A HELE-SHAW problem for tumor growth

We consider weak solutions to a problem modeling tumor growth. Under certain conditions on the initial data, solutions can be obtained by passing to the stiff (incompressible) limit in a porous medium type problem with a Lotka-Volterra source term describing the evolution of the number density of cancerous cells. We prove that such limit solutions solve a free boundary problem of Hele-Shaw type. We also obtain regularity properties, both for the solution and for its free boundary. The main new difficulty arises from the competition between the growth due to the source, which keeps the initial singularities, and the free boundary which invades the domain with a regularizing effect. New islands can be generated at singular times.