Relaxation of Variational Functionals with Piecewise Constant Growth Conditions

We study the lower semicontinuous envelope of variational functionals given by $\int f(x, Du)\,dx$, for smooth functions $u$, and equal to $+\infty$ elsewhere, under nonstandard growth conditions of $(p,q)$-type: namely, we assume that $$\vert z\vert^{\px}\leq f(x,z)\leq L(1+\vert z\vert^{\px})\,.$$ If the growth exponent is piecewise constant, i.e., $\px\equiv p_i$\, 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 $\px$ uniformly continuous on each set of the partition. We finally give an example of energy concentration in the process of relaxation