Calculus of Variations and Geometric Measure Theory
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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.


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