Geodesic
Regularity for minimizers of non-autonomous non-quadratic functionals in the case $1 p 2$: an a priori estimate
Rendiconto della Accademia delle scienze fisiche e matematiche, Série 4, Tome 85 (2018) no. 1, pp. 185-200.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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.
Ricaviamo una stima a priori per le derivate seconde di minimi locali di funzionali integrali del tipo \begin{equation*}\mathcal{F}(\nu, \Omega) = \int_{\Omega} f(x, D\nu(x))\, dx \end{equation*} con integranda convessa rispetto alla variabile gradiente, assumendo che la funzione che misura l’oscillazione dell’integranda rispetto alla variabile $x$ appartenga ad un opportuno spazio di Sobolev. La novità, qui, è che si tratta del caso in cui l’integranda soddisfi condizioni di crescita subquadratica rispetto alla variabile gradiente. 
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Gentile, Andrea. Regularity for minimizers of non-autonomous non-quadratic functionals in the case $1 < p < 2$: an a priori estimate. Rendiconto della Accademia delle scienze fisiche e matematiche, Série 4, Tome 85 (2018) no. 1, pp. 185-200. http://geodesic.mathdoc.fr/item/RASFM_2018_4_85_1_a4/

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