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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 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import Mathlib
import OAI.Analysis.Conductivity.Variational.PhysicalOpenVoltage
import OAI.Analysis.Conductivity.Sobolev.OriginalSmoothMultiplier
import OAI.Analysis.Conductivity.Variational.OriginalGradientLocality
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology Matrix
attribute [local instance] Classical.propDecidable
namespace PhysicalFiniteEndingData
variable {s : Fin 3 → ℝ} (z : PhysicalFiniteEndingData s)
lemma source_patch_gradient {η : ℝ} (hη : 0<η) (hηc : 2*η<centralThickness)
(q W : H1) (j : Fin 2)
(hv : ∀ᵐ x∂ballMeasure,weakValue W x=
if WithLp.ofLp x∈physicalOpenBlock then weakValue q x-z.terminalValue 0 j
else z.terminalProfile η j (WithLp.ofLp x)) :
∀ᵐ x∂ballMeasure,weakGradient W x=
if WithLp.ofLp x∈physicalOpenBlock then weakGradient q x else 0 := by
let E : R3 ≃L[ℝ] Coord3 := PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 3 => ℝ)
let U : Set R3 := E ⁻¹' physicalOpenBlock
have hU : IsOpen U := physicalOpenBlock_open.preimage E.continuous
have hUb : U⊆ball := fun x hx => physicalOpenBlock_subset_ball hx
have hinside : ∀ᵐ x∂ballMeasure,x∈U → weakGradient W x=weakGradient q x := by
apply original_local_value_difference_gradient W q hU hUb (-z.terminalValue 0 j)
filter_upwards [hv] with x hx hm
change WithLp.ofLp x∈physicalOpenBlock at hm
rw [hx,ite_eq_left hm]
ring
let O : Set R3 := ball∩{x | terminalTime 0 (E x)<0}
have hO : IsOpen O := Metric.isOpen_ball.inter
(isOpen_lt ((terminalTime_locallyLipschitz 0).continuous.comp E.continuous) continuous_const)
have hout : ∀ᵐ x∂ballMeasure,x∈O → weakGradient W x=0 := by
apply original_local_constant_gradient W hO inter_subset_left 0
filter_upwards [hv] with x hx hm
have hn : E x∉physicalOpenBlock := by
intro hh
have hh := (sourceOpenBlock_time_iff (E x)).mp hh
exact (not_lt.mpr hm.2.le) hh.1
change WithLp.ofLp x∉physicalOpenBlock at hn
rw [hx,ite_eq_right hn]
exact z.terminalProfile_zero_outside hη hηc j hm.2.le
let C : Fin 2 → Set R3 := fun k => ball∩{x | 0<terminalTime k.succ (E x)}
have hC (k : Fin 2) : IsOpen (C k) := Metric.isOpen_ball.inter
(isOpen_lt continuous_const ((terminalTime_locallyLipschitz k.succ).continuous.comp E.continuous))
have hchild (k : Fin 2) : ∀ᵐ x∂ballMeasure,x∈C k → weakGradient W x=0 := by
apply original_local_constant_gradient W (hC k) inter_subset_left (z.terminalValue k.succ j-z.terminalValue 0 j)
filter_upwards [hv] with x hx hm
have hn : E x∉physicalOpenBlock := by
intro hh
have hh := (sourceOpenBlock_time_iff (E x)).mp hh
exact (not_lt.mpr hm.2.le) (hh.2 k)
change WithLp.ofLp x∉physicalOpenBlock at hn
have hm' : 0<terminalTime k.succ (WithLp.ofLp x) := hm.2
rw [hx,ite_eq_right hn,z.terminalProfile_child_band hη hηc j k (by linarith)]
rw [min_eq_right hm'.le]
ring
have h0 := sourceColevel_ae_ne (show (0:ℝ)∈Icc (-(1:ℝ)/100) (1/100) by norm_num)
have hh (i : Fin 3) : ∀ᵐ y : Coord3,terminalTime i y≠0 := by
refine Fin.cases h0 (fun k => ?_) i
exact (sourceChildInverse_quasi (actualChildSign k)).ae h0
have hne := ae_restrict_of_ae (s:=ball) ((PiLp.volume_preserving_ofLp (Fin 3)).quasiMeasurePreserving.ae
(ae_all_iff.mpr hh))
filter_upwards [hinside,hout,ae_all_iff.mpr hchild,hne,ae_restrict_mem Metric.isOpen_ball.measurableSet]
with x hi ho hc hn hx
by_cases hb : WithLp.ofLp x∈physicalOpenBlock
· rw [ite_eq_left hb]
exact hi hb
· rw [ite_eq_right hb]
by_cases hp : terminalTime 0 (E x)<0
· exact ho ⟨hx,hp⟩
· have hp' : 0<terminalTime 0 (E x) := lt_of_le_of_ne (le_of_not_gt hp) (hn 0).symm
have hnchild : ¬∀ k : Fin 2,terminalTime k.succ (E x)<0 := by
intro hh
exact hb ((sourceOpenBlock_time_iff (E x)).mpr ⟨hp',hh⟩)
push Not at hnchild
obtain ⟨k,hk⟩ := hnchild
exact hc k ⟨hx,lt_of_le_of_ne hk (hn k.succ).symm⟩
end PhysicalFiniteEndingData
end ScalarConductivity
end
end
end OAI
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