In this expository talk we introduce various PDE models which feature interfaces, i.e., lower dimensional manifolds across which the properties of a material and/or of the solutions vary. We distinguish two types: fixed and moving interfaces. The first is a fixed material interface, e.g., a thin region where the solutions exhibit large gradients, and thus the models feature a discontinuity in primary unknowns or fluxes. The second is a moving free interface (a free boundary) whose position is not known a-priori. Typically, the free boundary moves with a velocity proportional to, e.g., a jump of the fliuxes, such as in the well known Stefan problem for solid-liquid phase transitions, or is driven by the gradient of the solution such as in biofilm models. Studying PDEs with interfaces poses challenges which can be handled by a variety of regularization techniques or, e.g., by additional PDE models such as phase field (diffuse interface) models, or by using so called "broken" functional spaces. We overview some of the techniques from the literature and mention some of our recent work on the phase transition problems from applications.
The talk is connected to joint work with students and collaborators to be named in the talk.