What is the complexity of related elliptic, parabolic, and hyperbolic problems?

Abstract:

Traub and Woźniakowski have dealt with the complexity of some simple partial differential equations. They chose three model problems and showed that the parabolic problem considered had significantly lower complexity than the elliptic problem, which in turn had significantly lower complexity than the hyperbolic problem considered. They asked whether this is true in general. We show that this is not the case by proving that if L is a reasonably well-behaved elliptic operator, then the steady-state heat equation $Lu = f$, the heat equation ${\partial _t}u + Lu = f$, and the wave equation ${\partial _{tt}}u + Lu = f$ all have roughly the same worst-case complexity for f in the unit ball of a certain Sobolev space of smoothness r.