| import Mathlib |
| import OAI.Analysis.Conductivity.Model |
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| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology |
| open scoped Manifold |
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| variable {P E : Type*} [TopologicalSpace P] [TopologicalSpace E] |
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| lemma compact_parametric_abs_bound {f : P → E → ℝ} {p₀ : P} {K : Set E} |
| (hK : IsCompact K) (hf : Continuous (fun px : P×E => f px.1 px.2)) : |
| ∃ M : ℝ,0<M ∧ ∀ᶠ p in 𝓝 p₀,∀ x∈K,|f p x|≤M := by |
| have hbase : Continuous (fun x => f p₀ x) := hf.comp (continuous_const.prodMk continuous_id) |
| obtain ⟨C,hC⟩ := hK.exists_bound_of_continuousOn hbase.continuousOn |
| refine ⟨|C|+1,by positivity,?_⟩ |
| apply hK.eventually_forall_of_forall_eventually |
| intro x hx |
| have hlt : |f p₀ x| < |C|+1 := by |
| have h := hC x hx |
| rw [Real.norm_eq_abs] at h |
| linarith [le_abs_self C] |
| have hct : Tendsto (fun px : P×E => |f px.1 px.2|) (𝓝 (p₀,x)) (𝓝 |f p₀ x|) := hf.abs.continuousAt |
| exact (hct.eventually_lt_const hlt).mono (fun _ h => h.le) |
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| lemma compact_parametric_abs_lower {f : P → E → ℝ} {p₀ : P} {K : Set E} |
| (hK : IsCompact K) (hf : Continuous (fun px : P×E => f px.1 px.2)) |
| (hne : ∀ x∈K,f p₀ x≠0) : |
| ∃ κ : ℝ,0<κ ∧ ∀ᶠ p in 𝓝 p₀,∀ x∈K,κ≤|f p x| := by |
| have hbase : Continuous (fun x => |f p₀ x|) := hf.abs.comp (continuous_const.prodMk continuous_id) |
| obtain ⟨κ,hκ,hlo⟩ := hK.exists_forall_le' |
| hbase.continuousOn |
| (fun x hx => abs_pos.mpr (hne x hx)) |
| refine ⟨κ/2,by positivity,?_⟩ |
| apply hK.eventually_forall_of_forall_eventually |
| intro x hx |
| have hlt : κ/2 < |f p₀ x| := by have := hlo x hx; linarith |
| have hct : Tendsto (fun px : P×E => |f px.1 px.2|) (𝓝 (p₀,x)) (𝓝 |f p₀ x|) := hf.abs.continuousAt |
| exact (hct.eventually_const_lt hlt).mono (fun _ h => h.le) |
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| lemma exists_smooth_core_cutoff |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] |
| {K Ω : Set V} (hK : IsCompact K) (hΩ : IsOpen Ω) (hb : Bornology.IsBounded Ω) |
| (hsub : K⊆Ω) : |
| ∃ χ : V → ℝ,ContDiff ℝ (↑(⊤ : ℕ∞)) χ ∧ HasCompactSupport χ ∧ |
| tsupport χ⊆Ω ∧ (∀ x,χ x∈Icc 0 1) ∧ |
| (∀ x∈K,χ =ᶠ[𝓝 x] (fun _ => 1)) := by |
| obtain ⟨χ,hχ₀,hχ₁,hχb⟩ := exists_contMDiffMap_zero_one_nhds_of_isClosed (𝓘(ℝ,V)) |
| hΩ.isClosed_compl hK.isClosed (disjoint_left.mpr (fun _ hx hxK => hx (hsub hxK))) (n:=⊤) |
| have ht : tsupport (χ : V → ℝ)⊆Ω := by |
| intro x hx |
| by_contra hn |
| exact (notMem_tsupport_iff_eventuallyEq.mpr (hχ₀.filter_mono (nhds_le_nhdsSet hn))) hx |
| refine ⟨χ,contMDiff_iff_contDiff.mp χ.contMDiff, |
| (hb.subset ht).isCompact_closure.of_isClosed_subset (isClosed_tsupport _) subset_closure, |
| ht,hχb,?_⟩ |
| intro x hx |
| exact hχ₁.filter_mono (nhds_le_nhdsSet hx) |
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| end ScalarConductivity |
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| end |
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| end OAI |
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