In J.L Lions' book "Quelques méthodes de résolution des problèmes aux limites non linéaires" (page 196), the author considers a two-phase problem with moving boundary separating the interface. The equations governing the two phases $$\frac{du_1}{dt} - \sum_i \frac{d}{dx_i}\left(k(u_1)\frac{du_1}{dx_i}\right) = f \quad\text{for $u_1 < \lambda_0$}$$ $$\frac{du_2}{dt} - \sum_i \frac{d}{dx_i}\left(k(u_2)\frac{du_2}{dx_i}\right) = f \quad\text{for $u_2 > \lambda_0$}$$ are transformed by using a transformation of the form $K(\mu) = \int_0^{\mu}k(\lambda)d\lambda$.

Lions then introduces a multivalued map and then its inverse and proves well-posedness.

Can anyone recommend another source (book or paper or lecture notes) that addresses this topic (in particular the multivalued maps that arise from the transformation) in more detail (and in English)? Is it possible to summarise the state-of-the-art of an approach like this?

Thanks