On the singular set for Lipschitzian critical points of polyconvex functionals

Partial regularity is proved for Lipschitzian critical points of polyconvex functionals provided ‖Du‖L∞‖Du‖L∞ is small enough. In particular, the singular set for a Lipschitzian critical point has Hausdorff dimension strictly less than n   when ‖Du‖L∞‖Du‖L∞ is small enough. Model problems treated include∫Ω|∇u|2+|det∇u|2, where u:Ω(⊂R2)→R2u:Ω(⊂R2)→R2, and∫Ω|∇u|2+|∇u|s+|Ad∇u|s+|det∇u|s, where u:Ω(⊂R3)→R3 with s⩾2s⩾2. Moreover, it is shown that the singular set of a Lipschitzian global minimizer has Hausdorff dimension strictly less than n.