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110f4f3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 | import OAI.Analysis.BernoulliCone.HalfspaceDifferential
namespace OAI
section
noncomputable section
open MeasureTheory Set Filter Metric
open scoped Topology NNReal ENNReal ContDiff
namespace Bernoulli
open scoped _root_.Bernoulli
variable {d : ℕ}
lemma integral_gradient_compact {q : Space d → ℝ} {K : ℝ≥0}
(hq : LipschitzWith K q) (hc : HasCompactSupport q) (e : Space d) :
(∫ x, inner ℝ (gradient q x) e) = 0 := by
have hh := (LipschitzWith.const (1:ℝ)).integral_lineDeriv_mul_eq
(μ:=volume) hq hc (-e)
have he : (fun x => lineDeriv ℝ q x (-(-e)) * (1:ℝ)) =ᵐ[volume]
(fun x => inner ℝ (gradient q x) e) := by
filter_upwards [hq.ae_differentiableAt] with x hx
simp only [neg_neg,mul_one,hx.lineDeriv_eq_fderiv,inner_gradient_left]
rw [integral_congr_ae he] at hh
have hz (x : Space d) : lineDeriv ℝ (fun _ : Space d => (1:ℝ)) x (-e)=0 := by
rw [(differentiableAt_const (1:ℝ)).lineDeriv_eq_fderiv]
simp
simpa only [hz,zero_mul,integral_zero] using hh.symm
lemma setIntegral_gradient_compact {q : Space d → ℝ} {K : ℝ≥0}
(hq : LipschitzWith K q) (hc : HasCompactSupport q)
{D : Set (Space d)} (hD : tsupport q ⊆ D) (e : Space d) :
(∫ x in D, inner ℝ (gradient q x) e) = 0 := by
rw [setIntegral_eq_integral_of_forall_compl_eq_zero]
· exact integral_gradient_compact hq hc e
· intro x hx
have hn : x∉tsupport q := fun hh => hx (hD hh)
simp [gradient,fderiv_of_notMem_tsupport ℝ hn]
end Bernoulli
end
end
end OAI
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