2020
DOI: 10.1016/j.aim.2020.107380
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Rearrangement and polarization

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Cited by 7 publications
(15 citation statements)
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“…This follows easily from Lemma 8.1 below, first proved by Crowe, Zweibel, and Rosenbloom [11,Theorem 3] for Schwarz rearrangement without the assumption that F (s, 0) and F (0, t) decrease with s ≥ 0 and t ≥ 0, respectively. Variants of Proposition 3.7 are stated for general rearrangements by Brock and Solynin [7,Theorem 3.1] and by Van Schaftingen and Willem [43,Corollary 1], whose approaches to rearrangements differ from ours; see [6,Appendix] for a comparison.…”
Section: Appendixmentioning
confidence: 94%
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“…This follows easily from Lemma 8.1 below, first proved by Crowe, Zweibel, and Rosenbloom [11,Theorem 3] for Schwarz rearrangement without the assumption that F (s, 0) and F (0, t) decrease with s ≥ 0 and t ≥ 0, respectively. Variants of Proposition 3.7 are stated for general rearrangements by Brock and Solynin [7,Theorem 3.1] and by Van Schaftingen and Willem [43,Corollary 1], whose approaches to rearrangements differ from ours; see [6,Appendix] for a comparison.…”
Section: Appendixmentioning
confidence: 94%
“…essentially. This shows that by substituting ♦ * T for ♦ T in (3.10), we may take the supremum over R and thus bring the formula into line with those in [7] and [43]; see the discussion in [6,Appendix].…”
Section: (Smoothing and Kmentioning
confidence: 98%
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“…In this article, using polarization (also known as two-point symmetrization; cf. [8,13]), we prove various symmetries of an optimal triple on a more general class of domains (not necessarily simply connected) for the linear case (i.e., 𝑝 = 2 in (2)). To the best of our knowledge, there are no such results available in the literature regarding the symmetry properties of an optimal triple for the problem (2) on domains that are not simply connected, except a few counterexamples (for symmetry breaking) mentioned earlier.…”
Section: Introductionmentioning
confidence: 99%
“…We refer the reader to [11,61,67] for some examples and applications, as well as the article [53] by Milman and Rotem where the more general "size functionals" were studied. We also refer the reader to [9,14,17,25,32,40] for general background on rearrangements and symmetrizations of functions.…”
mentioning
confidence: 99%