Symmetrization of measures and the one-dimensional Poisson equation with Dirichlet boundary conditions

Let $\mu$ be a finite Borel measure on $(-\pi ,\pi )$. Consider the one-dimensional Poisson equation $-u^{\mathrm{\prime }\mathrm{\prime }}=\mu$, where equality holds in the sense of distributions, with Dirichlet boundary conditions $u(\pm \pi )=0$. In this paper, we define measures that are transformations of $\mu$, we compare the convex integral means of the original solutions $u_{\mu }$ and the transformed ones, and we prove the uniqueness of a solution that maximizes the convex integral means.