  We develop a new boundary condition for the weak inverse mean curvature flow,
which gives canonical and non-trivial solutions in bounded domains. Roughly
speaking, the boundary of the domain serves as an outer obstacle, and the
evolving hypersurfaces are assumed to stick tangentially to the boundary upon
contact. In smooth bounded domains, we prove an existence and uniqueness
theorem for weak solutions, and establish $C^{1,\alpha}$ regularity of the
level sets up to the obstacle. The proof combines various techniques, including
elliptic regularization, blow-up analysis, and certain parabolic estimates. As
an analytic application, we address the well-posedness problem for the usual
weak inverse mean curvature flow, showing in a broad generality that the
initial value problem always admits a unique maximal (or innermost) weak
solution.
