
Journal of Convex Analysis 10 (2003), No. 2, 295324 Copyright Heldermann Verlag 2003 Relaxation of Variational Functionals with Piecewise Constant Growth Conditions Domenico Mucci Dip. di Matematica, Università di Parma, Via D'Azeglio 85/A, 43100 Parma, Italy, domenico.mucci@unipr.it [Abstractpdf] 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. FullTextpdf (754 K) 