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Elsevier, Journal of Differential Equations, 1(205), p. 253-269, 2004

DOI: 10.1016/j.jde.2004.03.024

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Oscillating minimizers of a fourth-order problem invariant under scaling

Journal article published in 2004 by Rafael D. Benguria, Isabelle Catto, Jean Dolbeault ORCID, Régis Monneau
This paper is made freely available by the publisher.
This paper is made freely available by the publisher.

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Abstract

By variational methods, we prove the inequality: $ ∫_{ℝ} u''{}^2 dx-∫_{ℝ} u'' u^2 dx≥ I ∫_{ℝ} u^4 dx\quad ∀ u𝟄 L^4({ℝ}) {such that} u''𝟄 L^2({ℝ}) $ for some constant $I𝟄 (-9/64,-1/4)$. This inequality is connected to Lieb-Thirring type problems and has interesting scaling properties. The best constant is achieved by sign changing minimizers of a problem on periodic functions, but does not depend on the period. Moreover, we completely characterize the minimizers of the periodic problem. ; Comment: 19 pages, 2 figures