Schwarz rearrangement does not decrease the energy of the pseudo p-Laplacian operator - doi: 10.5269/bspm.v29i1.10428

Résumé

It is well known that the Schwarz symmetrization decrease the energy of the \textit{p}-Laplacian operator, i.e $$\int_{\overline{\Omega}}|\nabla u|^p\, dx\geq\int_{\overline{\Omega}^{\star}}|\nabla u^{\star}|^p\, dx.$$where $u^{\star}$ is the Schwarz rearranged function of $u$, for appropriate $u$ and $\Omega$. In this note, we shall proof that the Schwarz rearrangement does not decrease the energy of the pseudo \textit{p}-Laplacian operator, i.e there exist a function $u$ sucht that,$$ \int_{\Omega^{\star}}\sum_{i=1}^n\left|\frac{\partial u^{\star}}{\partial x_i}\right|^{p}\, dx\geq  \int_{\Omega}\sum_{i=1}^n\left|\frac{\partial u}{\partial x_i}\right|^{p}\, dx.$$