Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry

In this article we investigate the possible losses of regularity of the solution for hyperbolic boundary value problems defined in the strip $ \mathbb{R}^{d-1}\times \left[0,1 \right] $.

This question has already been widely studied in the half-space geometry in which a full characterization is almost completed (see [16,7,6]). In this setting it is known that several behaviours are possible, for example, a loss of a derivative on the boundary only or a loss of a derivative on the boundary combined with one or a half loss in the interior.

Crudely speaking the question addressed here is "can several boundaries make the situation becomes worse?".

Here we focus our attention to one special case of loss (namely the elliptic degeneracy of [16]) and we show that (in terms of losses of regularity) the situation is exactly the same as the one described in the half-space, meaning that the instability does not meet the geometry. This result has to be compared with the one of [2] in which the geometry has a real impact on the behaviour of the solution.