Geodesic
Differentiability for minimizers of anisotropic integrals
Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 4, pp. 685-696.

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We consider a function $u:\Omega \to \Bbb R^N$, $\Omega \subset \Bbb R^n$, minimizing the integral $\int_\Omega(|D_1 u|^2 + \dots +|D_{n-1}u|^2 +|D_n u|^p)\,dx$, $2(n+1)/(n+3)\leq p2$, 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 $D(D_n u) \in L^p$.
Classification : 35J50, 35J60, 49N60
Keywords: regularity; minimizers; integral functionals; anisotropic growth 
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     author = {Cavaliere, P. and D'Ottavio, A. and Leonetti, F. and Longobardi, M.},
     title = {Differentiability for minimizers of anisotropic integrals},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {685--696},
     publisher = {mathdoc},
     volume = {39},
     number = {4},
     year = {1998},
     mrnumber = {1715458},
     zbl = {1060.49507},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1998__39_4_a4/}
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Cavaliere, P.; D'Ottavio, A.; Leonetti, F.; Longobardi, M. Differentiability for minimizers of anisotropic integrals. Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 4, pp. 685-696. http://geodesic.mathdoc.fr/item/CMUC_1998__39_4_a4/