Oscillating minimizers of a fourth order problem invariant under scaling
Résumé
By variational methods, we prove the inequality ∫\Ru″2dx−∫\Ru″u2dx≥I∫\Ru4dx∀u∈L4(\R)such thatu″∈L2(\R)
for some constant I∈(−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.