  We consider four-dimensional Euclidean gravity in a finite cavity. Dirichlet
conditions do not yield a well-posed elliptic system, and Anderson has
suggested boundary conditions that do. Here we point out that there exists a
one-parameter family of boundary conditions, parameterized by a constant $p$,
where a suitably Weyl rescaled boundary metric is fixed, and all give a
well-posed elliptic system. Anderson and Dirichlet boundary conditions can be
seen as the limits $p \to 0$ and $\infty$ of these. Focussing on static
Euclidean solutions, we derive a thermodynamic first law. Restricting to a
spherical spatial boundary, the infillings are flat space or the Schwarzschild
solution, and have similar thermodynamics to the Dirichlet case. We consider
smooth Euclidean fluctuations about the flat space saddle; for $p > 1/6$ the
spectrum of the Lichnerowicz operator is stable -- its eigenvalues have
positive real part. Thus we may regard large $p$ as a regularization of the
ill-posed Dirichlet boundary conditions. However for $p < 1/6$ there are
unstable modes, even in the spherically symmetric and static sector. We then
turn to Lorentzian signature. For $p < 1/6$ we may understand this spherical
Euclidean instability as being paired with a Lorentzian instability associated
with the dynamics of the boundary itself. However, a mystery emerges when we
consider perturbations that break spherical symmetry. Here we find a plethora
of dynamically unstable modes even for $p > 1/6$, contrasting starkly with the
Euclidean stability we found. Thus we seemingly obtain a system with stable
thermodynamics, but unstable dynamics, calling into question the standard
assumption of smoothness that we have implemented when discussing the Euclidean
theory.
