A Direct Method for Time-Periodic L^p Estimates

Inbetween elliptic PDEs, which do not depend on time (think of the Poisson equation), and honest parabolic PDEs, which do depend on time and are started at a given initial value (think of the heat equation), there are time-periodic parabolic PDEs: On the one hand, time-independent solutions to the elliptic PDE are also trivially time-periodic, which gives periodic problems an elliptic touch, on the other hand solutions to the initial value problem which are not constant in time might very well be periodic.

I want to advocate for time-periodic problems not being the little sister of either elliptic or parabolic problems, but being a connector between the two and a class of its own right, by introducing a direct method for showing a priori $L^p$ estimates for time-periodic, linear, partial differential equations. The method is generic and can be applied to a wide range of problems. In the talk, I intend to demonstrate it on the heat equation and on boundary value problems of Agmon-Douglas-Nirenberg type.

The talk is based on joint works with Yasunori Maekawa and Mads Kyed.