  The free boundary problem for a two-dimensional fluid filtered in porous
media is studied. This is known as the one-phase Muskat problem and is
mathematically equivalent to the vertical Hele-Shaw problem driven by gravity
force. We prove that if the initial free boundary is the graph of a periodic
Lipschitz function, then there exists a global-in-time Lipschitz solution in
the strong $L^\infty_t L^2_x$ sense and it is the unique viscosity solution.
The proof requires quantitative estimates for layer potentials and pointwise
elliptic regularity in Lipschitz domains. This is the first construction of
unique global strong solutions for the Muskat problem with initial data of
arbitrary size.
