The course primarily focuses on elliptic variational inequalities (VIs) of obstacle type and their control. VIs are nonsmooth and nonlinear structures that are ubiquitous in applied mathematics. Starting with existence results and basic properties and theory, we'll move onto approximation by solutions of PDEs and related concepts. Sensitivity analysis and directional differentiability of solution maps of VIs will also be touched on, as well as optimal control with VI constraints: existence of controls and stationarity conditions in particular. The concluding part of the course will cover quasi-variational inequalities (QVIs), an exciting generalization of VIs with many interesting properties and an active area of ​​research. Machine learning aspects to do with solving VIs may also be presented depending on the interests of the audience.

Basic functional analysis. Knowledge of Sobolev spaces and theory of weak solutions of PDEs would be helpful, but I'll likely recap this in lectures. Feel free to contact me in case of questions.

Lectures: every Friday, 1100-1300 in room 2.006 (Seminarraum 20), Johann von Neumann House
Exercises: every second Friday, 0900-1100 in room 2.006 (Seminarraum 20), Johann von Neumann House

Sheet 1 (18th Oct)
Sheet 2 (2nd Nov)
Sheet 3 (16th Nov)

Regarding VI formulations of free boundary problems:
Coupled Bulk-Surface Free Boundary Problems Arising from a Mathematical Model of Receptor-Ligand Dynamics (section 9)
A variational inequality approach to Hele-Shaw flow with a moving boundary
The variational inequality approach to the one-phase Stefan problem
Variational Methods in the Stefan Problem

You can email me at amal (dot) alphonse (at) wias-berlin.de.