# A new partial regularity result for non autonomous convex integrals with non standard growth conditions

created by passarell on 31 May 2010
modified by demaria on 05 May 2011

[BibTeX]

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.