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 
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/
@article{RASFM_2018_4_85_1_a4,
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     title = {Regularity for minimizers of non-autonomous non-quadratic functionals in the case $1 < p < 2$: an a priori estimate},
     journal = {Rendiconto della Accademia delle scienze fisiche e matematiche},
     pages = {185--200},
     year = {2018},
     volume = {Ser. 4, 85},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RASFM_2018_4_85_1_a4/}
}
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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.