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.
Publié le : 2004-07-05
Classification:
Minimization,
Inequalities,
Fourth-order operators,
Loss of compactness,
Scaling invariance,
Euler–Lagrange equation,
Lagrange multiplier,
Lieb–Thirring inequalities,
Commutator method for Lieb– Thirring inequalities,
Shooting method,
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
@article{hal-00157514,
author = {Catto, Isabelle and Dolbeault, Jean and Benguria, Rafael and Monneau, R\'egis},
title = {Oscillating minimizers of a fourth order problem invariant under scaling},
journal = {HAL},
volume = {2004},
number = {0},
year = {2004},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00157514}
}
Catto, Isabelle; Dolbeault, Jean; Benguria, Rafael; Monneau, Régis. Oscillating minimizers of a fourth order problem invariant under scaling. HAL, Tome 2004 (2004) no. 0, . http://gdmltest.u-ga.fr/item/hal-00157514/