Construction of innitely many singular weak solutions to the equations of nonlinear elasticity

Radial deformations of a ball composed of a nonlinear elastic ma-terial and corresponding to cavitation have been much studied (see [7]). In this note we use rescalings to show that each such deformation can be used to construct innitely many non-symmetric, singular weak solutions of the equations of nonlinear elasticity for the same displace-ment boundary-value problem. Surprisingly, this property appears to have been unnoticed in the literature to date. Let IRn denote the reference state of a nonlinearly elastic body, let p 1, and let u:! IRn; u 2 W 1;p(; IRn); detru> 0 a:e:; denote a deformation of the body. In hyperelasticity we associate with each deformation the energy E(u) = Z W (ru(x)) dx; (1