We discuss the stability of "critical" or "equilibrium" shapes ofa shape-dependent energy functional. We analyze a problem arising whenlooking at the positivity of the second derivative in order to provethat a critical shape is an optimal shape. Indeed, often whenpositivity -or coercivity- holds, it does for a weaker norm than thenorm for which the functional is twice differentiable and localoptimality cannot be a priori deduced. We solve this problem for aparticular but significant example. We prove "weak-coercivity" ofthe second derivative uniformly in a "strong" neighborhood of theequilibrium shape.
