Linear deformations as global minimizers in nonlinear elasticity
of a ball B c K , composed of a homogeneous, isotropic, compressible, hyperelastic material, among those deformations f e WX'X{B, R3) that satisfy f(x) = Ax for x e dB. Ball showed that for A sufficiently large the smooth solution of the corresponding equilibrium (Euler-Lagrange) equations, f0(x) := Ax, is not the global minimizer of the total energy £ for a large class of stored energy functions W. Ball also showed that among those deformations that are radial, orientation preserving, and one-to-one, a global minimizer exists and exhibits a spherical hole at the center of the deformed body. A result of Knops and Stuart [8] shows that f0(x) = Ax is the global minimizer among smooth deformations. Thus functions that are not smooth must be considered if one wants to determine the global minimizer for such problems. Motivated by Ball's work others1 have shown that both radial minimizers and radial solutions of the equilibrium equations can exhibit spherical cavities under a wide range of constitutive hypotheses. It is not known, however, whether any such radial deformation is an absolute minimizer of the energy. In fact one does not even know that an absolute minimizer does indeed exist.2 In this note we give constitutive hypotheses under which the linear deformations f0(x) := Fqx are minimizers of the total energy among those deformations that are one-to-one, have the same boundary-values as f0, and whose range is contained in