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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 92 93 94 | import Mathlib
import OAI.Analysis.Conductivity.Walls.HadamardWall
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set Filter Topology
lemma realCLM_invertible {L : ℝ →L[ℝ] ℝ} (hL : L 1≠0) : L.IsInvertible := by
apply ContinuousLinearMap.IsInvertible.of_inverse
(g := (L 1)⁻¹ • ContinuousLinearMap.id ℝ ℝ)
· apply ContinuousLinearMap.ext_ring
simp only [ContinuousLinearMap.comp_apply,smul_apply,
ContinuousLinearMap.id_apply,smul_eq_mul]
rw [←smul_eq_mul, map_smul,smul_eq_mul,inv_mul_cancel₀ hL]
· apply ContinuousLinearMap.ext_ring
simp only [ContinuousLinearMap.comp_apply,smul_apply,
ContinuousLinearMap.id_apply,smul_eq_mul]
exact inv_mul_cancel₀ hL
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
lemma real_section_hasDeriv {g : E × ℝ → ℝ} (hg : Differentiable ℝ g)
(p : E) (z : ℝ) : HasDerivAt (fun t => g (p,t)) (wallDerivative g (p,z)) z := by
simpa only [Function.comp_def,wallDerivative] using
(hg (p,z)).hasFDerivAt.comp_hasDerivAt z ((hasDerivAt_const z p).prodMk (hasDerivAt_id z))
lemma real_section_strictMono {g : E × ℝ → ℝ} (hg : Differentiable ℝ g)
{p : E} {a b : ℝ} (hd : ∀ z∈Icc a b, 0<wallDerivative g (p,z)) :
StrictMonoOn (fun z => g (p,z)) (Icc a b) := by
apply strictMonoOn_of_deriv_pos (convex_Icc a b)
((hg.continuous.comp (continuous_const.prodMk continuous_id)).continuousOn)
intro z hz
change 0<deriv (fun t => g (p,t)) z
rw [(real_section_hasDeriv hg p z).deriv]
exact hd z (interior_subset hz)
variable [CompleteSpace E]
theorem exists_smooth_simple_wall {g : E × ℝ → ℝ}
(hg : ContDiff ℝ (↑(⊤ : ℕ∞)) g) {U : Set E} (hU : IsOpen U)
{a b : ℝ} (hab : a<b)
(hleft : ∀ p∈U, g (p,a)<0) (hright : ∀ p∈U, 0<g (p,b))
(hder : ∀ p∈U, ∀ z∈Icc a b, 0<wallDerivative g (p,z)) :
∃ ρ : E → ℝ, ContDiffOn ℝ (↑(⊤ : ℕ∞)) ρ U ∧
(∀ p∈U, ρ p∈Ioo a b ∧ g (p,ρ p)=0) ∧
∀ p∈U, ∀ z∈Icc a b, g (p,z)=0 → z=ρ p := by
classical
have hd := hg.differentiable (by simp)
have he (p : E) (hp : p∈U) : ∃ z∈Ioo a b, g (p,z)=0 := by
obtain ⟨z,hz,hgz⟩ := intermediate_value_Icc hab.le
((hg.continuous.comp (continuous_const.prodMk continuous_id)).continuousOn (s := Icc a b))
(show (0:ℝ)∈Icc (g (p,a)) (g (p,b)) from ⟨(hleft p hp).le,(hright p hp).le⟩)
change g (p,z)=0 at hgz
refine ⟨z,⟨lt_of_le_of_ne hz.1 ?_,lt_of_le_of_ne hz.2 ?_⟩,hgz⟩
· intro h; subst z; linarith [hleft p hp]
· intro h; subst z; linarith [hright p hp]
choose ρ₀ hρ₀ using he
let ρ : E → ℝ := fun p => if hp : p∈U then ρ₀ p hp else 0
have hρ (p : E) (hp : p∈U) : ρ p∈Ioo a b ∧ g (p,ρ p)=0 := by
simpa only [ρ,dite_eq_left hp] using hρ₀ p hp
have huniq (p : E) (hp : p∈U) (z : ℝ) (hz : z∈Icc a b) (hgz : g (p,z)=0) : z=ρ p := by
apply (real_section_strictMono hd (hder p hp)).injOn hz (Ioo_subset_Icc_self (hρ p hp).1)
exact hgz.trans (hρ p hp).2.symm
refine ⟨ρ,?_,hρ,huniq⟩
intro p hp
have hc : ContDiffAt ℝ (↑(⊤ : ℕ∞)) g (p,ρ p) := hg.contDiffAt
have hinv : (fderiv ℝ g (p,ρ p) ∘L ContinuousLinearMap.inr ℝ E ℝ).IsInvertible := by
apply realCLM_invertible
change wallDerivative g (p,ρ p)≠0
exact (hder p hp (ρ p) (Ioo_subset_Icc_self (hρ p hp).1)).ne'
let ψ := hc.implicitFunction (by simp) hinv
have hψsmooth : ContDiffAt ℝ (↑(⊤ : ℕ∞)) ψ p := hc.contDiffAt_implicitFunction (by simp) hinv
have hψself : ψ p=ρ p := hc.implicitFunction_apply_self (by simp) hinv
have hψeq : ∀ᶠ q in 𝓝 p, g (q,ψ q)=0 := by
have hh := hc.eventually_apply_implicitFunction (by simp) hinv
simpa only [(hρ p hp).2] using hh
have hψin : ∀ᶠ q in 𝓝 p, ψ q∈Ioo a b :=
hψsmooth.continuousAt (isOpen_Ioo.mem_nhds (by simpa only [hψself] using (hρ p hp).1))
have hident : ψ =ᶠ[𝓝 p] ρ := by
filter_upwards [hU.mem_nhds hp,hψeq,hψin] with q hq he hi
exact huniq q hq _ (Ioo_subset_Icc_self hi) he
exact (hψsmooth.congr_of_eventuallyEq hident.symm).contDiffWithinAt
end ScalarConductivity
end
end
end OAI
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