15. Elliptic Regularity Theory

Elliptic Regularity Theory, (ERT) is all about the smoothness of solutions to elliptic partial differential equations.  Turns out when you solve an elliptic PDE in a region where you've specified the values on the boundary, the solution is really smooth inside the region even if the values on the boundary aren't that nice.

This probably seems super basic to all of you, but it took me awhile to get the hang of ERT. I spent most of the afternoon reading the seminal work on the topic by Gilbarg and Trudinger.  Elliptic PDE are ubiquitous so the regularity of their solutions is really important. Put the time in and learn more about ERT.  The sky's the limit, when your heart is in it.