Symmetrisierung für Funktionen mehrerer reeller Variablen

Abstract

Imagine a potter's lathe with a lump of clay on the disc and on turning the wheel mould the lump into a perfectly symmetric shape, while retaining every particle of clay in its constant height over the disc. Now regard the lump of clay as a smooth, realvalued, nonnegative function u with compact support in ℝⁿ, and the symmetric shape obtained from it as its so called “symmetrized u^*”. The results are: The measures induced on ℝ₊ by u and u^* from the n-dimensional Lebesgue measure are identical and$\left\| {\nabla u^ * } \right\|_{Lp(\mathbb{R}^n )} \leqslant \left\| {\nabla u} \right\|_{Lp(\mathbb{R}^n )}$ for every 1≤p≤∞ This symmetrization process is applicable to proving Sobolew inequalities.