Relaxation of Variational Functionals with Piecewise Constant Growth Conditions

We study the lower semicontinuous envelope of variational functionals given by

, for smooth functions

u u

, and equal to

+\infty + ∞

elsewhere, under nonstandard growth conditions of

(p,q) ( p , q )

-type: namely, we assume that

\vert z\vert^{p(x)}\leq f(x,z)\leq L(1+\vert z\vert^{p(x)})\,. ∣ z ∣ p ( x ) ≤ f ( x , z ) ≤ L ( 1 + ∣ z ∣ p ( x ) )   .

If the growth exponent is piecewise constant, i.e.,

p(x)\equiv p_i 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) p ( x )

uniformly continuous on each set of the partition. We finally give an example of energy concentration in the process of relaxation