Relaxation of variational functionals with piecewise constant growth conditions

We study the lower semicontinuous envelope of variational functionals under nonstandard growth conditions of (p; q)-type. If the growth exponent is piecewise constant, i.e., p(x) = p_i on each set of a smooth partition of the domain, we prove measure and representation property of the relaxed functional. We then extend the previous results by considering p(x) uniformly continuous on each set of the partition. We finally give an example of energy concentration in the process of relaxation.