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.