  We consider first-order differential operators with locally bounded
measurable coefficients on vector bundles with measurable coefficient metrics.
Under a mild set of assumptions, we demonstrate the equivalence between the
essential self-adjointness of such operators to a negligible boundary property.
When the operator possesses higher regularity coefficients, we show that higher
powers are essentially self-adjoint if and only if this condition is satisfied.
In the case that the low-regularity Riemannian metric induces a complete length
space, we demonstrate essential self-adjointness of the operator and its higher
powers up to the regularity of its coefficients. We also present applications
to Dirac operators on Dirac bundles when the metric is non-smooth.
