File size: 3,235 Bytes
159357f | 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 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 | import Mathlib
import OAI.Analysis.Conductivity.Model
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology Metric
variable {E P G : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [ProperSpace E]
[NormedAddCommGroup P] [NormedSpace ℝ P] [SecondCountableTopology P]
[MeasurableSpace P] [BorelSpace P] [NormedAddCommGroup G] [NormedSpace ℝ G]
[CompleteSpace G] {μ : Measure P} [IsLocallyFiniteMeasure μ]
omit [NormedSpace ℝ P] [CompleteSpace G] in
lemma hasFDerivAt_compact_integral {F : E × P → G} {D : E × P → E →L[ℝ] G}
(hF : Continuous F) (hD : Continuous D)
(hderiv : ∀ x p, HasFDerivAt (fun y => F (y,p)) (D (x,p)) x)
{K : Set P} (hK : IsCompact K) (x : E) :
HasFDerivAt (fun y => ∫ p in K, F (y,p) ∂μ) (∫ p in K, D (x,p) ∂μ) x := by
obtain ⟨C,hC⟩ := ((isCompact_closedBall x 1).prod hK).exists_bound_of_continuousOn
hD.continuousOn
have hFd (y : E) : Continuous (fun p => F (y,p)) :=
hF.comp (continuous_const.prodMk continuous_id)
have hDd (y : E) : Continuous (fun p => D (y,p)) :=
hD.comp (continuous_const.prodMk continuous_id)
have hi : Integrable (fun _ : P => C) (μ.restrict K) :=
ContinuousOn.integrableOn_compact hK continuous_const.continuousOn
apply hasFDerivAt_integral_of_dominated_of_fderiv_le
(μ := μ.restrict K) (s := closedBall x 1) (bound := fun _ => C)
(closedBall_mem_nhds x (by norm_num))
(Filter.Eventually.of_forall fun y => (hFd y).aestronglyMeasurable)
(ContinuousOn.integrableOn_compact hK (hFd x).continuousOn)
(hDd x).aestronglyMeasurable _ hi
· exact Filter.Eventually.of_forall (fun p y _ => hderiv y p)
· filter_upwards [ae_restrict_mem hK.measurableSet] with p hp
intro y hy
exact hC (y,p) ⟨hy,hp⟩
lemma contDiff_compact_integral_nat (n : ℕ) {F : E × P → G}
(hF : ContDiff ℝ n F) {K : Set P} (hK : IsCompact K) :
ContDiff ℝ n (fun x => ∫ p in K, F (x,p) ∂μ) := by
induction n generalizing G with
| zero =>
exact contDiff_zero.mpr (continuous_parametric_integral_of_continuous (f := fun x p => F (x,p)) hF.continuous hK)
| succ n ih =>
let D : E × P → E →L[ℝ] G := fun z =>
(fderiv ℝ F z).comp (ContinuousLinearMap.inl ℝ E P)
have hD : ContDiff ℝ n D := by
exact (hF.fderiv_right (by norm_cast)).clm_comp contDiff_const
have hderiv : ∀ x p, HasFDerivAt (fun y => F (y,p)) (D (x,p)) x := by
intro x p
have hf := (hF.differentiable (by norm_cast)).differentiableAt.hasFDerivAt
(x := (x,p))
exact hf.comp x ((hasFDerivAt_id x).prodMk (hasFDerivAt_const p x))
rw [Nat.cast_add, Nat.cast_one]
apply contDiff_succ_iff_hasFDerivAt.mpr
exact ⟨fun x => ∫ p in K, D (x,p) ∂μ, ih hD,
hasFDerivAt_compact_integral hF.continuous hD.continuous hderiv hK⟩
lemma contDiff_compact_integral {F : E × P → G}
(hF : ContDiff ℝ (↑(⊤ : ℕ∞)) F) {K : Set P} (hK : IsCompact K) :
ContDiff ℝ (↑(⊤ : ℕ∞)) (fun x => ∫ p in K, F (x,p) ∂μ) := by
apply contDiff_infty.mpr
intro n
exact contDiff_compact_integral_nat n (contDiff_infty.mp hF n) hK
end ScalarConductivity
end
end OAI
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