| import OAI.Analysis.BernoulliCone.HalfspaceDifferential |
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| namespace OAI |
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| section |
| noncomputable section |
| open MeasureTheory Set Filter Metric |
| open scoped Topology NNReal ENNReal ContDiff |
| namespace Bernoulli |
| open scoped _root_.Bernoulli |
| variable {d : ℕ} |
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| 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 |
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| 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 |
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| end |
| end |
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| end OAI |
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