Two singular limits: nonlinear diffusion and a free boundary problem

In the past decade there have been many results on the relationship between PDEs of porous medium type and variants of the Hele-Shaw problem. In particular, the latter arises in the limit of large diffusion exponent of the former. This is called the incompressible limit. In the first half of the talk, I will present a recent result along these lines; the novelty is that the limiting problem exhibits heterogeneity on the free boundary. This motivates the topic of the second half of the talk: stochastic homogenization of a Hele-Shaw problem with oscillation on both the interior and the free boundary. The result that I will describe holds only in one spatial dimension; however, it is new even in the setting of periodic coefficients. Based on joint works with Anthony Sulak and with Paul Yuming Zhang.