A Poincaré type inequality with three constraints
Résumé
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
$$
\inf_{u\in\mathcal{W}}
\frac{\displaystyle\int_{-\pi}^{\pi}[(u')^2-u^2]d\theta}{\displaystyle\left[\int_{-\pi}^{\pi}
|u| d\theta\right]^2}
$$
where $u\in \mathcal{W}$ is a $H^1(-\pi,\pi)$ periodic function, with
zero average on $(-\pi,\pi)$
and orthogonal to sine and cosine.
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