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

Abstract

<p style='text-indent:20px;'>In this article we investigate the possible losses of regularity of the solution for hyperbolic boundary value problems defined in the strip <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^{d-1}\times \left[0,1 \right] $\end{document}</tex-math></inline-formula>. <p style='text-indent:20px;'>This question has already been widely studied in the half-space geometry in which a full characterization is almost completed (see [<xref ref-type="bibr" rid="b16">16</xref>,<xref ref-type="bibr" rid="b7">7</xref>,<xref ref-type="bibr" rid="b6">6</xref>]). 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. <p style='text-indent:20px;'>Crudely speaking the question addressed here is "can several boundaries make the situation becomes worse?". <p style='text-indent:20px;'>Here we focus our attention to one special case of loss (namely the elliptic degeneracy of [<xref ref-type="bibr" rid="b16">16</xref>]) 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 [<xref ref-type="bibr" rid="b2">2</xref>] in which the geometry has a real impact on the behaviour of the solution.