The linear $H^1$-formulation of Strong Cosmic Censorship for AdS black holes

Let $\psi$ be a linear perturbation solving the wave equation on Reissner--Nordström--AdS or Kerr--AdS, arising from generic Cauchy data with Dirichlet boundary conditions imposed at infinity $\mathcal I$. Let $\mathcal{CH}$ denote the Cauchy horizon, and let $H^1_{\mathrm{loc}}$ denote the local Sobolev space of functions with locally square-integrable first derivatives. Linear analog of the $H^1$-formulation of Strong Cosmic Censorship. The perturbation has infinite local energy along hypersurfaces intersecting transversally the Cauchy horizon; equivalently, it fails to belong to $H^1_{\mathrm{loc}}$ around every point of $\mathcal{CH}$.

This is the weaker $H^1$ analogue of the $C^0$ formulation of Strong Cosmic Censorship. The paper establishes the corresponding result for Reissner--Nordström--AdS and formulates the more general Kerr--AdS statement as an open conjecture.