Student Analysis Seminar

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An Introduction to Obstacle and Alt-Caffarelli Free Boundary Problems.

Speaker: Mark Ma, Columbia University

Location: Warren Weaver Hall 201

Date: Monday, April 20, 2026, 12:30 p.m.

Synopsis:

In this talk, we'll introduce the basic setup and strategy to study the Obstacle and Alt-Caffarelli (One Phase Bernoulli) Free Boundary Problems. We'll start with a discussion on `solutions': basic properties of minimizers, the optimal regular, non-degeneracy. Then we look at the Free Boundary, which is the boundary of the positive set, with a focus on the Regularity of the Free Boundary and its Singular set. Along the way we introduce the Weiss Monotonicity formula and characterize the homogeneity of blowup limits. We shall also discuss potential classification of blowup profiles, and emphasize the difference between the two problems.