The incompressible limit of an inhomogeneous model of tissue growth

We study a porous medium equation that models tissue growth in a heterogeneous environment. We show that, in the incompressible limit, solutions converge to those of a weak form of a Hele-Shaw type free boundary problem. To obtain enough compactness to take the limit, we establish an $L^4$ bound on the gradient of the pressure and an estimate of Aronson-Bénilan type.