  Let $Q$ be a first-order differential operator on a compact, smooth oriented
Riemannian manifold with smooth boundary. Then, Friedrichs' extension lemma
states that the minimal closed extension $Q_{min}$ (the closure of the graph)
and the maximal closed extension $Q_{max}$ (in the sense of distributions) of
$Q$ in $L^p$-spaces ($1\leq p<\infty$) coincide. In the present paper, we show
that the same is true for boundary values with respect to $Q_{min}$ and
$Q_{max}$. This gives a useful characterization of weak boundary values,
particularly for $Q=d-bar$ the Cauchy-Riemann operator. As an application, we
derive the Bochner-Martinelli-Koppelman formula for $L^p$-forms with weak
d-bar-boundary values.
