We consider a function $u:\Omega \to \Bbb R^N$, $\Omega \subset \Bbb R^n$, minimizing the integral $\int_\Omega(|D_1 u|^2 + \dots +|D_{n-1}u|^2 +|D_n u|^p)\,dx$, $2(n+1)/(n+3)\leq p<2$, where $D_i u = \partial u/ \partial x_i$, or some more general functional with the same behaviour; we prove the existence of second weak derivatives $D(D_1 u), \dots , D(D_{n-1} u) \in L^2$ and $D(D_n u) \in L^p$.
@article{119044,
author = {Paola Cavaliere and Anna D'Ottavio and Francesco Leonetti and Maria Longobardi},
title = {Differentiability for minimizers of anisotropic integrals},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {39},
year = {1998},
pages = {685-696},
zbl = {1060.49507},
mrnumber = {1715458},
language = {en},
url = {http://dml.mathdoc.fr/item/119044}
}
Cavaliere, Paola; D'Ottavio, Anna; Leonetti, Francesco; Longobardi, Maria. Differentiability for minimizers of anisotropic integrals. Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) pp. 685-696. http://gdmltest.u-ga.fr/item/119044/
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