Smoothness and stability in the Alt-Phillips problem.

Area of Science:

• Free boundary problems
• Partial differential equations
• Geometric analysis

Background:

• The Alt-Phillips free boundary problem is a significant area of research in geometric analysis.
• Understanding the behavior of free boundaries is crucial for various applications.
• Previous studies have explored positive exponents, but the negative exponent case presents unique challenges.

Purpose of the Study:

• To establish smoothness of C^(1,α)-regular free boundaries for the Alt-Phillips problem with negative exponents (γ ∈ (-2, 0)).
• To derive a new stability condition for the Alt-Phillips problem in the negative exponent regime.
• To investigate the existence of nontrivial axially symmetric stable cones in low dimensions.

Main Methods:

• Reduction of the problem to degenerate quasilinear PDEs.
• Establishment of Schauder estimates for the degenerate quasilinear PDEs.
• Exploitation of higher regularity of solutions to derive stability conditions.

Main Results:

• Unified proof of C^(1,α)-regularity for free boundaries in the negative exponent regime.
• A new stability condition is derived, ruling out nontrivial axially symmetric stable cones in low dimensions.
• A variational criterion for cone stability is provided, consistent with minimal surfaces as γ → -2.

Conclusions:

• The study successfully demonstrates the smoothness of free boundaries for the Alt-Phillips problem with negative exponents.
• The derived stability condition offers new insights into the geometric properties of solutions.
• The findings contribute to a deeper understanding of free boundary problems and their stability criteria.