  This paper has its motivation in the study of the Fractional Quantum Hall
Effect. We consider 2D quantum particles submitted to a strong perpendicular
magnetic field, reducing admissible wave functions to those of the Lowest
Landau Level. When repulsive interactions are strong enough in this model,
highly correlated states emerge, built on Laughlin's famous wave function. We
investigate a model for the response of such strongly correlated ground states
to variations of an external potential. This leads to a family of variational
problems of a new type. Our main results are rigorous energy estimates
demonstrating a strong rigidity of the response of strongly correlated states
to the external potential. In particular we obtain estimates indicating that
there is a universal bound on the maximum local density of these states in the
limit of large particle number. We refer to these as incompressibility
estimates.
