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In this chapter, we will study the continuity of the map, \(u\in W^{1,p}({\Omega })\to u^{\prime }_{*}\in L^p({\Omega }_{*},k)\) . The motivations for this question have already been discussed in Chap. 2 , but in addition, the results of this chapter will be essential for the next chapters. The answer to the question was provided by Coron, Almgren and Lieb and is quite surprising.

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Continuity of the Derivative Map of the Monotone Rearrangement

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  • 发表日期：

    2025年

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  • Jean Michel Rakotoson

摘要

In this chapter, we will study the continuity of the map, \(u\in W^{1,p}({\Omega })\to u^{\prime }_{*}\in L^p({\Omega }_{*},k)\) . The motivations for this question have already been discussed in Chap. 2 , but in addition, the results of this chapter will be essential for the next chapters. The answer to the question was provided by Coron, Almgren and Lieb and is quite surprising.