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Oscillating minimizers of a fourth order problem invariant under scaling

Catto, Isabelle; Dolbeault, Jean; Benguria, Rafael; Monneau, Régis (2004), Oscillating minimizers of a fourth order problem invariant under scaling, Journal of Differential Equations, 205, 1, p. 253-269. http://dx.doi.org/10.1016/j.jde.2004.03.024

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Type
Article accepté pour publication ou publié
Date
2004
Journal name
Journal of Differential Equations
Volume
205
Number
1
Pages
253-269
Publication identifier
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Author(s)
Catto, Isabelle cc
Dolbeault, Jean cc
Benguria, Rafael
Monneau, Régis
Abstract (EN)
By variational methods, we prove the inequality $$\int_\R u''{}^2\,dx-\int_\R u''\,u^2\,dx\geq I\,\int_\R u^4\,dx\quad \forall\; u\in L^4(\R)\;\mbox{such that}\; u''\in L^2(\R) $$ for some constant $I\in (-9/64,-1/4)$. 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.
Subjects / Keywords
Fourth-order operators; Loss of compactness; Inequalities; Minimization; Scaling invariance; Euler–Lagrange equation; Lagrange multiplier; Shooting method; Commutator method for Lieb– Thirring inequalities; Lieb–Thirring inequalities