  In this work our main objective is to establish various (high frequency-)
uniqueness criteria. Initially, we consider $p-$Dirichlet type functionals on a
suitable class of measure preserving maps $u: B\subset \mathbb{R}^2 \mapsto
\mathbb{R}^2,$ $B$ being the unit disk, and subject to suitable boundary
conditions. In the second part we focus on a very similar situations only
exchanging the previous functionals by a suitable class of $p-$growing
polyconvex functionals and allowing the maps to be arbitrary. In both cases a
particular emphasis is laid on high pressure situations, where only uniqueness
for a subclass, containing solely of variations with high enough Fourier-modes,
can be obtained.
