A remark on a stability criterion for the radial cavitating map in nonlinear elasticity

We study the integral functional I(w):=∫B|adj∇w(w|w|3)|qdx on suitable maps w:B⊂R3→R3 and where 2q∈(2,3). The inequality I(w)≥I(i), which we establish on a subclass of the admissible maps, was first proposed in [13] as one of two possible necessary conditions for the stability, i.e. local minimality, of the radial cavitating map in nonlinear elasticity. Here, i is the identity map. Admissible maps w either do not vanish (and in this case possess a single discontinuity x0 in B which produces a cavity about the origin), or vanish at exactly one point x0 in B, in which case w is a diffeomorphism in a neighbourhood of x0. We show that I(⋅) behaves like a polyconvex functional and associate with it another functional, K(⋅), satisfying I(w)≥I(i)+q(K(w)−K(i)). We give conditions under which K(w)=K(i), and from these infer I(w)≥I(i). It is also shown that (i) K is strictly decreasing along paths of admissible functions that move x0 away from the origin and (ii) K(w) exhibits some quite pathological behaviour when w is sufficiently close to i.