Homogenization of Fucik eigenvalues by optimal partition methods

TL;DR

This paper investigates the asymptotic behavior of Fucik eigenvalues in a bounded domain with periodic weights, providing precise convergence rate bounds as the parameter approaches zero.

Contribution

It introduces a method to accurately estimate the convergence rates of eigenvalue curves for homogenized Fucik problems with periodic weights.

Findings

Established bounds on convergence rates of eigenvalue curves

Demonstrated homogenization effects in Fucik eigenvalues

Extended understanding of eigenvalue asymptotics in periodic media

Abstract

Given a bounded domain $\Omega$ in $\mathbb{R}^N$, $N\geq 1$ we study the asymptotic behavior as $\varepsilon \to 0$ of the eigencurves of $$ -\Delta_p u_\varepsilon=\alpha_\varepsilon m(\tfrac{x}{\varepsilon})(u_\varepsilon^+ )^{p-1} - \beta_\varepsilon n(\tfrac{x}{\varepsilon})(u_\varepsilon^- )^{p-1} \quad \textrm{ in } \Omega $$ with Dirichlet boundary conditions, where $m$ and $n$ are bounded periodic weights. In this work we obtain accurate bounds of the convergence rates of these curves to some limit curves as $\varepsilon \to 0$.