Laminated Metamaterials - Divergence Form Equations and Homogenisation

Explore divergence form equations with sign-indefinite, real-valued coefficients in d dimensions in this 23-minute talk from the Workshop on "Spectral Theory of Differential Operators in Quantum Theory" at the Erwin Schrödinger International Institute. Discover solution criteria for the induced divergence-form problem in L_2 and learn how, for any given piecewise constant coefficient α depending on x_1, there exists only a countable, nowhere dense set Λ⊆ℝ where div(α-λ)grad fails to be continuously invertible in L_2. Examine homogeneous Dirichlet boundary conditions on the boundary of Ω=(0,1)×Ω̂, with Ω̂⊆ℝ^(d-1) open and bounded. Delve into an associated homogenisation problem and a generalised homogenisation method for highly oscillatory ill-posed problems. Understand how homogenised coefficients can lead to 4th order nonlocal operators in certain cases, despite starting with second-order, local problems. Based on arXiv:2210.04650, this talk by Marcus Waurick provides insights into laminated metamaterials and their mathematical properties.