M.Sc Thesis
M.Sc Student Gover Shlomi The Lady in the Lake Revisited Department of Mathematics Professor Gershon Wolansky

Abstract

A lady swims in a circular lake and a monster runs on the shore faster than the lady can swim. What is the minimal speed which allows the lady to exit the lake without getting caught by the monster? And how exactly should she swim in order to do so? This problem has been known for decades, although its answer has not been rigorously proved. In this thesis, we propose a way to prove that the conjectured answer is true by changing the game to a new one and using the theory of viscosity solutions which is closely related to these kinds of problems.

This kind of game is called a “differential game” and its solution is a function called “value function” which is known to be a viscosity solution of a partial differential equation (PDE) determined by the game (if the value function is uniformly continuous). The PDE of the original game does not have a unique solution, but by changing the game so that the exit time of the lady is minimized, the PDE of the new game has a unique solution, hence finding a viscosity solution leads us to the value function. We prove that the value function of the new game converges to the value function of the original game as the time factor goes to zero and conclude that the value of the new game helps us in our goal to find the value function of the original game.