Oscillating minimizers of a fourth order problem invariant under scaling

Abstract

By variational methods, we prove the inequality: $${\int }_{\mathbb{R}}{u}^{\prime \prime }{}^{2}dx-{\int }_{\mathbb{R}}{u}^{\prime \prime }{u}^{2}dx\ge I{\int }_{\mathbb{R}}{u}^{4}dx\forall u\in L^4(\mathbb{R})suchthat{u}^{\prime \prime }\in L^2(\mathbb{R})$$ 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