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

TL;DR

This paper investigates conditions under which drift-diffusion operators on domains in Euclidean space are essentially self-adjoint, introducing a new weak anisotropic Hardy inequality that aids in understanding boundary behavior of coefficients.

Contribution

It provides new criteria linking coefficient behavior near boundaries to self-adjointness and proves a novel weak anisotropic Hardy inequality of independent interest.

Findings

Criteria for essential self-adjointness based on boundary coefficient behavior

Introduction of a weak anisotropic Hardy inequality

Application to drift-diffusion operators on domains in alf6d

Abstract

We consider the problem of essential self-adjointness of the drift-diffusion operator $H=-\frac{1}{\rho}\nabla\cdot \rho \mathbb D\nabla +V$ 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 give criteria showing how the behavior as $x \rightarrow \partial \Omega$ of these coefficients balances to ensure essential self-adjointness of $H$. On the way we prove a weak anisotropic Hardy inequality which is of independent interest.