摘要：Let $(L,B)$ denote a hyperbolic differential operator $L$ together with a boundary condition $B$ imposed along the lateral boundary. Given $L$, there are three topologically stable classes of boundary conditions $B$: (i) the uniform Lopatinskii condition is fulfilled, (ii) the weak Lopatinskii condition is violated, and (iii) $(L,B)$ belongs to the WR class.
The cases (i) and (ii) are well known. In case (i), there is a well established solution theory, where solutions possess the same regularity as the solutions to the Cauchy problem. In case (ii), the situation is instable and there is no hope for any linear estimates, let alone a solution theory. Following work of my former doctoral student Santiago Correa (now back to Colombia), I am going to discuss case (iii). This case occurs widely in applications. Here, one losses one derivative as compared to the Cauchy problem when working in Sobolev spaces, which is best possible. This, however, is not good enough for a direct application of the linear estimates to non-linear problems (and one is compelled to use techniques such as the Nash-Moser-H\"ormander iteration scheme). I am going to show how to obtain estimates without a loss of regularity. The idea is to employ suitable micro-local cutoffs (in the form of well-chosen pseudo-differential operators).
