For the Alt-Phillips free boundary problem, the paper proves smoothness of regular free boundaries for all exponents, derives a stability inequality for negative exponents, and rules out nontrivial axially symmetric stable cones in dimensions d≤6 (and d=7 for γ below about -0.717).
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Smoothness and stability in the Alt-Phillips problem
For the Alt-Phillips free boundary problem, the paper proves smoothness of regular free boundaries for all exponents, derives a stability inequality for negative exponents, and rules out nontrivial axially symmetric stable cones in dimensions d≤6 (and d=7 for γ below about -0.717).