Optimization of the principal eigenvalue of the Neumann Laplacian with indefinite weight and monotonicity of minimizers in cylinders

TL;DR

This paper investigates the optimization and properties of the principal eigenvalue of the Neumann Laplacian with indefinite weights, including existence, characterization of minimizers, and monotonicity in cylindrical domains, with applications to population dynamics.

Contribution

It provides new results on the continuity, convexity, and differentiability of the eigenvalue map, characterizes minimizers under certain conditions, and proves monotonicity of minimizers in cylindrical geometries.

Findings

Existence and characterization of eigenvalue minimizers.

Non-existence of eigenvalue maximizers under given conditions.

Monotonicity of minimizers in cylindrical domains.

Abstract

Let $\Omega\subset\mathbb{R}^N$, $N\geq 1$, be an open bounded connected set. We consider the indefinite weighted eigenvalue problem $-\Delta u =\lambda m u$ in $\Omega$ with $\lambda \in \mathbb{R}$, $m\in L^\infty(\Omega)$ and with homogeneous Neumann boundary conditions. We study weak* continuity, convexity and G\^ateaux differentiability of the map $m\mapsto1/\lambda_1(m)$, where $\lambda_1(m)$ is the principal eigenvalue. Then, denoting by $\mathcal{G}(m_0)$ the class of rearrangements of a fixed weight $m_0$, under the assumptions that $m_0$ is positive on a set of positive Lebesgue measure and $\int_\Omega m\,dx<0$, we prove the existence and a characterization of minimizers of $\lambda_1(m)$ and the non-existence of maximizers. Finally, we show that, if $\Omega$ is a cylinder, then every minimizer is monotone with respect to the direction of the generatrix. In the context of the population dynamics, this kind of problems arise from the question of determining the optimal spatial location of favourable and unfavourable habitats for a population to survive.