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.