  We investigate the non-perturbative stability of asymptotically anti-de
Sitter gravity coupled to tachyonic scalar fields with mass near the
Breitenlohner-Freedman bound. Such scalars are characterized by power-law
radial decay near the AdS boundary, and typical boundary conditions are
"Dirichlet" (fix the slower fall-off mode) or "Neumann" (fix the faster
fall-off mode). More generally though, these "designer gravity" theories admit
a large class of boundary conditions defined by a functional relation between
the two modes. While previous stability proofs have considered boundary
conditions that are deformations of the Neumann theory, the goal of this paper
is to analyze stability in designer gravity with boundary conditions that are
irrelevant deformations of the Dirichlet theory. We obtain a lower bound on the
energy using spinor charge methods and show that for the most interesting class
of such boundary conditions, the theory is always stable. We argue that the
deformed theory flows to a new fixed point in the ultraviolet, which is just
the Neumann theory. We also derive a corresponding "effective potential" that
implies stability if it has a global minimum.
