Steiner symmetry in the minimization of the principal positive eigenvalue of an eigenvalue problem with indefinite weight
Claudia Anedda, Fabrizio Cuccu · arXiv (Cornell University) · 2020
In \cite{CC} the authors, investigating a model of population dynamics, find the following result. Let $\Omega\subset \mathbb{R}^N$, $N\geq 1$, be a bounded smooth domain. The weighted eigenvalue problem $-\Delta u =\lambda m u $ in $\Omega$ under homogeneous Dirichlet boundary conditions, where $\lambda \in \mathbb{R}$ and $m\in L^\infty(\Omega)$, is considered. The authors prove the existence of minimizers $\check m$ of the principal positive eigenvalue $\lambda_1(m)$ when $m$ varies in a class $\mathcal{M}$ of functions where average, maximum, and minimum values are given. A similar result is obtained in \cite{CCP} when $m$ is in the class $\mathcal{G}(m_0)$ of rearrangements of a fixed $m_0\in L^\infty(\Omega)$. In our work we establish that, if $\Omega$ is Steiner symmetric, then every minimizer in \cite{CC,CCP} inherits the same kind of symmetry.