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,$$ \int_{\Omega^{\star}}\sum_{i=1}^n\left|\frac{\partial u^{\star}}{\partial x_i}\right|^{p}\, dx\geq \int_{\Omega}\sum_{i=1}^n\left|\frac{\partial u}{\partial x_i}\right|^{p}\, dx.$$
@article{10428,
title = {Schwarz rearrangement does not decrease the energy of the pseudo p-Laplacian operator - doi: 10.5269/bspm.v29i1.10428},
journal = {Boletim da Sociedade Paranaense de Matem\'atica},
volume = {28},
year = {2010},
doi = {10.5269/bspm.v29i1.10428},
language = {EN},
url = {http://dml.mathdoc.fr/item/10428}
}
mohammed, moussa. Schwarz rearrangement does not decrease the energy of the pseudo p-Laplacian operator - doi: 10.5269/bspm.v29i1.10428. Boletim da Sociedade Paranaense de Matemática, Tome 28 (2010) . doi : 10.5269/bspm.v29i1.10428. http://gdmltest.u-ga.fr/item/10428/