  We study the Muskat problem for one fluid or two fluids, with or without
viscosity jump, with or without rigid boundaries, and in arbitrary space
dimension $d$ of the interface. The Muskat problem is scaling invariant in the
Sobolev space $H^{s_c}(\mathbb{R}^d)$ where $s_c=1+\frac{d}{2}$. Employing a
paradifferential approach, we prove local well-posedness for large data in any
subcritical Sobolev spaces $H^s(\mathbb{R}^d)$, $s>s_c$. Moreover, the rigid
boundaries are only required to be Lipschitz and can have arbitrarily large
variation. The Rayleigh-Taylor stability condition is assumed for the case of
two fluids with viscosity jump but is proved to be automatically satisfied for
the case of one fluid. The starting point of this work is a reformulation
solely in terms of the Drichlet-Neumann operator. The key elements of proofs
are new paralinearization and contraction results for the Drichlet-Neumann
operator in rough domains.
