On the Singular Set of Free Interface in an Optimal Partition Problem

Abstract

We study the singular set of free interface in an optimal partition problem for the Dirichlet eigenvalues. We prove that its upper (n − 2)‐dimensional Minkowski content, and consequently its (n − 2)‐dimensional Hausdorff measure, are locally finite. We also show that the singular set is countably (n − 2)‐rectifiable; namely, it can be covered by countably many C1‐manifolds of dimension (n − 2), up to a set of (n − 2)‐dimensional Hausdorff measure zero. Our results hold for optimal partitions on Riemannian manifolds and harmonic maps into homogeneous trees as well. © 2019 Wiley Periodicals, Inc.

Rectifiability of the Singular Set of Harmonic Maps into Buildings

“…Because the support of µ x is on B 1 (x) \ B 1/2 (x), and since x is on the line segment between x 1 and x 2 , we have that |x − x ℓ | ≤ 1 4 for ℓ = 1, 2 and we see that…”

“…Otherwise, we include them in B(1) (and we will repeat the refining process shortly). We define C(1) = G(1) ∪ B (1). Observe that Thus, if we choose ρ to be smaller than ρ 0 (m) := 1 2C(m) , which is a purely dimensional constant, we can guarantee that C(m)ρ ≤ 1 2 .…”

“…Proof. First, without loss of generality we may assume that τ = 1 8 and x = 0. Although this may increase I φ (0, 64), it will still be bounded in terms of Λ by Lemma 4.2.…”

“…[9,10]. The paper of Naber and Valtorta provides a framework which de Lellis, Marchese, Spadaro, and Valtorta's have used to study Q-valued functions in [8], and which Alper has used in [1] to study an optimal 2020 Mathematics Subject Classification. Primary 58E20, Secondary 49N60.…”

“…partition problem. We draw special attention to this work of Alper, [1], because it shows that the singular sets of harmonic maps into one-dimensional buildings are (m − 2)-rectifiable. In this paper, we adapt the framework of Naber-Valtorta to our special case, closely following the arguments of [8].…”