Calculus of Variations and Geometric Measure Theory
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E. Acerbi - G. Mingione

Regularity results for a class of quasiconvex functionals with nonstandard growth

created on 30 Nov 2001
modified on 06 Jul 2002

[BibTeX]

Published Paper

Inserted: 30 nov 2001
Last Updated: 6 jul 2002

Journal: Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)
Volume: 30
Number: 2
Pages: 311-339
Year: 2001

Abstract:

We consider the integral functional $\int f(x,Du)\,dx$ under non standard growth assumptions of $(p,q)$-type: namely, we assume that $$
z
{p(x)}\le f(x,z)\le L(1+
z
{p(x)})$$ for some function $p(x) >1$, a condition appearing in several models from mathematical physics. Under sharp assumptions on the continuous function $p(x)$ we prove partial regularity of minimizers in the vector-valued case $u: \Omega \ (\subset R^{n}) \to R^{N}$, allowing for quasiconvex energy densities. This is, to our knowledge, the first regularity theorem for quasiconvex functionals under non standard growth conditions. The proof relies on a careful, intrisic, blow-up technique which is dictated by the oscillations of the function $p(x)$ and the size of $Du$ itself.

Keywords: Partial regularity, quasiconvexity, integral functionals, Nonstandard growth, Minimizers

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