Local Boundedness for Minimizers of Some Polyconvex Integrals

We give a regularity result for local minimizers $u:Omega subset mathbb{R}^3 o mathbb{R}^3$ of a special class of polyconvex functionals. Under some structure assumptions on the energy density, we prove that local minimizers $u$ are locally bounded. For each component $u^{alpha}$ of $u$, we first prove a Caccioppoli's inequality and then apply De Giorgi's iteration method to get the boundedness of $u^{alpha}$. Our result can be applied to the polyconvex integral [int_Omega (sum_{alpha = 1}^{3} |D u^alpha|^{p} + |adj_2 Du|^q + |det Du|^{r}) dx] with suitable p,q,r >1.