  We derive conditions that ensure the existence of a bounded
$H_\infty$-calculus in weighted $L_p$-Sobolev spaces for closed extensions
$\underline{A}_T$ of a differential operator $A$ on a conic manifold with
boundary, subject to differential boundary conditions $T$. In general, these
conditions ask for a particular pseudodifferential structure of the resolvent
$(\lambda-\underline{A}_T)^{-1}$ in a sector $\Lambda\subset\mathbf{C}$. In
case of the minimal extension they reduce to parameter-ellipticity of the
boundary value problem $(A,T)$. Examples concern the Dirichlet and Neumann
Laplacians.
