The characterisation of the bulk energy density of the relaxation in W1, P(Ω; ℝd) of a functional
is obtained for p > q – q/N, where u ∈ W1, P(Ω; ℝd), and f is a continuous function on the set of d × N matrices verifying
for some constant C > 0 and 1 ≦ q < + ∞. Typical examples may be found in cavitation and related theories. Standard techniques cannot be used due to the gap between the exponent q of the growth condition and the exponent p of the integrability of the macroscopic strain ∇u. A recently introduced global method for relaxation and fine Sobolev trace and extension theorems are applied.