Existence of minimizers for polyconvex and nonpolyconvex problems

We study the existence of Lipschitz minimizers of integral functionals I(u) = integral(Omega)(rho)(x, detDu(x)) dx, where Omega is an open subset of R-N with Lipschitz boundary, rho: O x(0,+infinity) --> [0,+infinity) is a continuous function, and u is an element of W-1,W- N(Omega, R-N), u(x) = x on partial derivative Omega. We consider both the cases of rho convex and nonconvex with respect to the last variable. The attainment results are obtained passing through the minimization of an auxiliary functional and the solution of a prescribed Jacobian equation.