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 $$\underset{u\in \mathcal{W}}{\inf }\frac{{\int }_{-\pi }^{\pi }[(u^{\prime }{)}^2-u^2]d\theta }{{\left[{\int }_{-\pi }^{\pi }|u|d\theta \right]}^2}$$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.