Abstract

We study the lower semicontinuous envelope of variational functionals given by ∫f(x,Du) dx\int f(x, Du)\,dx, for smooth functions uu, and equal to +∞+\infty elsewhere, under nonstandard growth conditions of (p,q)(p,q)-type: namely, we assume that ∣z∣p(x)≤f(x,z)≤L(1+∣z∣p(x)) .\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)≡pip(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)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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Domenico Mucci

Dip. di Matematica, Università di Parma, Via D'Azeglio 85/A, 43100 Parma, Italy,

domenico.mucci@unipr.it

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Suggested citation

D. Mucci. “Relaxation of Variational Functionals with Piecewise Constant Growth Conditions.” Journal of Convex Analysis 10 (2003), No. 2, 295–324. https://doi.org/10.68381/jca1018

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Published by Heldermann Verlag, 2003. Rights now held by Banach Press LLC.