Inserted: 9 apr 2002
Last Updated: 12 oct 2005
Journal: Communications in Partial Differential Equations
We introduce a variational approach to the Hele-Shaw flow $D_t k=\triangle u+f\chi$, $f\geq 0$ in $R^N$, where $k$ is the characteristic function of an open set $O(t)$ in $R^N$ and $u(t,\cdot)$ in $H^1_0(O(t))$ solves $-\triangle u(t,\cdot)=f$ in $O(t)$.
By choosing a time step $\tau_j=\tau/2^j$ and iteratively solving a variational problem in $R^N$, we construct a staircase family of opens sets and a corresponding family of functions: as $j\to\infty$, both sets and functions converge increasingly, at fixed time, to a weak solution of the problem. When the latter is not unique, the solution thus obtained is characterized by a minimality property, with respect to set inclusion, at fixed time.
We also prove several monotonicity results of the solutions thus obtained, with respect to both the initial set and the forcing term $f$. In particular, these monotonicity properties imply that $O(t)$ has finite perimeter for every $t$, provided that $O(0)$ has finite perimeter.
Finally, under very mild assumptions, we prove that the number of connected components in non increasing, that $O(t)$ is connected for large $t$ and that it tends to fill the whole of $R^N$.
Keywords: hele-shaw, evolution equation