About stability of equilibrium shapes

We discuss the stability of "critical" or "equilibrium" shapes of a shape-d ependent energy functional. We analyze a problem arising when looking at th e positivity of the second derivative in order to prove that a critical sha pe is an optimal shape. Indeed, often when positivity - or coercivity - hol ds, it does for a weaker norm than the norm for which the functional is twi ce differentiable and local optimality cannot be a priori deduced. We solve this problem for a particular but significant example. We prove "weak-coer civity" of the second derivative uniformly in a "strong" neighborhood of th e equilibrium shape.