File size: 2,905 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 | import Mathlib
import OAI.Analysis.Conductivity.Model
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set Filter Topology
open scoped Manifold
variable {P E : Type*} [TopologicalSpace P] [TopologicalSpace E]
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)
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)
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)
end ScalarConductivity
end
end OAI
|