In the talk we consider divergence form equations with sign-indefinite, real-valued coefficients in $d$ dimensions. We derive solution criteria for the induced divergence-form problem in $L_2$. With this we show that for any given piecewise constant coefficient $\alpha$ depending on $x_1$, only, there exists only a countable, nowhere dense set $\Lambda\subseteq \mathbb{R}$ such that $\operatorname{div} (\alpha-\lambda) \operatorname{grad}$ fails to be continuously invertible in $L_2$, where we understand that any solution is subject to homogeneous Dirichlet boundary conditions on the boundary of $\Omega =(0,1)\times \hat{\Omega}$, $\hat{\Omega}\subseteq \mathbb{R}^{d-1}$ open and bounded. Time permitting, we will also address an associated homogenisation problem and provide a generalised homogenisation method to address highly oscillatory ill-posed problems. In certain cases, the homogenised coefficients might lead to 4th order nonlocal operators, whereas the problems one started out with are second order and local only.
The talk is based on arXiv:2210.04650