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Functions of bounded variation and rearrangements. (English) Zbl 1028.49035
The authors study the properties of the symmetric rearrangement \(u^*\) of a function \(u\) when \(u\) is of bounded variation in \({\mathbb R}^n\). Among these properties, the continuity and the approximate differentiability of \(u^*\) on the level sets \(\{u^*=t\}\) is investigated. For nonnegative compactly supported functions belonging to \(\text{BV}({\mathbb R}^n)\), the total variation of \(Du^*\) is bounded by the total variation of \(Du\). A similar result is proved for \(D^su^*\), the singular part (with respect to the Lebesgue measure) of the gradient, and for \(D^ju^*\) the most concentrated part of \(D^su^*\). Other properties are established, such as \(J_A(u^*) \leq J_A(u)\) where \(J_A\) is an integral functional associated with a Young function \(A\).

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
49N60 Regularity of solutions in optimal control
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
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