| Popis: |
We consider the higher differentiability of solutions to the problem of minimising \begin{document}$\int_{Ω} [L(\nabla v(x))+g(x, v(x))]dx~~~ \hbox {on}~~~ u^0+W^{1, p}_0(Ω)$ \end{document} where \begin{document}$\Omega\subset \mathbb R^N$\end{document} , \begin{document}$L(ξ) = l(|ξ|) = \frac{1}{p}|ξ|^p$\end{document} and \begin{document}$ u^0∈ W^{1, p}(Ω)$\end{document} and hence, in particular, the higher differentiability of weak solution to the equation \begin{document}${\rm div }(|\nabla u|^ {p-2}\nabla u) = f.$ \end{document} We show that, for \begin{document}$3≤ p , under suitable assumptions on \begin{document}$g$\end{document} , there exists a solution \begin{document}$ u^*$\end{document} to the Euler-Lagrange equation associated to the minimisation problem, such that \begin{document}$\nabla u^*∈ W^{s, 2}_{loc}(\Omega)$ \end{document} for \begin{document}$0 . In particular, for \begin{document}$p = 3$\end{document} , we show that the solution \begin{document}$u^*$\end{document} is such that \begin{document}$\nabla u^*∈ W^{s, 2}_{loc}(\Omega)$\end{document} for every \begin{document}$s . This result is independent of \begin{document}$N$\end{document} . We present an example for \begin{document}$N = 1$\end{document} and \begin{document}$p = 3$\end{document} whose solution \begin{document}$u$\end{document} is such that \begin{document}$\nabla u^*$\end{document} is not in \begin{document}$W^{1, 2}_{loc}(\Omega)$\end{document} , thus showing that our result is sharp. |
