A Note on Semiconcave Function |
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Authors: | Renjun Duan Changjiang Zhu |
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Institution: | 1. Laboratory of Nonlinear Analysis, Department of Mathematics , Central China Normal University , Wuhan, 430079, P.R. China;2. Wuhan Institute of Physics and Mathematics , The Chinese Academy of Sciences , Wuhan, 430071, P.R. China;3. Laboratory of Nonlinear Analysis, Department of Mathematics , Central China Normal University , Wuhan, 430079, P.R. China |
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Abstract: | This article gives a simple proof of an equivalent proposition on semiconcave function (see L.C. Evans (1998). Partial Differential Equations. American Mathematical Society; p. 130]). The proof of sufficiency of the proposition can be easily obtained. We prove its necessity by three steps: First, we prove that the equivalent proposition holds for discrete points <artwork name="GAPA31045ei1">; Secondly, we obtain continuity of semiconcave function; Finally, by using the fact that the sequences λm k are dense in the interval (0, 1), we prove that the equivalent proposition holds for each λ ∈ (0, 1). |
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Keywords: | Semiconcave Closed Interval Noose Induction Ams Subject Classification: 26b25 |
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