We introduce a generalized version of the Jang equation, designed
for the general case of the Penrose Inequality in the setting of
an asymptotically flat space-like hypersurface of a spacetime
satisfying the dominant energy condition. The appropriate
existence and regularity results are established in the special
case of spherically symmetric Cauchy data, and are applied to give
a new proof of the general Penrose Inequality for these data sets.
When appropriately coupled with an inverse mean curvature flow,
analogous existence and regularity results for the associated
system of equations in the nonspherical setting would yield a
proof of the full Penrose Conjecture. Thus it remains as an
important and challenging open problem to determine whether this
system does indeed admit the desired solutions.