Oscillating minimizers of a fourth-order problem invariant under scaling
Copyright policy )Oscillating minimizers of a fourth-order problem invariant under scaling
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.
19 pages, 2 figures
- Panthéon-Assas University France
- French National Centre for Scientific Research France
- University of Paris France
- PSL Research University France
- Paris Dauphine University France
Commutator method for Lieb–Thirring inequalities, Euler–Lagrange equation, fourth-order operators, Existence theories for free problems in one independent variable, FOS: Physical sciences, Lagrange multiplier, 510, 35J35; 26D20; 47J20; 49J40, Mathematics - Analysis of PDEs, 515, inequalities, Commutator method for Lieb– Thirring inequalities, Scaling invariance, Ecuaciones de Lagrange, Differential inequalities involving functions of a single real variable, Lieb--Thirring inequalities, FOS: Mathematics, minimization, 47J20, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 26D20, 49J40, Mathematical Physics, Matemática física y química, Shooting method, commutator method for Lieb--Thirring inequalities, shooting method, Mathematical Physics (math-ph), Desigualdades (Matemáticas), Analyse, Minimization, loss of compactness, Fourth-order operators, scaling invariance, Euler--Lagrange equation, Lieb–Thirring inequalities, 35J35, Inequalities, Loss of compactness, Analysis, Analysis of PDEs (math.AP)
Commutator method for Lieb–Thirring inequalities, Euler–Lagrange equation, fourth-order operators, Existence theories for free problems in one independent variable, FOS: Physical sciences, Lagrange multiplier, 510, 35J35; 26D20; 47J20; 49J40, Mathematics - Analysis of PDEs, 515, inequalities, Commutator method for Lieb– Thirring inequalities, Scaling invariance, Ecuaciones de Lagrange, Differential inequalities involving functions of a single real variable, Lieb--Thirring inequalities, FOS: Mathematics, minimization, 47J20, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 26D20, 49J40, Mathematical Physics, Matemática física y química, Shooting method, commutator method for Lieb--Thirring inequalities, shooting method, Mathematical Physics (math-ph), Desigualdades (Matemáticas), Analyse, Minimization, loss of compactness, Fourth-order operators, scaling invariance, Euler--Lagrange equation, Lieb–Thirring inequalities, 35J35, Inequalities, Loss of compactness, Analysis, Analysis of PDEs (math.AP)
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