An existence result in a problem of the vectorial case of the calculus of variations. (English) Zbl 0822.49009
Summary: We prove that the problem
\[
\text{Minimize}\quad \int_ \Omega g(\Phi(\nabla T(x))) dx,\quad T\in T_ B+ W^{1, \infty}_ 0 (\Omega, \mathbb{R}^ n)
\]
admits at least one solution for any lower-semicontinuous extended-valued function \(g\), for any quasi-affine real-valued function \(\Phi\), and for any piecewise-affine boundary datum \(T_ B\) such that \(\Phi(\nabla T_ B)\) is constant.
MSC:
49J45 | Methods involving semicontinuity and convergence; relaxation |