# Sharp regularity for functionals with (p,q) growth

created on 17 Dec 2001
modified on 16 Aug 2004

[BibTeX]

Published Paper

Inserted: 17 dec 2001
Last Updated: 16 aug 2004

Journal: J. Differential Equations
Volume: 204
Number: 1
Pages: 5-55
Year: 2004

Abstract:

We prove $W^{1,q}_{loc}$ regularity theorems for minimizers of functionals: $$\int{\Omega} f(x,Du) dx$$ where the integrand satisfies $(p,q)$ growth conditions: $z ^p < f(x,z) < L(1+ z ^q)$, with $p<q$. The main point here is the explicit dependence on the variable $x$ of the integrand. Energies of this type naturally arise in models for different physical situations (electrorheological fluids, thermistor problems, complex rheologies, homogenization). Differently from the case $p=q$, the regularity of minimizers depends on a subtle interaction between the growth of $f$ with respect to $z$ and its regularity with respect to the variable $x$. Indeed we prove that a sufficient condition for regularity is: $$qp < (n+\alpha)n$$ where $f$, roughly, is $\alpha$-Hölder continuous with respect to $x$ and $\Omega \subset R^n$. Such a condition is also sharp, as we show by mean of a counterexample. The results are carried out via a careful analysis of the Lavrentiev phenomenon associated to such functionals. We also solve a problem posed by Marcellini (J. Diff. Equ., 1991) showing a solution to a scalar variational problem that exhibits an isolated singularity in the interior.

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