My main interests are in the mathematical analysis of free boundary and interface problems arising in continuum theories. These are nonlinear partial differential equations (PDE) problems set on a unknown evolving domain moved by the unknown velocity of the fluid. They arise in different physical contexts: compressible Euler equations in gas dynamics, incompressible Euler equations in fluid dynamics, interaction between a fluid phase and an elastic body. The question of the well-posedness in Sobolev spaces for these problems is an outstanding question in PDE analysis, as geometric quantities linked to the unknown domain are here a leading order term.