THE EFFECTIVE BULK ENERGY OF THE RELAXED ENERGY OF MULTIPLE INTEGRALSBELOW THE GROWTH EXPONENT

The characterisation of the bulk energy density of the relaxation in W -1,W-p(Omega; R-d) of a functional F(u, Omega):= integral(Omega) f(del u) dx is obtained for p > q - q/N,where u epsilon W-1,W-p(Omega; R-d) , and f is a continuous function on the set of d x N matrices verifyin g 0 less than or equal to f(xi)less than or equal to C(1 + \xi\q) for some constant C > 0 and 1 less than or equal to q < + infinity. Typica l examples may be found in cavitation and related theories. Standard t echniques cannot be used due to the gap between the exponent q of the growth condition and the exponent p of the integrability of the macros copic strain del u. A recently introduced global method for relaxation and fine Sobolev trace and extension theorems are applied.