Free Boundary Regularity Problems in Harmonic Analysis

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ABSTRACTThe theme of this proposal is the strong relationship that exists between the questions of how the geometry of a domain can be recovered from the regularity of its harmonic measure, and free boundary regularity problems. Remarkably the analogies become more apparent when examined under a geometric measure theory magnifying glass. The core of this proposal addresses the question of describing in detail the singular set of the free boundary in problems which are similar to the optimal design question described below. The PI and N. Wickramasekera propose to use tools from geometric measure theory, which have been very successful in similar situations in geometric analysis. They have remarked that Weiss' new monotonicity formulas yield similar information to the one provided by the monotonicity formulas for energy minimizing harmonic maps and area minimizing surfaces. The proposed program follows the schemes which have lead to the successful resolution of these 2 major problems.Two-phase free boundary regularity problems arise naturally in physics, chemistry and engineering. In optimal design and optimal control problems within the context of electrochemical machining with threshold current, the surface between the anode and the electrolytic solution plays the role of the free boundary. In the symmetric plane flow model of jets with two fluids the outer jet is surrounded by air, so the flow region has two free boundaries: the boundary of the outer fluid and the boundary between the two fluids. The central problem of characterizing the regularity of the free boundary has been studied by many authors. Over the last 3 years the PI and C. Kenig have studied the case in which one has information about the ratio of the speeds of the two fluids at the free boundary. This question had not been addressed before. They found a global criteria which guarantees the regularity of the free boundary. One the themes of this proposal is the study of several free boundary regularity problems using these new tools. The proposed research provides a different outlet for geometric measure theory (GMT), a field of Mathematics that has contributed greatly to the development of the calculus of variations and geometric analysis. The success of the project would provide a road map for a new generation interested in applying GMT techniques to more concrete problems. An important feature of the proposed work is that, while some results have already been obtained, there is great potential for expansion. In particular, we expect the active participation of graduate students and junior mathematicians.