Free Boundary Problems

Around one third of all the differential equation models arising in industrial applications are free boundary problems. These are ones in which the region in which the equation has to be solved is not known in advance, but rather has to be found as part of the solution. Famous examples are the Stefan problem for melting ice or freezing water, or the optimal exercise strategy for American options, but bubbles, jets, shock waves, flames, tumour growth, crack propagation and contact problems, can also be classified in this heading. Mathematics can contribute most when it suggests a way of formulating the problem so that the free boundary only appears implicitly. This means that whether a numerical or analytical approach is attempted, the free boundary can be allowed to undergo major geometrical convolutions. This is necessary for example in wave breaking or in dendritic growth during solidification or in unstable "Hele-Shaw flow" between closely spaced parallel plates, or in secondary oil recovery.

People working in this area within OCIAM are