Free Boundary Regularity
We discuss level surfaces of solutions $u$ to singular semilinear elliptic equations such as $\Delta u = \delta(u)$. The level surface $u=0$ can be interpreted as a solution to a free boundary problem, the problem of finding the optimal shape of insulating material. A variant of this equation, the Prandtl-Batchelor equation, describes the wake created by a boat. We will prove that stable free boundaries are smooth in three-dimensional space. Numerical methods are needed to explore counterexamples in higher dimensions and to establish a more robust regularity theory with numerically effective bounds.
David Jerison, Massachusetts Institute of Technology