Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity

Abstract

It is still not known if the radial cavitating minimizers obtained by Ball [J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. R. Soc. Lond. A 306 (1982) 557–611] (and subsequently by many others) are global minimizers of any physically reasonable nonlinearly elastic energy. We therefore consider in this paper the related problem of obtaining necessary conditions for these radial solutions to be minimizers with respect to nonradial perturbations. A standard blowup argument applied to either an inner or an outer variation yields an apparently new inequality that, for most constitutive relations, has yet to be verified. However, in the special case of a compressible neo-Hookean material, $W(\mathbf{F})=\frac{\mu }{2}|\mathbf{F}{|}^2+h(\det \mathbf{F})$, we show that the inequality produced by an outer variation clearly holds whilst that produced by an inner variation is a well-known inequality (first proven by Brezis, Coron, and Lieb [H. Brezis, J.-M. Coron, E.H. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107 (1986) 649–705] ) which arises in the theory of nematic liquid crystals:

for all $\mathbf{n}\in W^{1,2}(B,\partial B)$ (so that $|\mathbf{n}|=1$ a.e.) that satisfy $\mathbf{n}(\mathbf{x})=\mathbf{x}/|\mathbf{x}|$ on $\partial B$, where $B$ is the unit ball in ${\mathbb{R}}^3$.