The Poincaré inequality for functions of bounded variation #
Let Ω be a convex open set of a finite-dimensional real inner product space E of dimension
n, with an additive Haar measure μ, and let u be locally integrable on Ω. This file proves
the L¹ Poincaré inequality for u in terms of its variational total variation
|Du|(Ω) = TauCeti.totalVariationOn μ Ω u: for every S ⊆ Ω of finite positive measure,
∫_Ω |u - ⨍_S u| ≤ (∫_{Ω - Ω} ‖h‖ dh) / μ(S) · |Du|(Ω).
For bounded Ω the constant is at most μ(B(0, 1)) (diam Ω) ^ (n + 1) / μ(S), and on a ball of
radius R, averaging over the ball itself, it is at most 2 ^ (n + 1) R. These are the constants
of the Poincaré–Wirtinger inequality on W^{1,p},
TauCeti.W1p.eLpNorm_value_sub_setAverage_le_of_convex, whose case p = 1 the present statements
extend to functions of bounded variation: no derivative, no integrability on Ω and no
finiteness of |Du|(Ω) is assumed, both sides may be infinite, and S need not be measurable.
For a W^{1,1} function the variation is the L¹ norm of the weak gradient
(TauCeti.HasWeakFDerivOn.totalVariationOn_eq_setLIntegral_enorm).
The proof averages the translation estimate
TauCeti.setLIntegral_enorm_comp_add_sub_le_mul_totalVariationOn. Writing y = x + h,
∫_Ω ∫_Ω |u x - u y| dy dx = ∫_{Ω - Ω} ∫_{Ω ∩ (Ω - h)} |u (x + h) - u x| dx dh,
and by convexity of Ω the segment from x to x + h lies in Ω whenever both endpoints do, so
the inner integral is at most ‖h‖ |Du|(Ω). Averaging |u x - u y| over y ∈ S then bounds the
deviation of u x from the mean of u over S.
For bounded Ω, S ⊆ Ω of positive measure and u locally integrable on Ω, the file also
proves the inequality localized to an arbitrary subset A ⊆ Ω:
∫_A |u - ⨍_S u| ≤ μ(B(0, 1)) ^ (1 - 1/n) (diam Ω) ^ n / μ(S) μ(A) ^ (1/n) |Du|(Ω).
The factor μ(A) ^ (1/n) makes it scale-invariant: it is a weak-type L^{n/(n-1)} Poincaré
inequality, implied by the strong L^{n/(n-1)} Sobolev–Poincaré inequality through Hölder's
inequality. Its indicator case is the relative isoperimetric inequality
(TauCeti.min_measure_inter_measure_sdiff_rpow_le_perimeter). The C¹ case
(TauCeti.setLIntegral_enorm_sub_setAverage_le_mul_measure_rpow_mul_of_convex) integrates the
Riesz potential estimate over A and bounds the Riesz kernel on A by its value on a ball of the
same measure. It passes to u through mollifications on convex open sets with compact closures in
Ω, which converge in L¹ and eventually do not increase the variation
(TauCeti.exists_mollifier_seq_of_isCompact_closure), and through an
exhaustion of Ω by such sets
(TauCeti.exists_seq_isOpen_convex_isCompact_closure_subset_iUnion_eq).
Main results #
TauCeti.setLIntegral_setLIntegral_enorm_sub_le_mul_totalVariationOn: the double-integral estimate∫_Ω ∫_Ω |u x - u y| ≤ (∫_{Ω - Ω} ‖h‖) |Du|(Ω)on a convex open set.TauCeti.setLIntegral_enorm_sub_setAverage_le_mul_totalVariationOn: the Poincaré inequality on a convex open set, with constant(∫_{Ω - Ω} ‖h‖) / μ(S).TauCeti.setLIntegral_enorm_sub_setAverage_le_mul_totalVariationOn_of_convex: the Poincaré inequality on a bounded convex open set, with constantμ(B(0, 1)) (diam Ω) ^ (n + 1) / μ(S).TauCeti.integrableOn_of_totalVariationOn_ne_top_of_convex: a locally integrable function of finite total variation on a bounded convex open set is integrable there.TauCeti.setLIntegral_enorm_sub_setAverage_le_mul_totalVariationOn_of_eq_ball: the Poincaré inequality on a ball of radiusR, with constant2 ^ (n + 1) R.TauCeti.setLIntegral_enorm_sub_setAverage_le_mul_measure_rpow_mul_totalVariationOn_of_convex: the inequality localized to a subsetA, on a bounded convex open set.
References #
- L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Remark 3.50.
- L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions, revised edition, §5.6.2 (Poincaré's inequality for functions of bounded variation).
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, equation (7.45), for the constant on convex sets, and Lemmas 7.12 and 7.16.
The double-integral estimate. On a convex open set Ω, a locally integrable u
satisfies ∫_Ω ∫_Ω |u x - u y| ≤ (∫_{Ω - Ω} ‖h‖) |Du|(Ω), where |Du|(Ω) is the total
variation of u on Ω. Both sides may be infinite.
The Poincaré inequality for functions of bounded variation on a convex set. Let u be
locally integrable on a convex open set Ω and let S ⊆ Ω have finite positive measure. Then
∫_Ω |u - ⨍_S u| ≤ (∫_{Ω - Ω} ‖h‖) / μ(S) · |Du|(Ω). Both sides may be infinite.
The Poincaré inequality for functions of bounded variation on a bounded convex set. Let
u be locally integrable on a bounded convex open set Ω of dimension n, and let S ⊆ Ω have
positive measure. Then u deviates from its mean over S in L¹(Ω) by at most
μ(B(0, 1)) (diam Ω) ^ (n + 1) / μ(S) times its total variation on Ω.
The Poincaré inequality for functions of bounded variation on a ball. A function locally
integrable on a ball of radius R in a space of dimension n deviates from its mean over the
ball in L¹ by at most 2 ^ (n + 1) R times its total variation on the ball. The constant is
proportional to R, as the scaling of both sides forces.
The localized inequality #
A function locally integrable on a bounded convex open set Ω and of finite total variation
there is integrable on Ω. This is the integrability half of membership in BV(Ω).
The Poincaré–Wirtinger inequality for functions of bounded variation, localized to a
subset. Let Ω be a bounded convex open set in a finite-dimensional real inner product space of
dimension n, and let S ⊆ Ω have positive measure. For every u locally integrable on Ω
and every A ⊆ Ω, the integral over A of the deviation of u from its mean over S is at most
μ(B(0, 1)) ^ (1 - 1/n) * (diam Ω) ^ n / μ(S) * μ(A) ^ (1/n) times the total variation
|Du|(Ω), finite or infinite.