Three Courses on Partial Differential Equations

Modeling, in particular with partial differential equations, plays an ever growing role in the applied sciences. Hence its mathematical understanding is an important issue for today's research. This book provides an introduction to three different topics in partial differential equations stemming from the applications for students and researchers with a basic background in this subject. Michel Chipot investigates equilibrium positions of several disks rolling on a wire. In particular, he discusses existence and uniqueness of, and the exact position for an equilibrium. Josselin Garnier considers problems arising from acoustics and geophysics where waves propagate in complicated media, the properties of which can only be described statistically. He shows in particular that if the different scales present in the problem can be separated, there exists a deterministic result. Otared Kavian is interested in so-called inverse problems, where one or several parameters of a partial differential equation need to be determined using, for example, measurements on the boundary of the domain. The question that arises naturally is what information is necessary to determine the unknown parameters. This lecture answers this question in different settings.