Oscillating minimizers of a fourth order problem invariant under scaling

Rafael Benguria P. Universidad Catolica de Chile, Chile

By variational methods we prove the inequality

∫_{ℜ}(u”)^{2} dx − ∫_{ℜ} u”u^{2} dx ≥ I ∫_{ℜ} u^{4} dx,

for all u ∈ L^{4}(ℜ, dx), with u” ∈ L^{2}(ℜ, dx), for some 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 it does not depend on the period. This is joint work with I. Catto, J. Dolbeault, R. Monneau.