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.
Highlights
We consider the two-phase functional Jtp defined, for every open set D ⊂ Rd and every function u : D → R, as Jtp(u, D) := |∇u|2 d x + λ2+| + u ∩ D| + λ2−| − u D|, (TP)
Jtp u, ≤ Jtp v, for all open sets and functions v : D → R such that ⊂ D and v = u on this paper we aim to study the regularity of the free boundary ∂
Let u : D → R be a local minimizer of Jtp in the open set D ⊂ Rd
Summary
We consider the two-phase functional Jtp defined, for every open set D ⊂ Rd and every function u : D → R, as. Let u : D → R be a local minimizer of Jtp in the open set D ⊂ Rd. for every two-phase point x0 ∈ tp ∩ D, there exists a radius r0 > 0 (depending on x0) such that. Combining Theorem 1.1 with the known regularity theory for one-phase problem, one obtains the following result, which provides a full description of the free boundary of local minimizers of Jtp. Corollary 1.2 (Regularity of the free boundary). (i) The regular part Reg(∂ i ) is a relatively open subset of ∂ i and is locally the graph of a C1,η-regular function, for some η > 0 ∗)-rectifiable subset i where d∗ ∈ {5, 6, 7} is the critical dimension from Corollary 1.2
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