Finding nicer mappings with lower or equal energy in nonlinear elasticity

In the framework of nonlinear elasticity, we consider the model energy F(u)=∫Ω[|Du(x)|p+h(det⁡Du(x))]dx, where u:Ω⊂Rn→Rn with det⁡Du>0 and h:(0,+∞)→[0,+∞) is convex; moreover h(t) blows up when t→0+. We study the problem: fix the mapping u with finite energy F(u) and find a mapping v with the same boundary values, with det⁡Dv>0 and energy F(v) not higher than F(u), such that every component vβ of v enjoys a kind of maximum and minimum principle. Due to the constraint det⁡Dv>0, a truncation argument does not work. On the contrary, the constraint det⁡Dv>0 makes the problem easy when p≥n since it is known that every component is weakly monotone, so we can take v=u. In the present work we address the case 2≤p