Published Paper

Inserted: 31 may 2010
Last Updated: 5 may 2011

Journal: J. Differential Equations
Volume: 250
Number: 3
Year: 2011

Abstract:

We establish $C^{1,\gamma}$-partial regularity of minimizers of non autonomous convex integral functionals of the type: $ \mathcal{F}(u; \Omega):=\int_{\Omega}f(x, Du)\ dx , $ with non standard growth conditions into the gradient variable $$ \frac{1}{L}
\xi
^{p} \leq f(x,\xi) \leq L (1+
\xi
^{q}) $$ for a couple of exponents $p,q$ such that $$ 1< p\leq q< \min\left\{p \frac{n}{n-1}, p+1\right\} , $$ and $\alpha$- Hölder continuous dependence with respect to the $x$ variable. The significant point here is that the distance between the exponents $p$ and $q$ is independent of $\alpha$. Moreover this bound on the gap between the growth and the coercitivity exponents improves previous results in this setting.

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