| Abstract: |
| It is known that the classical Obstacle and Alt-Caffarelli Problems can be regarded as two endpoint cases for the family of free boundary problems, known as the Alt-Phillips Problem. Their minimizers are non-negative, subharmonic, and solves an elliptic-type equation in the positive set with an a priori unknown positive-zero set interface, which we call the `Free Boundary'. In proof of the Optimal Regularity, the classical approaches involve showing the optimal growth rate near the boundary, then combining with the interior equation to gain full regularity. In this work, we discuss a novel dichotomy argument initiated by De Silva - Savin. The key idea is to say, for the size of the minimizer u sufficiently large at unit scale, either the size of u shrinks by 1/2, or u is sufficiently close to its harmonic replacement, as one goes to the next scale. We also discuss a generalization to the vectorial N-Membrane Alt-Phillips Free Boundary Problem, and study how the same argument transfers. In particular, we point out the key difference between the Alt-Phillips with parameter in (0, 1), and the endpoint cases. |
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