On a new class of elastic deformations not allowing for cavitation (Q1904208)

The divergence identity \[ \partial_j\{(g^i\circ u)(\text{adj }Du)^j_i\}= (\text{div }g)\circ\text{det }Du \] is proved under weak assumptions. Also, an integrability result for \(\text{det }Du\) is proved. As an application, a degree formula is proved in the class \(A_{p,q}(\Omega)\). As a further application, the existence of an absolutely continuous minimizer of \[ I(u)=\int_\Omega W(x,u,Du)dx \] is proved if \(W\) is polyconvex and satisfies the inequality \(W(x,u,F)\geq a(|F|^p+|\text{adj }F|^q)\) with \(p\geq 2\) and \(q\geq 3/2\).