Partial Regularity for Minimizers of Degenerate Polyconvex Energies

We prove partial regularity of minimizers for a class of polyconvex integral functionals

where

f f

is degenerate convex. Our class includes the model case

\int_\Omega (|Du|^p + |\text{Ad}\, Du|^p + |\text{det}\, Du|^p)\, dx. ∫ Ω ( ∣ D u ∣ p + ∣ Ad   D u ∣ p + ∣ det   D u ∣ p )   d x .

The method of proof involves a blow-up technique combined with a suitable asymptotic analysis of the degeneration nature of the first term

\int_\Omega |Du|^p\, dx ∫ Ω ∣ D u ∣ p   d x

.