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We consider the problem of essential self-adjointness of the drift-diffusionoperator $H=-\frac{1}{\rho}\nabla\cdot \rho \mathbb D\nabla +V$ on domains$\Omega \subset \mathbb{R}^d$ with $\mathcal{C}^2$-boundary $\partial \Omega$and for large classes of coefficients $\rho,\; \mathbb{D}$ and $V$. We givecriteria showing how the behavior as $x \rightarrow \partial \Omega$ of thesecoefficients balances to ensure essential self-adjointness of $H$. On the waywe prove a weak anisotropic Hardy inequality which is of independent interest.

Weak anisotropic Hardy inequality: essential self-adjointness of drift-diffusion operators on domains in $\mathbb{R}^d$, revisited