Existence of Minimizers for Polyconvex and Nonpolyconvex Problems

We study the existence of Lipschitz minimizers of integral functionals I(u)=(integral)(Omega) (phi)(x,det,Du(x)) where (Omega) is an open subset of R^N with Lipschitz boundary, (phi):(Omega)*(0,+(infinity)(right arrow)(0,+(infinity)) is a continuous function, and u in W^1,N((Omega),R^N), u(x)=x on (partial)(Omega). We consider both the cases of (phi) 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.