The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent

Abstract: "The characterization of the bulk energy density of the relaxation in W[superscript 1,p]([omega];R[superscript d]) of a functional F(u,[omega]) := [integral subscript omega] f([delta]u)dx is obtained for p > q-q/N, where u [element of] W[superscript 1,p]([omega];R[superscript d]), and f is a continuous function on the set of d x N matrices verifying 0[< or =] f([Xi]) [< or =] C(1 + [absolute value Xi][superscript q] for some constant C > 0 and 1 [< or =] q < +[infniity]. 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 integrability of the macroscopic strain [delta inverse]u. A recently introduced global method for relaxation and fine Sobolev trace and extension theorems are applied."