Homogenization of Fucik eigenvalues by optimal partition methods

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 $\varepsilon \to 0$.