Free Boundaries and Extremal Inequalities

A free boundary is an interface between two materials like oil and water. Another example is the curve outlining the wake of a boat. Yet another is the interface between plasma and ordinary matter in a fusion reactor. Remarkably, the same mathematics of free boundaries that describes these physical phenomena can be used to design optimal shapes. For example, when one wants to enclose an oven or pipe with insulating material, there is a shape that is optimal in the sense that the sum of the cost of insulation and the cost due to heat loss is minimized. Moreover, even farther from physics, one can seek optimal ways to divide data sets into yes/no regions according to rules that minimize the errors, that is, false positive or false negative identifications or diagnoses. The overarching goal of this project is to reduce the complexity of the problem of searching for these optimal shapes. The expectation is that there are broad classes of situations in which the optimal divider (free boundary) resembles a straight line or plane at an appropriate scale. In those cases, one can be confident of finding a near optimal shape quickly. In addition to conducting his own research, the PI has served and will serve as faculty advisor for research projects by dozens of undergraduates and high school students in programs at MIT. Moreover, he has posted videos and lecture notes of a widely viewed single variable calculus course on MIT's Open Courseware site. He is currently working on an on-line course to be disseminated by MITx.

Free boundaries arise as the interface between materials in which the materials retain some energy. Typically, space is divided into level sets of some quantity like temperature or pressure. In contrast, the interface represented by a minimal surface lives in an ambient space that is empty. Despite this difference between these two types of interfaces, there are profound connections between them. The main goal of this project is to show that interfaces and level sets of least energy for a wide variety of problems are as simple as possible. The PI proposes that the level sets of optimizers resemble parallel planes in that these surfaces are connected and cleanly separated. Usually, methods from the more developed theory of minimal surfaces have guided the study of free boundaries, but here ideas from the theory of free boundaries will guide the study of minimal surfaces. The proposal also gives a pathway to proving analogous simple behavior of level sets of the least energy Neumann eigenfunction for a convex symmetric domain. This would yield an important case of the longstanding ``hot spots'' conjecture of J. Rauch. A second project is to identify the cases of equality in the celebrated Alexandrov-Fenchel inequalities in convex geometry. The PI will use a geometric approach based on establishing new properties, of independent interest, of the Brenier (optimal transportation) mapping. A third project is aimed at developing a highly accurate description of localization of eigenfunctions and quantum tunneling, relevant to the design of LEDs.