Lower semicontinuous extension of the fundamental problem of calculus of variations (Q914139)

The multidimensional variational problem \[ (P)\quad J(x)=\int_{\Omega}f(t,x(t),x'(t))dt\quad \to \quad Inf,\quad x(\cdot)|_{\Gamma}=\phi \] is considered, where \(x(\cdot)=(x^ 1(\cdot),...,x^ m(\cdot))\) is a vector function in \(\bar W^ 1_{\infty}(\Omega,R^ m)=W^ 1_{\infty}(\Omega,R^ m)\cap C^ m({\bar \Omega})\) and \(x'(t)=\{\partial x^ i(t)/\partial t_ j\}.\) The author gives the definitions of quasiconvexification of a function f and lower semicontinuous extension of the problem (P). In the main theorem a sufficient condition for the quasiconvexification of f and the lower semicontinuous extension of (P) is given.