Schwarz rearrangement does not decrease the energy for the pseudo p-Laplacian operator

Abstract

It is well known that the Schwarz symmetrization decrease the energyfor the p-Laplacian operator, i.eint_{overline{Omega}}|∇u|^p dx ≥ int_{overline{Omega^*}}|∇u^⋆|p dx,where u⋆ 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 for the pseudo p-Laplacian operator, that is, there exist a bounded domain Omega ⊂ R^N and a function u ∈ W_0^{1,p}(Omega)such thatint_{Omega^*}sum_{i=1}^N |partial_{x_i}u^*|^p dx geq int_{Omega}sum_{i=1}^N |partial_{x_i}u|^p dx.

Keywords

• Schwarz symmetrization
• pseudo p-Laplacian operator.