization theory.
When the period is indeed of order E, homogenization theory allows us to take account of the oscillations in the environment without fault, notably by introducing periodic correctors. The problem considered is therefore twofold : i) confirm (or deny) the fact that the presence of the fault does not modify the boundary problem (typically the homogenized problem) ; ii) determine, where possible, the behavior of solutions, in the presence of faults, by specified correctors.
To conclude this introduction, let's mention that, in a way, our approach gives a precise meaning to what multiscale methods in scientific computing do (or should do...). On the other hand, the program outlined above has been carried out for classical stationary models (elliptic equations and systems, quasilinear equations, Hamilton-Jacobi equations and geometrical optics, completely nonlinear equations...) with easy adaptations to time-dependent models (heat-type equations, Schrödinger equation or wave equations - in " low-frequency regimes "). The main open problem concerns the propagation of waves with a wavelength of order E, also : in this case, complex phenomena are expected that have not yet been rigorously addressed.