A theorem on improving regularity of minimizing sequences by reverse Hölder inequalities (Q1380989)

Let \(M^{n\times m}\) denote the space of all real \(n\times m\)-matrices and consider a closed subset \(K\) of \(M^{n\times m}\) which satisfies \[ d_k(\lambda X)\leq k_0(d_k(X)+ 1) \] for all \(X\in M^{n\times m}\) and \(0\leq\lambda\leq 1\). Here \(k_0\) is some positive constant and \(d_k(\cdot)\) is the distance function of the set \(K\). On the Sobolev space \(W^1_p(D,\mathbb{R}^n)\), \(D\subset\mathbb{R}^m\), the author now introduces the energy \[ I_p(u, D)= \int_D d^p_k(\nabla u)dx, \] and shows that any weakly convergent minimizing sequence from \(W^1_p(D,\mathbb{R}^n)\) can be replaced by a minimizing sequence with higher integrability and better convergence properties. This result has applications in the theory of weakly almost conformal mappings.