Topological spectral gap, multiscale Weyl's law, and homogenization in high-contrast PDEs

Abstract

This paper introduces a unified abstract variational and topological framework to characterize the spectral gap and eigenvalue distribution in high-contrast multiscale partial differential equations (PDEs). We rigorously prove that the exact location of the spectral gap is universally determined by the dimension of the local null space associated with the high-contrast inclusions. For systems with infinite-dimensional kernels, this location is strictly determined by the topological Betti numbers. Furthermore, we establish a multiscale Weyl's law via a spectral decoupling theorem, demonstrating that as the contrast approaches infinity, the multiscale spectrum bifurcates into two independent components: the Dirichlet spectrum of the background matrix and the internal Neumann spectrum of the inclusions. Using spectral homogenization theory, we also show that in the limit of vanishing periodicity, this expanding topological eigenspace asymptotically spans the entire spectral space of the macroscopic homogenized operator. These theoretical results are comprehensively verified through numerical experiments on diffusion, elasticity, fourth-order plate, Maxwell, and grad-div operators.