Boundary Conditions are of the Devil

Jean Jacque Lions was a remarkable French mathematician, with multiple students of note currently working the US, including Roger Temam at the University of Indiana, whose Navier-Stokes book is a classic of mathematical analysis. Lions is credited with this quote:

”Alan Turing is reported as saying that PDE’s are made by God, the
boundary conditions by the Devil ! The situation has changed, Devil
has changed places. . . We can say that the main chal lenges are in the
interfaces, with Devil not far away from them. . . ”
Jacques-Louis Lions (1928-2001)

The quote that he references from Turing is telling about the relationship between mathematical models and the way mathematical researchers feel about them. In coursework we only study the very simplest types of boundary conditions, and for good reason, since anything more complex than basic Dirichlet conditions quickly leads us into the realm of nonuniqueness or even nonexistence. My own work has run across a situation where I would like to enforce unusual boundary conditions, and have been struggling with the theoretical portion, trying to prove existence (I already know uniqueness ain’t happenin’).

More interesting in the applied world is the implementation of boundary conditions in numerical methods. My experience has been that except in the simplest cases, there may be no optimal way to do this. Sometimes we impose boundary conditions with a penalty condition in the equation itself. Sometimes we impose boundary conditions in the function space (and hence in the basis) where our problem resides. There is some literature pointing out links between different formulations, and the relationships are not always straightforward.

My point in the discussion above is that our coursework doesn’t always prepare us for the challenges of real problems, especially when real problems have quirky boundary conditions. Both Turing and Lions were justified in feeling some angst towards these odd creatures.