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 $D(D_n u) \in L^p$.
Differentiability for minimizer of anisotropic integrals
CAVALIERE, Paola;
1998
Abstract
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 $D(D_n u) \in L^p$.File in questo prodotto:
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