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We present a novel technique well suited to study the ground state ofinhomogeneous fermionic matter in a wide range of different systems. The systemis described using a Fermionic Shadow wavefunction (FSWF) and the energy iscomputed by means of the Variational Monte Carlo technique. The general form ofFSWF is useful to describe many--body systems with the coexistence of differentphases as well in the presence of defects or impurities, but it requiresovercoming a significant sign problem. As an application, we studied the energyto activate vacancies in solid $^3$He.

Fermionic Shadow Wavefunction Variational calculations of the vacancy formation energy in $^3$He