Calculus of Variations and Geometric Measure Theory

D. Mucci

Relaxation of variational functionals with piecewise constant growth conditions

created on 20 Dec 2001
modified on 17 Dec 2003


Published Paper

Inserted: 20 dec 2001
Last Updated: 17 dec 2003

Journal: J. Convex Anal.
Volume: 10
Number: 2
Pages: 295-324
Year: 2003


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 $$ \vert z\vert{p(x)}\leq f(x,z)\leq L(1+\vert z\vert{p(x)})\,. $$ If the growth exponent is piecewise constant, i.e., $p(x)\equiv p_i$\, 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 $p(x)$ uniformly continuous on each set of the partition. We finally give an example of energy concentration in the process of relaxation.