| import OAI.Analysis.IntegralMeans.BoundaryTail |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory Filter Function InnerProductSpace |
| open scoped Topology ComplexConjugate Manifold NNReal ENNReal InnerProductSpace Classical |
| open MeasureTheory Function |
| open Set Filter |
| open Set MeasureTheory Filter Function |
| open Set MeasureTheory Filter Function InnerProductSpace |
| open TopologicalSpace |
| open scoped CompactlySupported |
| open scoped ENNReal |
| open scoped Manifold |
| open scoped Topology CompactlySupported ComplexConjugate |
| open scoped Topology ComplexConjugate Manifold NNReal ENNReal InnerProductSpace Classical |
| open scoped Topology ENNReal NNReal |
| namespace Brennan |
|
|
| attribute [local irreducible] classWeight |
| attribute [local irreducible] classFun |
|
|
| def pairLaplacian (g : ℝ × ℝ → ℝ) (p : ℝ × ℝ) : ℝ := |
| fderiv ℝ (fderiv ℝ g) p (1,0) (1,0) + fderiv ℝ (fderiv ℝ g) p (0,1) (0,1) |
|
|
| def traceFluxX (g : ℝ × ℝ → ℝ) (ε : ℝ) (p : ℝ × ℝ) : ℝ := |
| (p.2-ε)*fderiv ℝ g p (1,0) |
|
|
| def traceFluxY (g : ℝ × ℝ → ℝ) (ε : ℝ) (p : ℝ × ℝ) : ℝ := |
| (p.2-ε)*fderiv ℝ g p (0,1)-g p |
|
|
| lemma traceFlux_contDiffAt {g : ℝ × ℝ → ℝ} {p : ℝ × ℝ} |
| (hg : ContDiffAt ℝ 2 g p) (ε : ℝ) : |
| ContDiffAt ℝ 1 (traceFluxX g ε) p ∧ ContDiffAt ℝ 1 (traceFluxY g ε) p := by |
| have hd : ContDiffAt ℝ 1 (fderiv ℝ g) p := hg.fderiv_right (by norm_num) |
| have htr : ContDiffAt ℝ 1 g p := hg.of_le (by norm_num) |
| constructor |
| · unfold traceFluxX; fun_prop |
| · unfold traceFluxY; fun_prop |
|
|
| lemma divergence_traceFlux {g : ℝ × ℝ → ℝ} {p : ℝ × ℝ} |
| (hg : ContDiffAt ℝ 2 g p) (ε : ℝ) : |
| fderiv ℝ (traceFluxX g ε) p (1,0)+fderiv ℝ (traceFluxY g ε) p (0,1) = |
| (p.2-ε)*pairLaplacian g p := by |
| have hdg := hg.differentiableAt (by norm_num) |
| have hddg := (hg.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num) |
| have hx := hddg.hasFDerivAt.clm_apply (hasFDerivAt_const ((1,0) : ℝ × ℝ) p) |
| have hy := hddg.hasFDerivAt.clm_apply (hasFDerivAt_const ((0,1) : ℝ × ℝ) p) |
| have hs : HasFDerivAt (fun p : ℝ × ℝ => p.2-ε) (ContinuousLinearMap.snd ℝ ℝ ℝ) p := |
| (ContinuousLinearMap.snd ℝ ℝ ℝ).hasFDerivAt.sub_const ε |
| unfold traceFluxX traceFluxY |
| have hfx := (hs.mul hx).fderiv |
| have hfy := ((hs.mul hy).sub hdg.hasFDerivAt).fderiv |
| change fderiv ℝ (fun p => (p.2-ε)*fderiv ℝ g p (1,0)) p = _ at hfx |
| change fderiv ℝ (fun p => (p.2-ε)*fderiv ℝ g p (0,1)-g p) p = _ at hfy |
| rw [hfx,hfy] |
| simp [pairLaplacian,ContinuousLinearMap.flip_apply,mul_add] |
|
|
| lemma continuousAt_pairLaplacian {g : ℝ × ℝ → ℝ} {p : ℝ × ℝ} |
| (hg : ContDiffAt ℝ 2 g p) : ContinuousAt (pairLaplacian g) p := by |
| have hh : ContDiffAt ℝ 0 (fderiv ℝ (fderiv ℝ g)) p := |
| (hg.fderiv_right (m := 1) (by norm_num)).fderiv_right (by norm_num) |
| have hc := hh.continuousAt |
| unfold pairLaplacian |
| fun_prop |
|
|
| lemma rectangle_trace_identity {g : ℝ × ℝ → ℝ} {R ε Y : ℝ} |
| (hR : 0 ≤ R) (hεY : ε ≤ Y) |
| (hg : ∀ p ∈ Icc (-R,ε) (R,Y), ContDiffAt ℝ 2 g p) |
| (htop : ∀ x ∈ Icc (-R) R, g (x,Y) = 0 ∧ fderiv ℝ g (x,Y) (0,1) = 0) |
| (hside : ∀ y ∈ Icc ε Y, |
| fderiv ℝ g (-R,y) (1,0) = 0 ∧ fderiv ℝ g (R,y) (1,0) = 0) : |
| (∫ p in Icc (-R,ε) (R,Y), (p.2-ε)*pairLaplacian g p) = |
| ∫ x in -R..R, g (x,ε) := by |
| have hf : ContinuousOn (traceFluxX g ε) (Icc (-R,ε) (R,Y)) := |
| fun p hp => ((traceFlux_contDiffAt (hg p hp) ε).1.continuousAt).continuousWithinAt |
| have hh : ContinuousOn (traceFluxY g ε) (Icc (-R,ε) (R,Y)) := |
| fun p hp => ((traceFlux_contDiffAt (hg p hp) ε).2.continuousAt).continuousWithinAt |
| have hint : IntegrableOn (fun p : ℝ × ℝ => (p.2-ε)*pairLaplacian g p) |
| (Icc (-R,ε) (R,Y)) volume := by |
| apply ContinuousOn.integrableOn_compact isCompact_Icc |
| intro p hp |
| have hc : ContinuousAt (fun p : ℝ × ℝ => (p.2-ε)*pairLaplacian g p) p := |
| (continuous_snd.continuousAt.sub_const ε).mul (continuousAt_pairLaplacian (hg p hp)) |
| exact hc.continuousWithinAt |
| have heq (p : ℝ × ℝ) (hp : p ∈ Icc (-R,ε) (R,Y)) := divergence_traceFlux (hg p hp) ε |
| have hi : IntegrableOn (fun p => fderiv ℝ (traceFluxX g ε) p (1,0)+ |
| fderiv ℝ (traceFluxY g ε) p (0,1)) (Icc (-R,ε) (R,Y)) volume := |
| hint.congr (ae_restrict_of_forall_mem measurableSet_Icc (fun p hp => (heq p hp).symm)) |
| have hb := integral_divergence_prod_Icc_of_hasFDerivAt_of_le |
| (traceFluxX g ε) (traceFluxY g ε) (fderiv ℝ (traceFluxX g ε)) |
| (fderiv ℝ (traceFluxY g ε)) (-R,ε) (R,Y) ⟨by linarith,hεY⟩ hf hh |
| (fun p hp => ((traceFlux_contDiffAt (hg p ⟨⟨hp.1.1.le,hp.2.1.le⟩,⟨hp.1.2.le,hp.2.2.le⟩⟩) ε).1.differentiableAt (by norm_num)).hasFDerivAt) |
| (fun p hp => ((traceFlux_contDiffAt (hg p ⟨⟨hp.1.1.le,hp.2.1.le⟩,⟨hp.1.2.le,hp.2.2.le⟩⟩) ε).2.differentiableAt (by norm_num)).hasFDerivAt) hi |
| rw [setIntegral_congr_fun measurableSet_Icc heq] at hb |
| have ht : (∫ x in -R..R, traceFluxY g ε (x,Y)) = 0 := by |
| calc |
| _ = ∫ _ in -R..R, (0 : ℝ) := ?_ |
| _ = 0 := by simp |
| apply intervalIntegral.integral_congr |
| intro x hx |
| have hx' : x ∈ Icc (-R) R := by simpa only [uIcc_of_le (by linarith : -R ≤ R)] using hx |
| simp [traceFluxY,(htop x hx').1,(htop x hx').2] |
| have hl : (∫ y in ε..Y, traceFluxX g ε (-R,y)) = 0 := by |
| calc |
| _ = ∫ _ in ε..Y, (0 : ℝ) := ?_ |
| _ = 0 := by simp |
| apply intervalIntegral.integral_congr |
| intro y hy |
| simp [traceFluxX,(hside y (by simpa only [uIcc_of_le hεY] using hy)).1] |
| have hr : (∫ y in ε..Y, traceFluxX g ε (R,y)) = 0 := by |
| calc |
| _ = ∫ _ in ε..Y, (0 : ℝ) := ?_ |
| _ = 0 := by simp |
| apply intervalIntegral.integral_congr |
| intro y hy |
| simp [traceFluxX,(hside y (by simpa only [uIcc_of_le hεY] using hy)).2] |
| change _ = ((∫ x in -R..R, traceFluxY g ε (x,Y)) - ∫ x in -R..R, traceFluxY g ε (x,ε)) + |
| (∫ y in ε..Y, traceFluxX g ε (R,y)) - ∫ y in ε..Y, traceFluxX g ε (-R,y) at hb |
| rw [ht,hl,hr] at hb |
| simpa only [traceFluxY,sub_self,zero_mul,zero_sub, |
| intervalIntegral.integral_neg,sub_neg_eq_add,zero_add,add_zero,sub_zero] using hb |
|
|
| lemma half_trace_rectangle {g : ℝ × ℝ → ℝ} {ε : ℝ} (hε : 0 < ε) |
| (hc : ∀ p : ℝ × ℝ, 0 < p.2 → ContDiffAt ℝ 2 g p) |
| {K : Set (ℝ × ℝ)} (hK : IsCompact K) |
| (hs : ∀ p : ℝ × ℝ, ε/2 ≤ p.2 → g p ≠ 0 → p ∈ K) : |
| ∃ R : ℝ, 0 < R ∧ ε ≤ R ∧ Integrable (fun x : ℝ => g (x,ε)) ∧ |
| (∫ x : ℝ, g (x,ε)) = |
| ∫ p in Icc (-R,ε) (R,R), (p.2-ε)*pairLaplacian g p := by |
| obtain ⟨M,hM⟩ := hK.isBounded.subset_closedBall (0 : ℝ × ℝ) |
| let R := max M ε+1 |
| have hR : 0 < R := by dsimp only [R]; linarith [le_max_right M ε] |
| have hεR : ε ≤ R := by dsimp only [R]; linarith [le_max_right M ε] |
| have hMR : M < R := by dsimp only [R]; linarith [le_max_left M ε] |
| have hz (p : ℝ × ℝ) (hy : ε/2 ≤ p.2) (hn : M < ‖p‖) : g p = 0 := by |
| by_contra hg |
| have hh : ‖p‖ ≤ M := by simpa using hM (hs p hy hg) |
| exact (not_lt_of_ge hh) hn |
| have hznhd (p : ℝ × ℝ) (hy : ε/2 < p.2) (hn : M < ‖p‖) : |
| g =ᶠ[𝓝 p] (0 : ℝ × ℝ → ℝ) := by |
| have hy' : {q : ℝ × ℝ | ε/2 < q.2} ∈ 𝓝 p := |
| (isOpen_lt continuous_const continuous_snd).mem_nhds hy |
| have hn' : {q : ℝ × ℝ | M < ‖q‖} ∈ 𝓝 p := |
| (isOpen_lt continuous_const continuous_norm).mem_nhds hn |
| filter_upwards [hy',hn'] with q hq₁ hq₂ |
| exact hz q hq₁.le hq₂ |
| have hd (p : ℝ × ℝ) (hy : ε/2 < p.2) (hn : M < ‖p‖) : fderiv ℝ g p = 0 := by |
| rw [(hznhd p hy hn).fderiv_eq,fderiv_zero] |
| rfl |
| have htop (x : ℝ) (_hx : x ∈ Icc (-R) R) : |
| g (x,R) = 0 ∧ fderiv ℝ g (x,R) (0,1) = 0 := by |
| have hn : M < ‖(x,R)‖ := hMR.trans_le (by simpa only [Real.norm_eq_abs,abs_of_pos hR] using norm_snd_le (x,R)) |
| exact ⟨hz (x,R) (by linarith) hn,by rw [hd (x,R) (by linarith) hn]; rfl⟩ |
| have hside (y : ℝ) (hy : y ∈ Icc ε R) : |
| fderiv ℝ g (-R,y) (1,0) = 0 ∧ fderiv ℝ g (R,y) (1,0) = 0 := by |
| have h₁ : M < ‖(-R,y)‖ := hMR.trans_le (by simpa only [Real.norm_eq_abs,abs_neg,abs_of_pos hR] using norm_fst_le (-R,y)) |
| have h₂ : M < ‖(R,y)‖ := hMR.trans_le (by simpa only [Real.norm_eq_abs,abs_of_pos hR] using norm_fst_le (R,y)) |
| rw [hd (-R,y) (by linarith [hy.1]) h₁,hd (R,y) (by linarith [hy.1]) h₂] |
| simp |
| have hbox (p : ℝ × ℝ) (hp : p ∈ Icc (-R,ε) (R,R)) : ContDiffAt ℝ 2 g p := |
| hc p (hε.trans_le hp.1.2) |
| have hb := rectangle_trace_identity hR.le hεR hbox htop hside |
| have hzero (x : ℝ) (hx : x ∉ Icc (-R) R) : g (x,ε) = 0 := by |
| apply hz (x,ε) (by linarith) |
| have hh : R < |x| := by |
| have hh : x < -R ∨ R < x := by simpa only [mem_Icc,not_and_or,not_le] using hx |
| rcases hh with hh | hh |
| · exact (lt_neg_of_lt_neg hh).trans_le (neg_le_abs x) |
| · exact hh.trans_le (le_abs_self x) |
| exact hMR.trans (hh.trans_le (norm_fst_le (x,ε))) |
| have hcont : Continuous (fun x : ℝ => g (x,ε)) := by |
| apply continuous_iff_continuousAt.mpr |
| intro x |
| exact (hc (x,ε) hε).continuousAt.comp (f := fun x : ℝ => (x,ε)) |
| (continuous_id.prodMk continuous_const).continuousAt |
| have hint := hcont.continuousOn.integrableOn_compact (μ := volume) (isCompact_Icc (a := -R) (b := R)) |
| have hline : (∫ x in -R..R, g (x,ε)) = ∫ x : ℝ, g (x,ε) := by |
| rw [intervalIntegral.integral_of_le (by linarith : -R ≤ R), |
| setIntegral_congr_set Ioc_ae_eq_Icc] |
| exact setIntegral_eq_integral_of_forall_compl_eq_zero hzero |
| exact ⟨R,hR,hεR,hint.integrable_of_forall_notMem_eq_zero hzero,(hline.symm.trans hb.symm)⟩ |
|
|
| lemma ofReal_integral_le_lintegral_abs.{u_1} {α : Type u_1} [MeasurableSpace α] |
| {μ : Measure α} {f : α → ℝ} (hf : Integrable f μ) : |
| ENNReal.ofReal (∫ x, f x ∂μ) ≤ ∫⁻ x, ENNReal.ofReal |f x| ∂μ := by |
| rw [← ofReal_integral_eq_lintegral_ofReal hf.abs (Eventually.of_forall (fun x => abs_nonneg (f x)))] |
| exact ENNReal.ofReal_le_ofReal (integral_mono hf hf.abs (fun x => le_abs_self _)) |
|
|
| lemma half_trace_lintegral_le {g : ℝ × ℝ → ℝ} {ε : ℝ} (hε : 0 < ε) |
| (hc : ∀ p : ℝ × ℝ, 0 < p.2 → ContDiffAt ℝ 2 g p) |
| (hn : ∀ p : ℝ × ℝ, 0 ≤ g p) |
| {K : Set (ℝ × ℝ)} (hK : IsCompact K) |
| (hs : ∀ p : ℝ × ℝ, ε/2 ≤ p.2 → g p ≠ 0 → p ∈ K) : |
| (∫⁻ x : ℝ, ENNReal.ofReal (g (x,ε))) ≤ |
| ∫⁻ p in {p : ℝ × ℝ | ε ≤ p.2}, ENNReal.ofReal ((p.2-ε)*|pairLaplacian g p|) := by |
| obtain ⟨R,_,_,hi,he⟩ := half_trace_rectangle hε hc hK hs |
| rw [← ofReal_integral_eq_lintegral_ofReal hi (Eventually.of_forall (fun x => hn (x,ε))),he] |
| have hi₂ : IntegrableOn (fun p : ℝ × ℝ => (p.2-ε)*pairLaplacian g p) |
| (Icc (-R,ε) (R,R)) volume := by |
| apply ContinuousOn.integrableOn_compact isCompact_Icc |
| intro p hp |
| have hp' : 0 < p.2 := hε.trans_le hp.1.2 |
| have hh : ContinuousAt (fun p : ℝ × ℝ => (p.2-ε)*pairLaplacian g p) p := |
| (continuous_snd.continuousAt.sub_const ε).mul (continuousAt_pairLaplacian (hc p hp')) |
| exact hh.continuousWithinAt |
| refine (ofReal_integral_le_lintegral_abs hi₂).trans ?_ |
| calc |
| (∫⁻ p in Icc (-R,ε) (R,R), ENNReal.ofReal |(p.2-ε)*pairLaplacian g p|) = |
| ∫⁻ p in Icc (-R,ε) (R,R), ENNReal.ofReal ((p.2-ε)*|pairLaplacian g p|) := by |
| apply setLIntegral_congr_fun measurableSet_Icc |
| intro p hp |
| simp only [abs_mul,abs_of_nonneg (sub_nonneg.mpr hp.1.2)] |
| _ ≤ _ := lintegral_mono_set (fun p hp => hp.1.2) |
|
|
| def secondReal (f : ℂ → ℝ) (z u v : ℂ) : ℝ := fderiv ℝ (fderiv ℝ f) z u v |
|
|
| def complexSecondReal (f : ℂ → ℂ) (z u v : ℂ) : ℂ := fderiv ℝ (fderiv ℝ f) z u v |
|
|
| lemma secondReal_comp {ψ : ℂ → ℝ} {F : ℂ → ℂ} {z : ℂ} |
| (hψ : ContDiffAt ℝ 2 ψ (F z)) (hF : ContDiffAt ℝ 2 F z) (u v : ℂ) : |
| secondReal (ψ ∘ F) z u v = |
| fderiv ℝ ψ (F z) (complexSecondReal F z u v) + |
| secondReal ψ (F z) (fderiv ℝ F z u) (fderiv ℝ F z v) := by |
| have hdψ := hψ.differentiableAt (by norm_num) |
| have hdF := hF.differentiableAt (by norm_num) |
| have hddψ := (hψ.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num) |
| have hddF := (hF.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num) |
| have he : fderiv ℝ (ψ ∘ F) =ᶠ[𝓝 z] fun w => |
| (fderiv ℝ ψ (F w)).comp (fderiv ℝ F w) := by |
| filter_upwards [hF.eventually (by simp),hdF.continuousAt.eventually (hψ.eventually (by simp))] with w hFw hψw |
| exact fderiv_comp w (hψw.differentiableAt (by norm_num)) (hFw.differentiableAt (by norm_num)) |
| have hh := ((hddψ.hasFDerivAt.comp z hdF.hasFDerivAt).clm_comp hddF.hasFDerivAt).congr_of_eventuallyEq he |
| rw [secondReal,hh.fderiv] |
| simp [secondReal,complexSecondReal,ContinuousLinearMap.compL_apply] |
|
|
| lemma bilinear_complex_rotation (B : ℂ →L[ℝ] ℂ →L[ℝ] ℝ) (c : ℂ) : |
| B c c + B (c*Complex.I) (c*Complex.I) = ‖c‖^2*(B 1 1+B Complex.I Complex.I) := by |
| have hc : c = c.re • (1 : ℂ)+c.im • Complex.I := by simp |
| have hci : c*Complex.I = (-c.im) • (1 : ℂ)+c.re • Complex.I := by |
| apply Complex.ext <;> simp [Complex.mul_re,Complex.mul_im] |
| have hn : ‖c‖^2 = c.re^2+c.im^2 := by |
| rw [Complex.sq_norm] |
| simp [Complex.normSq_apply,pow_two] |
| rw [hci,hc] |
| simp only [map_add,map_smul,add_apply,smul_apply,smul_eq_mul] |
| rw [← hc,hn] |
| ring |
|
|
| lemma laplacian_eq_secondReal (f : ℂ → ℝ) (z : ℂ) : |
| Laplacian.laplacian f z = secondReal f z 1 1+secondReal f z Complex.I Complex.I := by |
| simp [laplacian_eq_iteratedFDeriv_complexPlane,iteratedFDeriv_two_apply,secondReal] |
|
|
| lemma laplacian_eq_complexSecondReal (f : ℂ → ℂ) (z : ℂ) : |
| Laplacian.laplacian f z = complexSecondReal f z 1 1+complexSecondReal f z Complex.I Complex.I := by |
| simp [laplacian_eq_iteratedFDeriv_complexPlane,iteratedFDeriv_two_apply,complexSecondReal] |
|
|
| lemma fderiv_real_complex_apply {F : ℂ → ℂ} {z : ℂ} |
| (hF : DifferentiableAt ℂ F z) (v : ℂ) : fderiv ℝ F z v = deriv F z*v := by |
| rw [(hF.hasDerivAt.hasFDerivAt.restrictScalars ℝ).fderiv] |
| simp [smul_eq_mul,mul_comm] |
|
|
| lemma laplacian_holomorphic_comp {ψ : ℂ → ℝ} {F : ℂ → ℂ} {z : ℂ} |
| (hψ : ContDiffAt ℝ 2 ψ (F z)) (hF : AnalyticAt ℂ F z) : |
| Laplacian.laplacian (ψ ∘ F) z = ‖deriv F z‖^2*Laplacian.laplacian ψ (F z) := by |
| rw [laplacian_eq_secondReal,secondReal_comp hψ (hF.contDiffAt.restrict_scalars ℝ), |
| secondReal_comp hψ (hF.contDiffAt.restrict_scalars ℝ)] |
| have hz := hF.harmonicAt.2.eq_of_nhds |
| rw [laplacian_eq_complexSecondReal] at hz |
| change complexSecondReal F z 1 1+complexSecondReal F z Complex.I Complex.I = 0 at hz |
| have he : fderiv ℝ ψ (F z) (complexSecondReal F z 1 1)+ |
| fderiv ℝ ψ (F z) (complexSecondReal F z Complex.I Complex.I) = 0 := by |
| rw [← map_add,hz,map_zero] |
| simp only [fderiv_real_complex_apply hF.differentiableAt,mul_one] |
| have hb := bilinear_complex_rotation (fderiv ℝ (fderiv ℝ ψ) (F z)) (deriv F z) |
| rw [laplacian_eq_secondReal] |
| dsimp only [secondReal] at * |
| linarith |
|
|
| lemma pairLaplacian_complex (f : ℂ → ℝ) (p : ℝ × ℝ) : |
| pairLaplacian (f ∘ Complex.equivRealProdCLM.symm) p = |
| Laplacian.laplacian f (Complex.equivRealProdCLM.symm p) := by |
| have he := Complex.equivRealProdCLM.symm.iteratedFDerivWithin_comp_right |
| f (s := univ) uniqueDiffOn_univ (mem_univ (Complex.equivRealProdCLM.symm p)) 2 |
| simp only [preimage_univ,iteratedFDerivWithin_univ] at he |
| have h₁ := congrArg (fun B => B ![((1,0) : ℝ × ℝ),(1,0)]) he |
| have h₂ := congrArg (fun B => B ![((0,1) : ℝ × ℝ),(0,1)]) he |
| simp only [iteratedFDeriv_two_apply,ContinuousMultilinearMap.compContinuousLinearMap_apply, |
| Matrix.cons_val_zero,Matrix.cons_val_one,ContinuousLinearEquiv.coe_coe] at h₁ h₂ |
| have hc₁ : Complex.equivRealProdCLM.symm (1,0) = 1 := by simp [Complex.equivRealProdCLM_symm_apply] |
| have hc₂ : Complex.equivRealProdCLM.symm (0,1) = Complex.I := by simp [Complex.equivRealProdCLM_symm_apply] |
| rw [hc₁] at h₁ |
| rw [hc₂] at h₂ |
| rw [pairLaplacian,h₁,h₂,laplacian_eq_secondReal] |
| rfl |
|
|
| lemma laplacian_eq_zero_outside_tsupport {ψ : ℂ → ℝ} {z : ℂ} |
| (hz : z ∉ tsupport ψ) : Laplacian.laplacian ψ z = 0 := by |
| have he := notMem_tsupport_iff_eventuallyEq.mp hz |
| have hh := laplacian_congr_nhds he |
| exact hh.eq_of_nhds.trans (by |
| change Laplacian.laplacian (fun _ : ℂ => (0 : ℝ)) z = 0 |
| rw [laplacian_const] |
| rfl) |
|
|
| lemma continuous_laplacian {ψ : ℂ → ℝ} (hψ : ContDiff ℝ 2 ψ) : |
| Continuous (Laplacian.laplacian ψ) := by |
| have hh : ContDiff ℝ 0 (fderiv ℝ (fderiv ℝ ψ)) := |
| (hψ.fderiv_right (m := 1) (by norm_num)).fderiv_right (by norm_num) |
| have hc := hh.continuous |
| have he : Laplacian.laplacian ψ = fun z => secondReal ψ z 1 1+secondReal ψ z Complex.I Complex.I := |
| funext (laplacian_eq_secondReal ψ) |
| rw [he] |
| unfold secondReal |
| fun_prop |
|
|
| lemma compactSupport_laplacian {ψ : ℂ → ℝ} (hψ : HasCompactSupport ψ) : |
| HasCompactSupport (Laplacian.laplacian ψ) := by |
| apply hψ.mono' |
| intro z hz |
| by_contra hn |
| exact hz (laplacian_eq_zero_outside_tsupport hn) |
|
|
| lemma conformal_lintegral_le {F : ℂ → ℂ} {s : Set ℂ} (hs : MeasurableSet s) |
| (hF : ∀ z ∈ s, DifferentiableAt ℂ F z) (hi : InjOn F s) (b : ℂ → ℝ≥0∞) : |
| (∫⁻ z in s, ENNReal.ofReal (‖deriv F z‖^2)*b (F z)) ≤ ∫⁻ w, b w := by |
| have he := lintegral_image_eq_lintegral_abs_det_fderiv_mul volume hs |
| (fun z hz => (hF z hz).hasFDerivAt.restrictScalars ℝ |>.hasFDerivWithinAt) hi b |
| have hh : (∫⁻ z in s, ENNReal.ofReal (‖deriv F z‖^2)*b (F z)) = ∫⁻ w in F '' s, b w := by |
| rw [he] |
| apply setLIntegral_congr_fun hs |
| intro z hz |
| have hd := (hF z hz).hasFDerivAt.restrictScalars ℝ |
| have hf : (ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ F z)) = fderiv ℝ F z := hd.fderiv.symm |
| dsimp only |
| rw [hf] |
| have hd' : (fderiv ℝ F z).det = ‖deriv F z‖^2 := det_fderiv_complex (hF z hz) |
| rw [hd',abs_of_nonneg (sq_nonneg _)] |
| rw [hh] |
| exact lintegral_mono' Measure.restrict_le_self le_rfl |
|
|
| lemma conformal_cutoff_trace_bound {F : ℂ → ℂ} {ψ : ℂ → ℝ} |
| (hF : ∀ z : ℂ, 0 < z.im → AnalyticAt ℂ F z) |
| (hi : InjOn F {z : ℂ | 0 < z.im}) |
| (hψ : ContDiff ℝ 2 ψ) (hn : ∀ w, 0 ≤ ψ w) (_hψK : HasCompactSupport ψ) |
| (hproper : ∀ η : ℝ, 0 < η → IsCompact ({z : ℂ | η ≤ z.im} ∩ F ⁻¹' tsupport ψ)) |
| {H : ℝ} (hH : 0 ≤ H) |
| (hheight : ∀ z : ℂ, 0 < z.im → F z ∈ tsupport ψ → z.im ≤ H) : |
| ∀ ε : ℝ, 0 < ε → (∫⁻ x : ℝ, ENNReal.ofReal (ψ (F (x+ε*Complex.I)))) ≤ |
| ENNReal.ofReal H * ∫⁻ w : ℂ, ENNReal.ofReal |Laplacian.laplacian ψ w| := by |
| intro ε hε |
| let C := Complex.equivRealProdCLM.symm |
| let g : ℝ × ℝ → ℝ := (ψ ∘ F) ∘ C |
| have hc (p : ℝ × ℝ) (hp : 0 < p.2) : ContDiffAt ℝ 2 g p := by |
| have hz : 0 < (C p).im := hp |
| exact (hψ.contDiffAt.comp _ ((hF (C p) hz).contDiffAt.restrict_scalars ℝ)).comp _ C.contDiff.contDiffAt |
| have hK := (hproper (ε/2) (by linarith)).image Complex.equivRealProdCLM.continuous |
| have hs (p : ℝ × ℝ) (hp : ε/2 ≤ p.2) (hg : g p ≠ 0) : |
| p ∈ Complex.equivRealProdCLM '' ({z : ℂ | ε/2 ≤ z.im} ∩ F ⁻¹' tsupport ψ) := by |
| refine ⟨C p,⟨hp,subset_closure hg⟩,?_⟩ |
| exact Complex.equivRealProdCLM.apply_symm_apply p |
| have ht := half_trace_lintegral_le hε hc (fun p => hn _) hK hs |
| have ht' : (∫⁻ x : ℝ, ENNReal.ofReal (ψ (F (x+ε*Complex.I)))) ≤ |
| ∫⁻ z in {z : ℂ | ε ≤ z.im}, ENNReal.ofReal ((z.im-ε)*|Laplacian.laplacian (ψ ∘ F) z|) := by |
| have he := Complex.volume_preserving_equiv_real_prod.setLIntegral_comp_preimage_emb |
| Complex.measurableEquivRealProd.measurableEmbedding |
| (fun p : ℝ × ℝ => ENNReal.ofReal ((p.2-ε)*|pairLaplacian g p|)) {p : ℝ × ℝ | ε ≤ p.2} |
| have hh : (∫⁻ p in {p : ℝ × ℝ | ε ≤ p.2}, ENNReal.ofReal ((p.2-ε)*|pairLaplacian g p|)) = |
| ∫⁻ z in {z : ℂ | ε ≤ z.im}, ENNReal.ofReal ((z.im-ε)*|Laplacian.laplacian (ψ ∘ F) z|) := by |
| rw [← he] |
| apply lintegral_congr |
| intro z |
| dsimp only [g,C] |
| rw [pairLaplacian_complex] |
| rfl |
| rw [hh] at ht |
| simpa only [g,C,Function.comp_apply,Complex.equivRealProdCLM_symm_apply] using ht |
| refine ht'.trans ?_ |
| have hmeas : MeasurableSet {z : ℂ | ε ≤ z.im} := isClosed_le continuous_const Complex.continuous_im |>.measurableSet |
| calc |
| (∫⁻ z in {z : ℂ | ε ≤ z.im}, ENNReal.ofReal ((z.im-ε)*|Laplacian.laplacian (ψ ∘ F) z|)) ≤ |
| ∫⁻ z in {z : ℂ | ε ≤ z.im}, ENNReal.ofReal H * |
| (ENNReal.ofReal (‖deriv F z‖^2)*ENNReal.ofReal |Laplacian.laplacian ψ (F z)|) := by |
| apply setLIntegral_mono' hmeas |
| intro z hz |
| rw [laplacian_holomorphic_comp hψ.contDiffAt (hF z (hε.trans_le hz)), |
| abs_mul,abs_of_nonneg (sq_nonneg _),← ENNReal.ofReal_mul (sq_nonneg _), |
| ← ENNReal.ofReal_mul hH] |
| apply ENNReal.ofReal_le_ofReal |
| by_cases hd : Laplacian.laplacian ψ (F z) = 0 |
| · simp [hd] |
| have hm : F z ∈ tsupport ψ := by |
| by_contra hh |
| exact hd (laplacian_eq_zero_outside_tsupport hh) |
| nlinarith [sq_nonneg ‖deriv F z‖,abs_nonneg (Laplacian.laplacian ψ (F z)), |
| hheight z (hε.trans_le hz) hm, |
| mul_nonneg (sq_nonneg ‖deriv F z‖) (abs_nonneg (Laplacian.laplacian ψ (F z)))] |
| _ = ENNReal.ofReal H * ∫⁻ z in {z : ℂ | ε ≤ z.im}, |
| ENNReal.ofReal (‖deriv F z‖^2)*ENNReal.ofReal |Laplacian.laplacian ψ (F z)| := |
| lintegral_const_mul' _ _ ENNReal.ofReal_ne_top |
| _ ≤ _ := by |
| exact mul_le_mul le_rfl (conformal_lintegral_le hmeas |
| (fun z hz => (hF z (hε.trans_le hz)).differentiableAt) |
| (hi.mono (fun z hz => hε.trans_le hz)) |
| (fun w => ENNReal.ofReal |Laplacian.laplacian ψ w|)) zero_le zero_le |
|
|
| lemma exists_trace_smooth_cutoff (K : Set ℂ) (hK : IsCompact K) : |
| ∃ ψ : ℂ → ℝ, ContDiff ℝ 2 ψ ∧ (∀ w, 0 ≤ ψ w) ∧ HasCompactSupport ψ ∧ |
| ∀ w ∈ K, ψ w = 1 := by |
| obtain ⟨R,hR⟩ := hK.isBounded.subset_closedBall (0 : ℂ) |
| let b : ContDiffBump (0 : ℂ) := ⟨max R 0+1,max R 0+2,by positivity,by linarith⟩ |
| refine ⟨b,b.contDiff,b.nonneg',b.hasCompactSupport,fun w hw => ?_⟩ |
| apply b.one_of_mem_closedBall |
| exact Metric.closedBall_subset_closedBall (by dsimp [b]; linarith [le_max_left R 0]) (hR hw) |
|
|
| lemma conformal_trace_bound {F : ℂ → ℂ} |
| (hF : ∀ z : ℂ, 0 < z.im → AnalyticAt ℂ F z) |
| (hi : InjOn F {z : ℂ | 0 < z.im}) |
| (hproper : ∀ K : Set ℂ, IsCompact K → ∀ η : ℝ, 0 < η → |
| IsCompact ({z : ℂ | η ≤ z.im} ∩ F ⁻¹' K)) |
| (hheight : ∀ K : Set ℂ, IsCompact K → ∃ H : ℝ, 0 < H ∧ |
| ∀ z : ℂ, 0 < z.im → F z ∈ K → z.im ≤ H) |
| {K : Set ℂ} (hK : IsCompact K) : |
| ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ ε : ℝ, 0 < ε → |
| volume {x : ℝ | F (x+ε*Complex.I) ∈ K} ≤ C := by |
| obtain ⟨ψ,hψ,hn,hψK,hψone⟩ := exists_trace_smooth_cutoff K hK |
| obtain ⟨H,hH,hHb⟩ := hheight (tsupport ψ) hψK |
| let C := ENNReal.ofReal H * ∫⁻ w : ℂ, ENNReal.ofReal |Laplacian.laplacian ψ w| |
| have hfin : C < ∞ := by |
| apply ENNReal.mul_lt_top ENNReal.ofReal_lt_top |
| have hi := (continuous_laplacian hψ).integrable_of_hasCompactSupport (μ := volume) (compactSupport_laplacian hψK) |
| simpa only [HasFiniteIntegral,← ofReal_norm,Real.norm_eq_abs] using hi.hasFiniteIntegral |
| refine ⟨C,hfin,fun ε hε => ?_⟩ |
| have hb := conformal_cutoff_trace_bound hF hi hψ hn hψK |
| (hproper (tsupport ψ) hψK) hH.le hHb ε hε |
| refine le_trans ?_ hb |
| have hcont : Continuous (fun x : ℝ => F (x+ε*Complex.I)) := by |
| apply continuous_iff_continuousAt.mpr |
| intro x |
| have hz : 0 < (x+ε*Complex.I : ℂ).im := by simpa using hε |
| exact (hF _ hz).continuousAt.comp (f := fun x : ℝ => (x+ε*Complex.I : ℂ)) |
| (by fun_prop) |
| have hm : MeasurableSet {x : ℝ | F (x+ε*Complex.I) ∈ K} := |
| hK.isClosed.measurableSet.preimage hcont.measurable |
| rw [← lintegral_indicator_one hm] |
| apply lintegral_mono |
| intro x |
| by_cases hx : F (x+ε*Complex.I) ∈ K |
| · simp [hx,hψone _ hx] |
| · simp [hx] |
|
|
| lemma class_trace_bound (g : DiskClass) {β : ℝ} (hβ : 1 < β) |
| (hfin : (∫⁻ z in outsideCore, |
| ENNReal.ofReal (z.im^(β+1)*‖classFun g z‖⁻¹^4)) < ∞) |
| {K : Set ℂ} (hK : IsCompact K) : |
| ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ ε : ℝ, 0 < ε → |
| volume {x : ℝ | classFun g (x+ε*Complex.I) ∈ K} ≤ C := by |
| apply conformal_trace_bound |
| (fun z hz => (classFun_schlicht g).1.1.analyticAt (isOpen_halfPlane.mem_nhds hz)) |
| (classFun_schlicht g).1.2 |
| (fun _ hK η hη => class_proper_preimage g hβ hfin hK hη) |
| (fun _ hK => class_preimage_height_bound g hβ hfin hK) hK |
|
|
| lemma uniform_relative_class_bound : |
| ∃ M : ℝ, 0 ≤ M ∧ ∀ (g : DiskClass) (w : ℂ), |w.re| ≤ 1 → |
| (1/2 : ℝ) ≤ w.im → w.im ≤ 1 → ‖classFun g w‖ ≤ M := by |
| let B : Set (ℝ × ℝ) := Icc (-1,1/2) (1,1) |
| let K : Set ℂ := Complex.equivRealProdCLM.symm '' B |
| have hK : IsCompact K := isCompact_Icc.image Complex.equivRealProdCLM.symm.continuous |
| have hKsub : K ⊆ Set.range (fun w : halfPlane => (w : ℂ)) := by |
| rintro _ ⟨p,hp,rfl⟩ |
| refine ⟨⟨Complex.equivRealProdCLM.symm p,?_⟩,rfl⟩ |
| change 0 < p.2 |
| linarith [hp.1.2] |
| have hKs : IsCompact ((fun w : halfPlane => (w : ℂ)) ⁻¹' K) := |
| Topology.IsEmbedding.subtypeVal.isInducing.isCompact_preimage' hK hKsub |
| have hprod := (isCompact_univ (X := DiskClass)).prod hKs |
| obtain ⟨M,hM⟩ := hprod.exists_bound_of_continuousOn continuous_classFun_eval.continuousOn |
| refine ⟨max M 0,le_max_right _ _,fun g w hx hy hyy => ?_⟩ |
| have hw : w ∈ halfPlane := by change 0 < w.im; linarith |
| have hmem : (g,⟨w,hw⟩) ∈ (univ : Set DiskClass) ×ˢ ((fun w : halfPlane => (w : ℂ)) ⁻¹' K) := by |
| refine ⟨mem_univ _,⟨(w.re,w.im),?_,?_⟩⟩ |
| · exact ⟨⟨(abs_le.mp hx).1,hy⟩,⟨(abs_le.mp hx).2,hyy⟩⟩ |
| · exact Complex.equivRealProdCLM.symm_apply_apply w |
| exact (hM _ hmem).trans (le_max_left _ _) |
|
|
| end Brennan |
|
|
| end |
|
|
| end OAI |
|
|
