Local existence of classical solutions to the well-posed Hele-Shaw problem. (Q1871551)

The authors prove the (locally in time) existence of a classical solution to the Hele-Shaw problem consisting of the equation \[ -\Delta p=f(x)\equiv\text{ div} F, \] \noindent defined in a two- or three-dimensional domain \(\Omega\), whose boundary is given by a finite number of connected moving (i.e. time dependent) and fixed components. The equation is also coupled with appropriate boundary conditions on the fixed part as well as on the moving part of the boundary \(\partial\Omega\). The novelty of the result, with respect to previously published ones, is due to the fact that general boundary conditions on the fixed part are allowed, and in particular to the method used. Indeed, the authors obtain the existence of a solution as limit of solutions of the approximating one-phase Stefan problems \[ \varepsilon{\partial\theta^\varepsilon\over\partial t}-\Delta\theta^\varepsilon=f, \] defined on a suitable domain \(\Omega_\varepsilon(t)\), and coupled with suitable boundary conditions.