On $L_2$-uniqueness of symmetric diffusion equations

We discuss the uniqueness of solutions of the parabolic evolution equation $\partial\varphi/\partial t+H\varphi=0$ where $H=-\mathrm{div}(C\nabla)$ is a second-order degenerate elliptic operator acting on a domain $\Omega$ in $\mathbf{R}^d$. Specifically the matrix of coefficients $C$ is strictly positive but one does not have a strong ellipticity bound $C\geq\mu I>0$. Instead we assume that the coefficients are Lipschitz continuous with  $C\sim  d_\Gamma^{\,\delta}$ and $\mathrm{div} C\sim d_\Gamma^{\,\delta-1}$ near the boundary $\Gamma$ of $\Omega$ where $d_\Gamma$ is the Euclidean distance to the boundary.

In earlier work with Sikora and Lehrb{\"a}ck we proved that for a large class of domains, including ones with fractal boundaries,  the diffusion equation has a unique $L_1$-solution if and only if $\delta\geq 2-(d-d_H)$ where $d_H$ is the Hausdorff dimension of $\Gamma$. In this talk I consider the more difficult question whether $\delta\geq 2-(d-d_H)/2$ is a criterion for $L_2$-uniqueness. Results at this time are restricted to domains with $C^2$-boundaries or boundaries consisting of isolated points.