Regularity of the free boundary for the two-phase Bernoulli problem
Abstract We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we also show regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian.
Authors:
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Award ID(s):
Publication Date:
NSF-PAR ID:
10283057
Journal Name:
Inventiones mathematicae
Volume:
225
Issue:
2
Page Range or eLocation-ID:
347 to 394
ISSN:
0020-9910
3. Abstract We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are $$C^{1, 1/2}$$ C 1 , 1 / 2 , and that near non-degenerate points the fracture set is $$C^{1, \alpha }$$ C 1 , α , for some $$\alpha \in (0, 1)$$ α ∈ ( 0 , 1 ) .