An Existence Result in a Problem of the Vectorial Case of the Calculus of Variations

Abstract

We prove that the problem \[{\text{Minimize}}\,\int_\Omega {g(\Phi (\nabla T(x)))dx,\quad T \in T_B + W_0^{1,\infty } (\Omega ,\mathbb{R}^n )} \] admits at least one solution for any lower-semicontinuous extended valued function g, for any quasiaffine real-valued function $\Phi $, and for any piecewise-affine boundary datum $T_B$ such that $\Phi (\nabla T_B )$ is constant.