The reduced boundary and the measure-theoretic outer normal #

Let Ω be an open set of a finite-dimensional real inner product space E and s a set of locally finite perimeter in Ω. The derivative of its indicator is the vector measure Dχ_s = σ |Dχ_s|, where |Dχ_s| is the perimeter measure (TauCeti.variationMeasure) and σ is the polar density (TauCeti.polarDensity). A point x ∈ Ω belongs to the reduced boundary ∂*s (TauCeti.reducedBoundary) when the averages

⨍_{closedBall x r} σ d|Dχ_s| = Dχ_s (closedBall x r) / |Dχ_s| (closedBall x r)

converge, as r → 0⁺, to a unit vector. The measure-theoretic outer normal ν_s x (TauCeti.outerNormal) is minus that limit, so that Dχ_s = -ν_s |Dχ_s|.

These are the objects of De Giorgi's structure theorem for sets of finite perimeter. This file proves their basic properties.

The perimeter measure is concentrated on the reduced boundary, and the outer normal is -σ at |Dχ_s|-almost every point (TauCeti.ae_mem_reducedBoundary, TauCeti.ae_outerNormal_eq). This is Lebesgue's differentiation theorem for σ against the Radon measure |Dχ_s| on Ω, which is available because Ω inherits the Besicovitch covering property (Besicovitch.ae_tendsto_average_closedBall). Consequently the Gauss–Green formula ∫_s div φ = ∫ ⟪φ, ν_s⟫ d|Dχ_s| holds for every C¹ vector field φ with compact support in Ω, where div φ = tr (Dφ) (TauCeti.integral_trace_fderiv_mul_indicator_eq_integral_inner_outerNormal). It is the case u = χ_s of integration by parts against C¹ vector fields, ∫ u div φ dμ = -∫ ⟪φ, σ_u⟫ d|Du| (TauCeti.integral_trace_fderiv_mul_eq_neg_integral_inner_polarDensity).
A point of the reduced boundary forces s to have locally finite perimeter in Ω (TauCeti.locallyIntegrableOn_and_totalVariationOn_ne_top_of_mem_reducedBoundary), and in particular to be null-measurable on Ω (TauCeti.nullMeasurableSet_restrict_of_mem_reducedBoundary), so otherwise ∂*s is empty (TauCeti.reducedBoundary_eq_empty_of_not).
Every point of the reduced boundary lies in the support of |Dχ_s| (TauCeti.variationMeasure_preimage_closedBall_ne_zero). Hence s has nonzero perimeter and positive measure in every small ball around it (TauCeti.perimeter_ball_ne_zero, TauCeti.measure_inter_ball_pos), and, when μ is an additive Haar measure, the point lies in the topological frontier of s (TauCeti.reducedBoundary_subset_frontier).
The outer normal concentrates at ν_s x in mean at every point x of the reduced boundary: ⨍_{closedBall x r} ‖ν_s - ν_s x‖ d|Dχ_s| → 0 (TauCeti.tendsto_setLIntegral_enorm_outerNormal_sub_div). The averages of the unit vectors ν_s tend to the unit vector ν_s x, and for unit vectors ‖a - ν‖² = 2 (1 - ⟪a, ν⟫).
The reduced boundary and the outer normal only depend on s almost everywhere on Ω (TauCeti.reducedBoundary_congr_ae, TauCeti.outerNormal_congr_ae); in particular ∂*(s ∩ Ω) = ∂*s (TauCeti.reducedBoundary_inter_opens). When μ is an additive Haar measure, the complement has the same reduced boundary with the opposite normal (TauCeti.reducedBoundary_compl, TauCeti.outerNormal_compl).

The definition uses closed balls, matching TauCeti.HasDensityAt; it does not require x to lie in the support of |Dχ_s|, since a unit limit forces this. When s does not have locally finite perimeter in Ω, the perimeter measure is zero by convention and the reduced boundary is empty.

References #

F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, §15.1 (Definition 15.1 and Proposition 15.5) and Chapter 15 (blow-ups at points of the reduced boundary).
L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions, revised edition, §5.7.
L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, §3.1.
F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, §12.1 (the Gauss–Green measure of a set of locally finite perimeter).

The reduced boundary and the measure-theoretic outer normal #

References #

Concentration of the outer normal #

Invariance under null modification and complements #