Relaxation of Variational Functionals with Piecewise Constant Growth Conditions
Domenico Mucci · Journal of convex analysis · 2003
We study the lower semicontinuous envelope of variational functionals given by \int f(x, Du)\,dx ∫ f ( x , D u ) d x , 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