Lower semicontinuous extension of multidimensional variational problems (Q1819357)

The following problem of calculus of variations is considered: \[ (P)\quad I(x(\cdot))=\int_{\Omega}f(t,x(t),\dot x(t))dt\to \inf,\quad x|_{\partial \Omega}=\xi. \] Above \(\Omega\) is an open, bounded and simply connected domain in \(R^ n\) with regular boundary \(\partial \Omega\), the integrand \(f: R^ n\times R^ m\times R^{nm}\to R\) and the bounded function \(\xi\) : \(\partial \Omega \to R^ m\) are given, and ẋ(t) stands for the Jacobian matrix: i.e. \(\dot x(t)=(\partial x^ i/\partial t_ j)\), \(i=1,...,m\), \(j=1,...,n\), where the partial derivatives are understood in the generalized sense. The important problem of semicontinuity of the functional I(x(\(\cdot))\) is investigated. The main result of the paper states that under some continuity and boundedness assumptions the lower semicontinuous extension (relaxation) of problem (P) in the space \(W^ 1_{\infty}(\Omega,R^ m)\) is the problem of the same type: \[ J(x(\cdot))=\int_{\Omega}\tilde f(t,x,\dot x)dt\to \inf,\quad x|_{\partial \Omega}=\xi, \] where \(\tilde f\) is the quasiconvexification in Morrey's sense of the integrand f with respect to the last argument. This theorem generalizes the known result in the one-dimensional case, where under some regularity assumptions the lower semicontinuity of the functional I is equivalent to the convexity of the integrand f with respect to the last variable.