Local Boundedness for Minimizers of Some Polyconvex Integrals

We give a regularity result for local minimizers u: Ω ⊂ R3→ R3 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 uα of u, we first prove a Caccioppoli’s inequality and then apply De Giorgi’s iteration method to get the boundedness of uα. Our result can be applied to the polyconvex integral ∫Ω(∑α=13|Duα|p+|adj2Du|q+|detDu|r)dxwith suitable p, q, r> 1.