Oscillating minimizers of a fourth order problem invariant under scaling
Benguria, R. ; Catto, I. ; Dolbeault, J. ; Monneau, R.
arXiv, 0311192 / Harvested from arXiv
Text on arXiv
By variational methods, we prove the inequality: $$ \int_{\mathbb{R}} u''{}^2 dx-\int_{\mathbb{R}} u'' u^2 dx\geq I \int_{\mathbb{R}} u^4 dx\quad \forall u\in L^4({\mathbb{R}}) {such that} u''\in L^2({\mathbb{R}}) $$ 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 : 2003-11-12
Classification:  Mathematics - Analysis of PDEs,  Mathematical Physics,  35J35,  26D20,  47J20,  49J40 
@article{0311192,
     author = {Benguria, R. and Catto, I. and Dolbeault, J. and Monneau, R.},
     title = {Oscillating minimizers of a fourth order problem invariant under scaling},
     journal = {arXiv},
     volume = {2003},
     number = {0},
     year = {2003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0311192}
}

Benguria, R.; Catto, I.; Dolbeault, J.; Monneau, R. Oscillating minimizers of a fourth order problem invariant under scaling. arXiv, Tome 2003 (2003) no. 0, . http://gdmltest.u-ga.fr/item/0311192/