When dealing with local perimeter minimizers, volume-constrained perimeter minimizers, and minimizers in prescribed mean curvature problems, the C1,γ- regularity theory from the previous chapters provides only preliminary information on the actual degree of regularity of reduced boundaries. In Section 27.2 we prove some higher regularity theorems, which are based on the fruitful connection between Euler–Lagrange equations for variational integrals and elliptic equations in divergence form presented in Section 27.1.

Elliptic equations for derivatives of Lipschitz minimizers

A convex function f ∈ C2(ℝn) is called locally uniformly convex if for every R > 0 there exists λ(R) > 0 such that

This is the case of the area integrand with

and (M(ξ)e) · e ≥ (1 + R2)−3/2∣e∣2 for every ∣ξ∣ ≤ R and e ∈ ℝn. As turns out, the regularity of local C1,γ minimizers of an integral functional defined by a locally uniformly convex integrand f, can be investigated through the classical Schauder theory for second order elliptic equations. The starting point here is the fact, proved in Theorem 23.4, that u is a solution to the weak Euler–Lagrange equation associated with f,

Recall that, if u is twice differentiable, then (27.3) takes the form

In turn, if both f and u are smooth, then we can differentiate in the xi direction the non-linear PDE (27.4), commute div and and find

This apparently complicated PDE has in fact a nice structure.

We finally prove the C1,γ-regularity theorems for local perimeter minimizers (Theorem 26.1), as well as for (Λ, r0)-perimeter minimizers (with Λ > 0; see Theorem 26.3). In the first case we prove C1,γ-regularity for every γ ∈ (0, 1), while in the second case the presence of the perturbation term Λr in the key estimate (25.6) will force the restriction γ ∈ (0, 1/2). In view of the higher regularity theory, local perimeter minimizers will in turn prove to be smooth, and in fact analytic. Hence, the real advantage in treating the two cases separately is just pedagogical. In Theorem 26.5, we apply these regularity results to prove the C1,γ-regularity of the reduced boundary, together with an important characterization of singular sets. This last result will be the starting point for the analysis of singularities carried on in Chapter 28. As a first application of the characterization result, in Theorem 26.6, Section 26.4, we shall refine conclusion (i) of Theorem 21.14, by showing a sort of “C1-convergence at regular points”-theorem for sequences of (Λ, r0)-perimeter minimizers.

The C1,γ-regularity theorem in the case Λ = 0

We now prove the C1,γ-regularity theorem for local perimeter minimizers. As usual, in order to simplify the notation we shall set

Below, C1(n) is as in Theorem 23.7 (Lipschitz approximation theorem).

Theorem 26.1 (C1,γ-regularity theorem for local minimizers) For every γ ∈ (0, 1) there exist positive constants ε4(n, γ), C5(n, γ) with the following property.

Synopsis

A cluster ε in ℝn is a finite disjoint family of sets of finite perimeter with finite and positive Lebesgue measure (note: the chambers ε(h) of ε are not assumed to be connected/indecomposable). By convention, denotes the exterior chamber of ε. The perimeter P(ε) of ε is defined as the total (n − 1)-dimensional Hausdorff measure of the interfaces of the cluster,

Denoting by b(ε) the vector in whose hth entry agrees with ∣ε(h)∣, we shall say that ε is a minimizing cluster in ℝn if spt με(h) = ε(h) for every h = 1,…, N, and, moreover, P(ε) ≤ P(ε′) whenever m(ε′) = m(ε). By a partitioning problem in ℝn, we mean any variational problem of the form

corresponding to the choice of some m. Proving the following theorem will be the main aim of Part IV. The existence and regularity parts will be addressed, respectively, in Chapter 29 and Chapter 30.

Theorem (Almgren's theorem) If n, N ≥ 2 and then there exist minimizers in the partitioning problem defined bym. If ε is an N-minimizing cluster in ℝn, then ε is bounded. If 0 ≤ h ≤ k ≤ N, then ε(h) ∩ ε(k) is an analytic constant mean curvature hypersurface in ℝn, relatively open inside ε(h) ∩ ε(k). Finally,

This existence and almost everywhere regularity theorem is one of the main results contained in the founding work for the theory of minimizing clusters and partitioning problems, that is Almgren's AMS Memoir [Alm76].

Let us now briefly sketch a picture of the situation after the results proved in the previous chapters. For the sake of clarity, in this overview, we shall directly refer to the case of a perimeter minimizer E in an open cylinder C(x0, r0), with x0 ∈ E (thus, Λ = 0). In the Lipschitz approximation theorem, Theorem 23.7, we have shown the existence of two constants ε1(n) and δ0(n) such that if e(E, x0, 9 r, υ) ≤ ε1(n) and 9r < r0, then there exists a Lipschitz function u: ℝn−1 → ℝ, which is “almost harmonic” on D(px0, r), and whose graph covers the “good” part M0 of C(x0, r, υ) ∩ E,

In turn, we were able to show that M0 covers a portion of C(x0, r, υ)∩E which is large in proportion to the smallness of e(E, x0, 9 r, υ). However, our final goal is showing that the two sets coincide: that is to say, we would like to show that, provided e(E, x0, 9 r, υ) is small enough, then one actually has

for every x ∈ C(x0, r, υ) ∩ E and s ∈ (0, 8 r). Clearly, this would imply that C(x0, r, υ) ∩ E coincides with the graph of u over D(px0, r).

In Theorem 26.5, we have proved the C1,γ-regularity of the reduced boundaries A∩E of any (Λ, r0)-perimeter minimizers E in some open set A. This chapter is devoted to the study of the singular set

aiming to provide estimates on its possible size. From the density estimates of Theorem 21.11, we already know that ℋn−1(Σ(E; A)) = 0. This result can be largely strengthened, as explained in the following theorem.

Theorem 28.1 (Dimensional estimates of singular sets of (Λ, r0)-perimeter minimizers) If E is a (Λ, r0)-perimeter minimizer in the open set A ⊂ ℝn, n ≥ 2, with Λr0 ≤ 1, then the following statements hold true:

1. (i) if 2 ≤ n ≤ 7, then Σ(E; A) is empty;

2. (ii) if n = 8, then Σ(E; A) has no accumulation points in A;

3. (iii) if n ≥ 9, then ℋs(Σ(E; A)) = 0 for every s > n − 8.

There exists a perimeter minimizer E in ℝ8with H0(Σ(E;∈8)) = 1. If n ≥ 9, then there exists a perimeter minimizer E in ℝn with Hn−8(Σ(E;ℝn)) = ∞.

Let us now sketch the proof of this deep theorem. We start by looking at the blow-ups Ex, r of a (Λ, r0)-minimizer E at a singular point x. In Theorem 28.6, we show that, loosely speaking, the Ex, r converge to a cone K with vertex at 0.