File size: 3,893 Bytes
2fca478 | 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 | import OAI.Geometry.Perimeter.Regularity.EllipticHilbert
import OAI.Geometry.Perimeter.Regularity.GradientContDiff
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal ContDiff InnerProductSpace
namespace RPISingularHarnack
variable {n : ℕ}
abbrev VectorL2 (n : ℕ) := Lp (Euc n) 2 (volume : Measure (Euc n))
def SmoothLocalTest (U : Set (Euc n)) :=
{φ : Euc n → ℝ // ContDiff ℝ ∞ φ ∧ HasCompactSupport φ ∧ tsupport φ ⊆ U}
def testGradientLp {U : Set (Euc n)} (φ : SmoothLocalTest U) : VectorL2 n :=
((gradient_contDiff φ.property.1).continuous.memLp_of_hasCompactSupport
(gradient_hasCompactSupport φ.property.2.1)).toLp (gradient φ.val)
theorem testGradientLp_ae {U : Set (Euc n)} (φ : SmoothLocalTest U) :
(testGradientLp φ : Euc n → Euc n)=ᵐ[volume] gradient φ.val :=
MemLp.coeFn_toLp _
def testGradientSpace (U : Set (Euc n)) : Submodule ℝ (VectorL2 n) :=
(Submodule.span ℝ (Set.range (testGradientLp (U := U)))).topologicalClosure
instance testGradientSpace_complete (U : Set (Euc n)) : CompleteSpace (testGradientSpace U) :=
(Submodule.isClosed_topologicalClosure _).completeSpace_coe
theorem testGradient_mem {U : Set (Euc n)} (φ : SmoothLocalTest U) :
testGradientLp φ∈testGradientSpace U :=
Submodule.le_topologicalClosure _ (Submodule.subset_span ⟨φ,rfl⟩)
theorem inner_zero_on_closed_span {H : Type*} [NormedAddCommGroup H]
[InnerProductSpace ℝ H] (s : Set H) {u : H} (h : ∀ w∈s, inner ℝ u w=0) :
∀ w∈(Submodule.span ℝ s).topologicalClosure, inner ℝ u w=0 := by
have hs : Submodule.span ℝ s ≤ (innerSL ℝ u).ker := by
apply Submodule.span_le.mpr
exact h
exact (Submodule.span ℝ s).topologicalClosure_minimal hs (innerSL ℝ u).isClosed_ker
theorem testGradient_pairing_ext {U : Set (Euc n)} {u e : VectorL2 n}
(h : ∀ φ : SmoothLocalTest U, inner ℝ u (testGradientLp φ)=inner ℝ e (testGradientLp φ)) :
∀ w∈testGradientSpace U, inner ℝ u w=inner ℝ e w := by
have hh := inner_zero_on_closed_span (Set.range (testGradientLp (U := U))) (u := u-e)
(by rintro _ ⟨φ,rfl⟩; rw [inner_sub_left,h φ,sub_self])
intro w hw
have hz := hh w hw
simpa only [inner_sub_left,sub_eq_zero] using hz
theorem testGradient_inner {U : Set (Euc n)} (u : VectorL2 n) (φ : SmoothLocalTest U) :
inner ℝ u (testGradientLp φ)=∫ x, inner ℝ (u x) (gradient φ.val x) := by
rw [MeasureTheory.L2.inner_def]
apply integral_congr_ae
filter_upwards [testGradientLp_ae φ] with x hx
rw [hx]
theorem testGradient_integrable_pairing (u : VectorL2 n) {U : Set (Euc n)}
(φ : SmoothLocalTest U) : Integrable (fun x ↦ inner ℝ (u x) (gradient φ.val x)) volume := by
apply (MeasureTheory.L2.integrable_inner (𝕜 := ℝ) u (testGradientLp φ)).congr
filter_upwards [testGradientLp_ae φ] with x hx
rw [hx]
def DivFreeLpOn (u : VectorL2 n) (U : Set (Euc n)) : Prop :=
∀ φ : SmoothLocalTest U, (∫ x, inner ℝ (u x) (gradient φ.val x))=0
theorem orthogonal_testGradient_divFree {U : Set (Euc n)} {u : VectorL2 n}
(hu : u∈(testGradientSpace U)ᗮ) : DivFreeLpOn u U := by
intro φ
rw [←testGradient_inner u φ]
exact Submodule.inner_left_of_mem_orthogonal (testGradient_mem φ) hu
theorem weak_harmonic_gradient_comparison {U : Set (Euc n)} {u e : VectorL2 n}
(h : ∀ φ : SmoothLocalTest U, (∫ x, inner ℝ (u x) (gradient φ.val x))=
∫ x, inner ℝ (e x) (gradient φ.val x)) :
∃ g : VectorL2 n, DivFreeLpOn g U ∧ u-g∈testGradientSpace U ∧ ‖u-g‖≤‖e‖ := by
obtain ⟨g,hg,hug,he⟩ := harmonic_projection_exists (testGradientSpace U)
(testGradient_pairing_ext (fun φ ↦ by simpa only [testGradient_inner] using h φ))
exact ⟨g,orthogonal_testGradient_divFree hg,hug,he⟩
end RPISingularHarnack
end
end OAI
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