| import OAI.Geometry.Perimeter.Regularity.SmoothFlatOrder |
| import OAI.Geometry.Perimeter.Regularity.EllipticTestSpace |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal ContDiff InnerProductSpace |
| namespace RPISingularHarnack |
| variable {n : ℕ} |
|
|
| def scalarDirection (φ : Euc n → ℝ) (a : Euc n) (x : Euc n) := fderiv ℝ φ x a |
|
|
| theorem scalarDirection_contDiff {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (a : Euc n) : |
| ContDiff ℝ ∞ (scalarDirection φ a) := |
| (hφ.fderiv_right (by simp)).clm_apply contDiff_const |
|
|
| theorem scalarDirection_tsupport (φ : Euc n → ℝ) (a : Euc n) : |
| tsupport (scalarDirection φ a) ⊆ tsupport φ := by |
| apply closure_minimal _ (isClosed_tsupport _) |
| intro x hx |
| by_contra hh |
| exact hx (by simp [scalarDirection,fderiv_of_notMem_tsupport ℝ hh]) |
|
|
| theorem scalarDirection_fderiv {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) |
| (a x b : Euc n) : |
| fderiv ℝ (scalarDirection φ a) x b=fderiv ℝ (fderiv ℝ φ) x b a := by |
| have hd : ContDiff ℝ ∞ (fderiv ℝ φ) := hφ.fderiv_right (by simp) |
| have h := ((ContinuousLinearMap.apply ℝ ℝ a).hasFDerivAt.comp x |
| ((hd.differentiable (by simp)) x).hasFDerivAt).fderiv |
| exact congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L b) h |
|
|
| def directionLocalTest {U : Set (Euc n)} (φ : SmoothLocalTest U) (a : Euc n) : SmoothLocalTest U := |
| ⟨scalarDirection φ.val a,scalarDirection_contDiff φ.property.1 a, |
| φ.property.2.1.of_isClosed_subset (isClosed_tsupport _) (scalarDirection_tsupport _ _), |
| (scalarDirection_tsupport _ _).trans φ.property.2.2⟩ |
|
|
| def curlTest (φ : Euc n → ℝ) (a b : Euc n) (x : Euc n) : Euc n := |
| scalarDirection φ a x • b - scalarDirection φ b x • a |
|
|
| theorem curlTest_contDiff {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (a b : Euc n) : |
| ContDiff ℝ ∞ (curlTest φ a b) := |
| ((scalarDirection_contDiff hφ a).smul contDiff_const).sub |
| ((scalarDirection_contDiff hφ b).smul contDiff_const) |
|
|
| theorem curlTest_tsupport (φ : Euc n → ℝ) (a b : Euc n) : |
| tsupport (curlTest φ a b) ⊆ tsupport φ := by |
| apply closure_minimal _ (isClosed_tsupport _) |
| intro x hx |
| by_contra hh |
| exact hx (by simp [curlTest,scalarDirection,fderiv_of_notMem_tsupport ℝ hh]) |
|
|
| theorem curlTest_divergence {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (a b : Euc n) : |
| divergence (curlTest φ a b)=0 := by |
| have ha := (scalarDirection_contDiff hφ a).differentiable (by simp) |
| have hb := (scalarDirection_contDiff hφ b).differentiable (by simp) |
| rw [show curlTest φ a b=(fun x ↦ scalarDirection φ a x • b)-(fun x ↦ scalarDirection φ b x • a) from rfl, |
| divergence_sub (ha.smul_const b) (hb.smul_const a),divergence_scalar_direction ha, |
| divergence_scalar_direction hb] |
| funext x |
| simp only [Pi.sub_apply,Pi.zero_apply,scalarDirection_fderiv hφ] |
| exact sub_eq_zero.mpr ((hφ.contDiffAt.isSymmSndFDerivAt (by simp only [minSmoothness_of_isRCLikeNormedField] |
|
|
| def curlTestLp {U : Set (Euc n)} (φ : SmoothLocalTest U) (a b : Euc n) : VectorL2 n := |
| ((curlTest_contDiff φ.property.1 a b).continuous.memLp_of_hasCompactSupport |
| (φ.property.2.1.of_isClosed_subset (isClosed_tsupport _) (curlTest_tsupport _ _ _))).toLp |
| (curlTest φ.val a b) |
| |
| theorem curlTestLp_ae {U : Set (Euc n)} (φ : SmoothLocalTest U) (a b : Euc n) : |
| (curlTestLp φ a b : Euc n → Euc n)=ᵐ[volume] curlTest φ.val a b := MemLp.coeFn_toLp _ |
|
|
| def CurlFreeLpOn (u : VectorL2 n) (U : Set (Euc n)) : Prop := |
| ∀ (φ : SmoothLocalTest U) (a b : Euc n), inner ℝ u (curlTestLp φ a b)=0 |
| |
| theorem testGradient_curlTest_inner {U V : Set (Euc n)} (ψ : SmoothLocalTest V) |
| (φ : SmoothLocalTest U) (a b : Euc n) : |
| inner ℝ (testGradientLp ψ) (curlTestLp φ a b)=0 := by |
| rw [MeasureTheory.L2.inner_def] |
| have he : (∫ x, inner ℝ (testGradientLp ψ x) (curlTestLp φ a b x))= |
| ∫ x, inner ℝ (gradient ψ.val x) (curlTest φ.val a b x) := by |
| apply integral_congr_ae |
| filter_upwards [testGradientLp_ae ψ,curlTestLp_ae φ a b] with x hx hy |
| rw [hx,hy] |
| rw [he] |
| have hi := integral_mul_divergence_eq_neg_inner_gradient_local ψ.property.1 |
| (curlTest_contDiff φ.property.1 a b) |
| (φ.property.2.1.of_isClosed_subset (isClosed_tsupport _) (curlTest_tsupport _ _ _)) |
| rw [curlTest_divergence φ.property.1] at hi |
| simpa only [Pi.zero_apply,mul_zero,integral_zero,eq_neg_iff_add_eq_zero,zero_add] using hi |
|
|
| theorem testGradientSpace_curlFree {U V : Set (Euc n)} {u : VectorL2 n} |
| (hu : u∈testGradientSpace V) : CurlFreeLpOn u U := by |
| intro φ a b |
| have hz := inner_zero_on_closed_span (Set.range (testGradientLp (U := V))) |
| (u := curlTestLp φ a b) (by |
| rintro _ ⟨ψ,rfl⟩ |
| rw [real_inner_comm] |
| exact testGradient_curlTest_inner ψ φ a b) u hu |
| rwa [real_inner_comm] at hz |
|
|
| theorem CurlFreeLpOn.sub {U : Set (Euc n)} {u v : VectorL2 n} |
| (hu : CurlFreeLpOn u U) (hv : CurlFreeLpOn v U) : CurlFreeLpOn (u-v) U := by |
| intro φ a b |
| rw [inner_sub_left,hu φ a b,hv φ a b,sub_self] |
| |
| theorem weak_curl_harmonic_comparison {U : Set (Euc n)} {u e : VectorL2 n} |
| (hu : CurlFreeLpOn u U) |
| (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 ∧ CurlFreeLpOn g U ∧ ‖u-g‖≤‖e‖ := by |
| obtain ⟨g,hg,hug,he⟩ := weak_harmonic_gradient_comparison h |
| have hcf := hu.sub (testGradientSpace_curlFree hug) |
| rw [sub_sub_cancel] at hcf |
| exact ⟨g,hg,hcf,he⟩ |
|
|
| end RPISingularHarnack |
|
|
| end |
|
|
| end OAI |
|
|
