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We saw in the previous chapter that if \({\Omega }\) is a connected bounded open set \(u\in W^{1,1}_{loc}({\Omega })\) , then its decreasing rearrangement \(u_*\) is in \(W^{1,1}_{loc}({\Omega }_*)\) . It turns out that when we know a little about the regularity of the boundary of \({\Omega }\) then we can relate the derivative \(u^{\prime }_*\) of \(u_*\) with the relative rearrangement of \(|\nabla u|\) with respect to u when \(u\in W^{1,1}({\Omega })\) , which leads us to an inequality of the form \(\displaystyle {} -\dfrac {du_*}{ds}(s)\leqslant K(s;{\Omega })\cdot |\nabla u|_{*u}(s)\ \ a.e\ in\ {\Omega }_*. \)

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Pointwise Inequalities and Sobolev Inclusions

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  • 发表日期：

    2025年

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  • Jean Michel Rakotoson

摘要

We saw in the previous chapter that if \({\Omega }\) is a connected bounded open set \(u\in W^{1,1}_{loc}({\Omega })\) , then its decreasing rearrangement \(u_*\) is in \(W^{1,1}_{loc}({\Omega }_*)\) . It turns out that when we know a little about the regularity of the boundary of \({\Omega }\) then we can relate the derivative \(u^{\prime }_*\) of \(u_*\) with the relative rearrangement of \(|\nabla u|\) with respect to u when \(u\in W^{1,1}({\Omega })\) , which leads us to an inequality of the form \(\displaystyle {} -\dfrac {du_*}{ds}(s)\leqslant K(s;{\Omega })\cdot |\nabla u|_{*u}(s)\ \ a.e\ in\ {\Omega }_*. \)