cool theorem

Consider the following game:

Let epsilon > 0. Let U be a domain in the plane, and define "the ball" to be a point x in U. There are two players, A and B. A turn consists of

1) Player A picks a unit vector v.
2) Player B sets b= +/- 1.
3) The ball is moved from x to x+b*sqrt(2)*epsilon*v.

The goal of A is to move the ball outside of U. The goal of B is prevent A from doing so.

It is not to hard to show that A can always win. In fact, if u_ep(x) = (epsilon)^2 k where k is the minimum number of turns in the optimal winning strategy for A and x is the starting point of the ball, then u_ep is bounded independently of x depending only the diameter of U. What is really interesting is that the function u defined by

u(x)=lim_(epsilon ->0)u_ep(x)

satisfies the pde

|grad(u)|div((grad u)/|grad u|) = -1 on U
u=0 on bdry U.

This is the same pde that governs the mean curvature flow. Creepy.