Regularity for minimizers of non-autonomous non-quadratic functionals in the case $1 p 2$: an a priori estimate

Gentile, Andrea

Rendiconto della Accademia delle scienze fisiche e matematiche, Série 4, Tome 85 (2018) no. 1, pp. 185-200

Résumé

We establish an a priori estimate for the second derivatives of local minimizers of integral functionals of the form \begin{equation*}\mathcal{F}(\nu, \Omega) = \int_{\Omega} f(x, D\nu(x))\, dx \end{equation*} with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the $x$ variable belongs to a suitable Sobolev space. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.