A Poincaré type inequality with three constraints

Abstract

In this paper, we consider a problem in calculus of variations motivated by a quantitative isoperimetric inequality in the plane. More precisely, the aim of this article is the computation of the minimum of the variational problem $$\begin{aligned}\inf _{u\in {\mathcal {W}}} \frac{\displaystyle \int _{-\pi }^{\pi }[(u')^2-u^2]d\theta }{\displaystyle \left[ \int _{-\pi }^{\pi } |u| d\theta \right] ^2} \end{aligned}$$ where a function $$u\in {\mathcal {W}}$$ is a $$H^1(-\pi ,\pi )$$ periodic function, with zero average on $$(-\pi ,\pi )$$ and orthogonal to sine and cosine.