Local solvability and turning for the inhomogeneous Muskat problem

In this work we study the evolution of the free boundary between two different fluids in a porous medium where the permeability is a two dimensional step function. The medium can fill the whole plane $\mathbb{R}^2$ or a bounded strip $S=\mathbb{R}\times(-\pi/2,\pi/2)$. The system is in the stable regime if the denser fluid is below the lighter one. First, we show local existence in Sobolev spaces by means of energy method when the system is in the stable regime. Then we prove the existence of curves such that they start in the stable regime and in finite time they reach the unstable one. This change of regime (turning) was first proven in \cite{ccfgl} for the homogeneus Muskat problem with infinite depth