Regularity of weakly well posed non characteristic boundary value problems

We consider a boundary value problem for a system of linear partial differential equations with non regular coefficients.We assume the problem to be “weakly” well posed, in the sense that a unique L2-solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of tangential regularity with respect to the data. This is the case of problems that do not satisfy the uniform Kreiss–Lopatinskii condition in the hyperbolic region of the frequency domain. Provided the data are sufficiently smooth, we obtain the regularity of solutions, in the natural framework of weighted Sobolev spaces.