import OAI.Analysis.HotSpots.Multiplier namespace OAI section ClosedChartTopology noncomputable section open Set Metric open scoped Topology namespace StrictHotSpots.Conformal.ClosedDiskChart open DiskH10 PlaneGreen variable {Ω : Set Plane} (c : ClosedDiskChart Ω) lemma complex_maps_disk {z : ℂ} (hz : z ∈ ball 0 1) : c.F (complexIso z) ∈ Ω := by have hx : complexIso z ∈ disk := by simpa only [disk,mem_ball_zero_iff,LinearIsometryEquiv.norm_map] using hz rw [← c.diskMap_agrees hx] exact c.diskMap.map_source hx lemma complex_maps_closed {z : ℂ} (hz : z ∈ closedBall 0 1) : c.F (complexIso z) ∈ closure Ω := by apply c.maps_closed simpa only [disk,closure_ball (0:Plane) (by norm_num : (1:ℝ) ≠ 0),mem_closedBall_zero_iff, LinearIsometryEquiv.norm_map] using hz lemma complex_maps_boundary (s : Circle) : c.F (complexIso (s:ℂ)) ∈ frontier Ω := by apply c.maps_boundary simp only [disk,frontier_ball (0:Plane) (by norm_num : (1:ℝ) ≠ 0),mem_sphere_zero_iff_norm, LinearIsometryEquiv.norm_map,Circle.norm_coe] lemma complex_inverse_disk {x : Plane} (hx : x ∈ Ω) : complexIso.symm (c.G x) ∈ ball 0 1 := by have hh := c.diskMap.map_target hx change c.diskMap.symm x ∈ disk at hh rw [c.diskMap_inverse_agrees hx] at hh simpa only [disk,mem_ball_zero_iff,LinearIsometryEquiv.norm_map] using hh lemma complex_inverse_boundary {x : Plane} (hx : x ∈ frontier Ω) : ‖complexIso.symm (c.G x)‖ = 1 := by have hh := c.inverse_maps_boundary hx simpa only [disk,frontier_ball (0:Plane) (by norm_num : (1:ℝ) ≠ 0),mem_sphere_zero_iff_norm, LinearIsometryEquiv.norm_map] using hh lemma complex_inverse_image (S : Set Plane) (hs : S ⊆ Ω) : (fun x => complexIso.symm (c.G x)) '' S = ball 0 1 ∩ (fun z => c.F (complexIso z)) ⁻¹' S := by ext z constructor · rintro ⟨x,hx,rfl⟩ refine ⟨c.complex_inverse_disk (hs hx),?_⟩ simpa only [mem_preimage,LinearIsometryEquiv.apply_symm_apply,c.right_inverse x (subset_closure (hs hx))] using hx · rintro ⟨hz,hx⟩ refine ⟨c.F (complexIso z),hx,?_⟩ have hd : complexIso z ∈ closure disk := by exact subset_closure (by simpa only [disk,mem_ball_zero_iff,LinearIsometryEquiv.norm_map] using hz) change complexIso.symm (c.G (c.F (complexIso z))) = z rw [c.left_inverse _ (c.source_eq.symm ▸ hd),LinearIsometryEquiv.symm_apply_apply] lemma connected_complex_pull {v : Plane → ℝ} {P : Set ℝ} (hcon : IsConnected (Ω ∩ v ⁻¹' P)) : IsConnected (ball (0:ℂ) 1 ∩ (fun z => v (c.F (complexIso z))) ⁻¹' P) := by have hh := hcon.image (fun x => complexIso.symm (c.G x)) (complexIso.symm.continuous.comp c.inverse_lipschitz.continuous).continuousOn rw [c.complex_inverse_image _ inter_subset_left] at hh convert hh using 1 ext z constructor · rintro ⟨hz,hv⟩ exact ⟨hz,c.complex_maps_disk hz,hv⟩ · rintro ⟨hz,_,hv⟩ exact ⟨hz,hv⟩ def circleBoundaryHomeomorph : Circle ≃ₜ frontier Ω := by classical let f : Circle → frontier Ω := fun s => ⟨c.F (complexIso (s:ℂ)),c.complex_maps_boundary s⟩ let g : frontier Ω → Circle := fun x => ⟨complexIso.symm (c.G x), mem_sphere_zero_iff_norm.mpr (c.complex_inverse_boundary x.property)⟩ have hleft : Function.LeftInverse g f := by intro s apply Subtype.ext have hd : complexIso (s:ℂ) ∈ closure disk := by simp only [disk,closure_ball (0:Plane) (by norm_num : (1:ℝ) ≠ 0),mem_closedBall_zero_iff, LinearIsometryEquiv.norm_map,Circle.norm_coe,le_refl] change complexIso.symm (c.G (c.F (complexIso (s:ℂ)))) = (s:ℂ) rw [c.left_inverse _ (c.source_eq.symm ▸ hd),LinearIsometryEquiv.symm_apply_apply] have hright : Function.RightInverse g f := by intro x apply Subtype.ext change c.F (complexIso (complexIso.symm (c.G x))) = x rw [LinearIsometryEquiv.apply_symm_apply,c.right_inverse x x.property.1] exact { toFun := f invFun := g left_inv := hleft right_inv := hright continuous_toFun := (c.lipschitz.continuous.comp (complexIso.continuous.comp continuous_subtype_val)).subtype_mk _ continuous_invFun := (complexIso.symm.continuous.comp (c.inverse_lipschitz.continuous.comp continuous_subtype_val)).subtype_mk _ } end StrictHotSpots.Conformal.ClosedDiskChart end end ClosedChartTopology section BoundaryCoordinateExtension noncomputable section open Set MeasureTheory Filter Metric Function open scoped NNReal Topology InnerProductSpace namespace StrictHotSpots.Conformal.ClosedDiskChart open PlaneGreen DiskH10 variable {Ω : Set Plane} (c : ClosedDiskChart Ω) theorem boundary_scalar_extension {f : Boundary → ℝ} {B : ℝ≥0} (hf : LipschitzWith B f) : ∃ (g : Plane → ℝ) (A : ℝ≥0), LipschitzWith A g ∧ ∀ s : Boundary, g (c.F s) = f s := by classical let f' (x : Plane) := if h : ‖x‖ = 1 then f ⟨x,h⟩ else 0 have hl : LipschitzOnWith B f' {x : Plane | ‖x‖ = 1} := by intro x hx y hy change ‖x‖ = 1 at hx change ‖y‖ = 1 at hy simpa only [f',dite_eq_left hx,dite_eq_left hy,Subtype.edist_eq] using hf ⟨x,hx⟩ ⟨y,hy⟩ obtain ⟨g,hg,he⟩ := hl.extend_real refine ⟨g ∘ c.G,B*c.InvLip,hg.comp c.inverse_lipschitz,?_⟩ intro s change g (c.G (c.F s)) = f s rw [c.left_inverse s (c.source_eq.symm ▸ (by simp only [disk,closure_ball (0:Plane) (by norm_num : (1:ℝ) ≠ 0),mem_closedBall_zero_iff,s.property,le_refl]))] exact (he s.property).symm.trans (by simp only [f',dite_eq_left s.property]) end StrictHotSpots.Conformal.ClosedDiskChart namespace StrictHotSpots.HilbertMultipliers lemma coordinate_lipschitz {H S ι : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [PseudoMetricSpace S] (e : HilbertBasis ι ℝ H) {b : S → H} {B : ℝ≥0} (hb : LipschitzWith B b) (i : ι) : LipschitzWith B (fun s => inner ℝ (e i) (b s)) := by apply LipschitzWith.of_dist_le_mul intro s t rw [Real.dist_eq,← inner_sub_right] calc |inner ℝ (e i) (b s-b t)| ≤ ‖e i‖*‖b s-b t‖ := abs_real_inner_le_norm _ _ _ = ‖b s-b t‖ := by rw [e.orthonormal.1 i,one_mul] _ ≤ B*dist s t := by simpa only [dist_eq_norm] using hb.dist_le_mul s t end StrictHotSpots.HilbertMultipliers end end BoundaryCoordinateExtension section GreenClosedContinuous noncomputable section open Set MeasureTheory Filter Metric open scoped Topology InnerProductSpace ENNReal namespace StrictHotSpots.PlaneGreen open DiskH10 lemma norm_le_of_mem_closure_disk {x : Plane} (hx : x ∈ closure disk) : ‖x‖ ≤ 1 := by simpa only [disk,closure_ball (0:Plane) (by norm_num : (1:ℝ) ≠ 0), mem_closedBall_zero_iff] using hx lemma mem_disk_of_norm_lt {x : Plane} (hx : ‖x‖ < 1) : x ∈ disk := by simpa only [disk,mem_ball_zero_iff] using hx lemma kernel_zero_left {x : Plane} (hx : ‖x‖ = 1) (y : Plane) : kernel x y = 0 := by simp [kernel_plane_formula,hx] lemma kernel_continuousAt_left_closed {x y : Plane} (hx : x ∈ closure disk) (hy : y ∈ disk) (hxy : x ≠ y) : ContinuousAt (fun z => kernel z y) x := by have hx1 := norm_le_of_mem_closure_disk hx have hy1 : ‖y‖ < 1 := by simpa only [disk,mem_ball_zero_iff] using hy have hx2 : 0 ≤ 1-‖x‖^2 := by nlinarith [norm_nonneg x] have hy2 : 0 ≤ 1-‖y‖^2 := by nlinarith [norm_nonneg y] have hn : ‖x-y‖^2 ≠ 0 := pow_ne_zero _ (norm_ne_zero_iff.mpr (sub_ne_zero.mpr hxy)) have ha : 1+(1-‖x‖^2)*(1-‖y‖^2)/‖x-y‖^2 ≠ 0 := by positivity simp_rw [kernel_plane_formula] fun_prop (disch := assumption) variable {ν : Measure Plane} {C : ℝ≥0∞} (hC : C ≠ (∞ : ℝ≥0∞)) (hν : ν ≤ C • volume.restrict disk) include hC hν in lemma kernel_memLp_closed (x : Plane) (hx : x ∈ closure disk) : MemLp (kernel x) 2 ν := by by_cases hi : ‖x‖ < 1 · exact kernel_memLp_weighted hC hν x (mem_disk_of_norm_lt hi) · have hn : ‖x‖ = 1 := le_antisymm (norm_le_of_mem_closure_disk hx) (not_lt.mp hi) have he : kernel x = 0 := funext (kernel_zero_left hn) rw [he] exact MemLp.zero include hC hν in lemma kernel_uniform_three_bound_closed : ∃ B : ℝ≥0∞, B ≠ ∞ ∧ ∀ x ∈ closure disk, eLpNorm (kernel x) 3 ν ≤ B := by obtain ⟨B,hB,h⟩ := kernel_uniform_three_bound_weighted hC hν refine ⟨B,hB,fun x hx => ?_⟩ by_cases hi : ‖x‖ < 1 · exact h x (mem_disk_of_norm_lt hi) · have hn : ‖x‖ = 1 := le_antisymm (norm_le_of_mem_closure_disk hx) (not_lt.mp hi) have he : kernel x = 0 := funext (kernel_zero_left hn) rw [he,eLpNorm_zero] exact bot_le def closedRow (x : closure disk) : Lp ℝ 2 ν := (kernel_memLp_closed hC hν x x.property).toLp (kernel x) lemma closedRow_coe_ae (x : closure disk) : closedRow hC hν x =ᵐ[ν] kernel x := (kernel_memLp_closed hC hν x x.property).coeFn_toLp lemma closedRow_continuous [IsFiniteMeasure ν] : Continuous (closedRow hC hν) := by apply continuous_iff_seqContinuous.mpr intro u x hu simp only [Function.comp_def,closedRow] apply (Lp.tendsto_Lp_iff_tendsto_eLpNorm'' (fun n => kernel (u n)) (fun n => kernel_memLp_closed hC hν _ (u n).property) (kernel x) (kernel_memLp_closed hC hν _ x.property)).mpr apply tendsto_Lp_finite_of_tendsto_ae (by norm_num) (by norm_num) (fun n => (kernel_memLp_closed hC hν _ (u n).property).aestronglyMeasurable) (kernel_memLp_closed hC hν _ x.property) · obtain ⟨B,hB,h⟩ := kernel_uniform_three_bound_closed hC hν exact L2Limits.unifIntegrable_two_of_three (fun n => (kernel_memLp_closed hC hν _ (u n).property).aestronglyMeasurable) hB (fun n => h _ (u n).property) · have hne : ∀ᵐ y ∂ν, y ≠ (x:Plane) := ((volume.restrict disk).ae_ne (x:Plane)).filter_mono (Measure.absolutelyContinuous_of_le_smul hν).ae_le filter_upwards [ae_mem_disk hν,hne] with y hyd hyx exact (kernel_continuousAt_left_closed x.property hyd hyx.symm).tendsto.comp ((continuous_subtype_val.tendsto x).comp hu) lemma green_integral_eq_closedInner (f : Lp ℝ 2 ν) (x : closure disk) : (∫ y, kernel x y*f y ∂ν) = inner ℝ (closedRow hC hν x) f := by rw [MeasurableL2Kernel.integral_mul_eq_inner (kernel_memLp_closed hC hν x x.property) (Lp.memLp f),Lp.toLp_coeFn] rfl include hC hν in lemma green_integral_continuousOn_closure [IsFiniteMeasure ν] (f : Lp ℝ 2 ν) : ContinuousOn (fun x : Plane => ∫ y, kernel x y*f y ∂ν) (closure disk) := by apply continuousOn_iff_continuous_domRestrict.mpr change Continuous (fun x : closure disk => ∫ y, kernel x y*f y ∂ν) simp_rw [green_integral_eq_closedInner hC hν] exact (closedRow_continuous hC hν).inner continuous_const lemma green_integral_boundary (f : Lp ℝ 2 ν) {x : Plane} (hx : x ∈ frontier disk) : (∫ y, kernel x y*f y ∂ν) = 0 := by have hn : ‖x‖ = 1 := by simpa only [disk,frontier_ball (0:Plane) (by norm_num : (1:ℝ) ≠ 0), mem_sphere_zero_iff_norm] using hx simp only [kernel_zero_left hn,zero_mul,integral_zero] end StrictHotSpots.PlaneGreen end end GreenClosedContinuous section BoundaryClosedSolution noncomputable section open Set MeasureTheory Filter Metric AddCircle open scoped Topology ContDiff InnerProductSpace ENNReal NNReal namespace StrictHotSpots.PlaneGreen open DiskH10 local instance : Fact (0 < 2*Real.pi) := ⟨by positivity⟩ local notation "τ" => (2*Real.pi) def boundaryClosedPoisson (f : C(Boundary,ℝ)) : Plane → ℝ := Douglas.closedPlanePoisson (boundaryAngleData f) lemma boundaryClosedPoisson_interior (f : C(Boundary,ℝ)) {x : Plane} (hx : x ∈ disk) : boundaryClosedPoisson f x = poissonExtension f x := (Douglas.closedPlanePoisson_eqOn (boundaryAngleData f) hx).trans (poissonExtension_eq_planePoisson f hx).symm lemma boundaryClosedPoisson_trace (f : C(Boundary,ℝ)) (s : Boundary) : boundaryClosedPoisson f s = f s := by obtain ⟨t,rfl⟩ := angleBoundary.surjective s rw [angleBoundary_coe] exact Douglas.closedPlanePoisson_boundary (boundaryAngleData f) t lemma boundaryClosedPoisson_continuousOn (f : C(Boundary,ℝ)) {A : ℝ≥0} (hf : LipschitzWith A f) : ContinuousOn (boundaryClosedPoisson f) (closure disk) := Douglas.closedPlanePoisson_continuousOn (boundaryAngleData f) A.coe_nonneg (boundaryAngleData_lipschitz f hf) variable {ν : Measure Plane} [IsFiniteMeasure ν] {C : ℝ≥0∞} (hC : C ≠ (∞ : ℝ≥0∞)) (hν : ν ≤ C • volume.restrict disk) def boundaryClosedSolution (f : C(Boundary,ℝ)) {B : ℝ} (hB : ∀ s, ‖f s‖ ≤ B) (x : Plane) : ℝ := boundaryClosedPoisson f x + ∫ y, kernel x y * (BanachResolvent.resolvent (weightedIntegralOperator hC hν) (poissonL2 hν f.continuous.measurable hB)) y ∂ν lemma boundaryClosedSolution_interior (f : C(Boundary,ℝ)) {B : ℝ} (hB : ∀ s, ‖f s‖ ≤ B) {x : Plane} (hx : x ∈ disk) : boundaryClosedSolution hC hν f hB x = boundarySolutionValue hC hν f hB x := by exact congrArg (fun a => a + ∫ y, kernel x y * (BanachResolvent.resolvent (weightedIntegralOperator hC hν) (poissonL2 hν f.continuous.measurable hB)) y ∂ν) (boundaryClosedPoisson_interior f hx) lemma boundaryClosedSolution_continuousOn (f : C(Boundary,ℝ)) {A : ℝ≥0} (hf : LipschitzWith A f) {B : ℝ} (hB : ∀ s, ‖f s‖ ≤ B) : ContinuousOn (boundaryClosedSolution hC hν f hB) (closure disk) := (boundaryClosedPoisson_continuousOn f hf).add (green_integral_continuousOn_closure hC hν _) lemma boundaryClosedSolution_trace (f : C(Boundary,ℝ)) {B : ℝ} (hB : ∀ s, ‖f s‖ ≤ B) (s : Boundary) : boundaryClosedSolution hC hν f hB s = f s := by have hs : (s:Plane) ∈ frontier disk := by simpa only [disk,frontier_ball (0:Plane) (by norm_num : (1:ℝ) ≠ 0), mem_sphere, dist_zero_right] using s.property rw [boundaryClosedSolution,green_integral_boundary _ hs,add_zero,boundaryClosedPoisson_trace] lemma boundaryClosedSolution_value_ae (f : C(Boundary,ℝ)) {A : ℝ≥0} (hf : LipschitzWith A f) {B : ℝ} (hB : ∀ s, ‖f s‖ ≤ B) : (H1.value (boundarySolution hC hν f hf) : Plane → ℝ) =ᵐ[volume.restrict disk] boundaryClosedSolution hC hν f hB := by filter_upwards [boundarySolution_value_ae hC hν f hf hB, ae_restrict_mem diskOpen.measurableSet] with x hx hxd exact hx.trans (boundaryClosedSolution_interior hC hν f hB hxd).symm lemma boundaryClosedSolution_hasH1Gradient (f : C(Boundary,ℝ)) {A : ℝ≥0} (hf : LipschitzWith A f) {B : ℝ} (hB : ∀ s, ‖f s‖ ≤ B) : HasH1Gradient disk (boundaryClosedSolution hC hν f hB) (H1.grad (boundarySolution hC hν f hf)) := (H1.hasH1Gradient _).congr (boundaryClosedSolution_value_ae hC hν f hf hB) Filter.EventuallyEq.rfl lemma boundaryClosedSolution_K (hT : ‖weightedIntegralOperator hC hν‖ < 1) (f : C(Boundary,ℝ)) {B : ℝ} (hB : ∀ s, ‖f s‖ ≤ B) (x : disk) : boundaryClosedSolution hC hν f hB x = ∫ s, fullBoundaryK hC hν x s*f s ∂boundaryMeasure := by rw [boundaryClosedSolution_interior hC hν f hB x.property] exact boundarySolution_K hC hν hT f hB x end StrictHotSpots.PlaneGreen end end BoundaryClosedSolution section PhysicalClosedSolution noncomputable section open Set MeasureTheory Filter open scoped ContDiff InnerProductSpace ENNReal NNReal Topology namespace StrictHotSpots.Conformal.ClosedDiskChart open DiskH10 PlaneGreen variable {Ω : Set Plane} (c : ClosedDiskChart Ω) lemma inverse_map_chartMeasure : (volume.restrict Ω).map c.G = chartMeasure c.diskMap c.d := by have hm : (chartMeasure c.diskMap c.d).map c.diskMap = volume.restrict Ω := map_chartMeasure c.diskMap c.d c.diskMap_deriv rw [← hm, AEMeasurable.map_map_of_aemeasurable c.inverse_lipschitz.continuous.aemeasurable (chart_aemeasurable c.diskMap c.d)] have he : c.G ∘ c.diskMap =ᵐ[chartMeasure c.diskMap c.d] id := by filter_upwards [ae_chart_source c.diskMap c.d] with x hx change c.G (c.diskMap x)=x rw [c.diskMap_agrees hx] exact c.left_inverse x (c.source_eq.symm ▸ subset_closure hx) rw [Measure.map_congr he,Measure.map_id] lemma pull_ae_to_physical {f g : Plane → ℝ} (h : f ∘ c.diskMap =ᵐ[chartMeasure c.diskMap c.d] g) : f =ᵐ[volume.restrict Ω] (g ∘ c.G) := by have hg : f ∘ c.diskMap =ᵐ[(volume.restrict Ω).map c.G] g := c.inverse_map_chartMeasure.symm ▸ h have ha := ae_of_ae_map c.inverse_lipschitz.continuous.aemeasurable hg filter_upwards [ha,ae_restrict_mem c.diskMap.open_target.measurableSet] with y hy hyΩ have hi : c.diskMap (c.G y)=y := by rw [← c.diskMap_inverse_agrees hyΩ] exact c.diskMap.right_inv hyΩ simpa only [Function.comp_apply,hi] using hy lemma physicalSolution_pull_ae (hb : Bornology.IsBounded Ω) (hs : SmoothBoundary Ω) (hμ : 0 ≤ firstPositiveNeumannValue Ω) {w : Plane → ℝ} (hw : ContDiffOn ℝ ∞ w (closure Ω)) {B : ℝ≥0} (hl : LipschitzWith B (c.datumTrace w hw.continuousOn)) : (H1.value (c.physicalSolution hb hs hμ hw hl) : Plane → ℝ) ∘ c.diskMap =ᵐ[chartMeasure c.diskMap c.d] H1.value (boundarySolution c.densityBound_ne_top (c.density_bound hμ) (c.datumTrace w hw.continuousOn) hl) := by have h := affine_value_ae c.diskMap c.diskMap_inverse_smooth c.d c.diskMap_deriv hb (smooth_closure_H1 (Ω:=Ω) (show IsOpen Ω from c.diskMap.open_target) hb hs hw) (c.pullDatum_comp_H1 hb hs hw) (c.datumCorrection hb hs hμ hw hl) have he : (c.pullDatum_comp_H1 hb hs hw).toH1+(c.datumCorrection hb hs hμ hw hl).val = boundarySolution c.densityBound_ne_top (c.density_bound hμ) (c.datumTrace w hw.continuousOn) hl := (congrArg (fun V : H1 disk => V+(c.datumCorrection hb hs hμ hw hl).val) (c.pullDatum_comp_toH1 hb hs hw)).trans (c.datumCorrection_spec hb hs hμ hw hl).symm exact (chartPull_ae c.diskMap c.d c.diskMap_deriv _).symm.trans (h.trans (EventuallyEq.of_eq (congrArg (fun V : H1 disk => (H1.value V : Plane → ℝ)) he))) lemma physicalSolution_closed_ae (hb : Bornology.IsBounded Ω) (hs : SmoothBoundary Ω) (hμ : 0 ≤ firstPositiveNeumannValue Ω) {w : Plane → ℝ} (hw : ContDiffOn ℝ ∞ w (closure Ω)) {B : ℝ≥0} (hl : LipschitzWith B (c.datumTrace w hw.continuousOn)) {M : ℝ} (hM : ∀ s : Boundary, ‖w (c.F s)‖ ≤ M) : let _ : IsFiniteMeasure c.potential := c.potential_finite hb (H1.value (c.physicalSolution hb hs hμ hw hl) : Plane → ℝ) =ᵐ[volume.restrict Ω] (boundaryClosedSolution c.densityBound_ne_top (c.density_bound hμ) (c.datumTrace w hw.continuousOn) hM) ∘ c.G := by let _ : IsFiniteMeasure c.potential := c.potential_finite hb apply c.pull_ae_to_physical exact (c.physicalSolution_pull_ae hb hs hμ hw hl).trans ((chartMeasure_ac c.diskMap c.d).ae_le (boundaryClosedSolution_value_ae c.densityBound_ne_top (c.density_bound hμ) (c.datumTrace w hw.continuousOn) hl hM)) end StrictHotSpots.Conformal.ClosedDiskChart end end PhysicalClosedSolution section BoundaryClosedConvergence noncomputable section open Set MeasureTheory Filter open scoped ContDiff InnerProductSpace ENNReal NNReal Topology namespace StrictHotSpots.PlaneGreen open DiskH10 variable {ν : Measure Plane} [IsFiniteMeasure ν] {C : ℝ≥0∞} (hC : C ≠ (∞ : ℝ≥0∞)) (hν : ν ≤ C • volume.restrict disk) lemma boundaryClosedSolution_tendsto (hT : ‖weightedIntegralOperator hC hν‖ < 1) (fn : ℕ → C(Boundary,ℝ)) (f : C(Boundary,ℝ)) {M B : ℝ} (hm : ∀ n s, ‖fn n s‖ ≤ M) (hb : ∀ s, ‖f s‖ ≤ B) (ht : ∀ s, Tendsto (fun n => fn n s) atTop (𝓝 (f s))) (p : disk) : Tendsto (fun n => boundaryClosedSolution hC hν (fn n) (hm n) p) atTop (𝓝 (boundaryClosedSolution hC hν f hb p)) := by obtain ⟨A,hA,hAb⟩ := fullBoundaryK_bounded hC hν p have hn (s : Boundary) : ‖fullBoundaryK hC hν p s‖ ≤ A := by rw [Real.norm_of_nonneg (fullBoundaryK_pos hC hν hT p s).le] exact hAb s simp only [boundaryClosedSolution_K hC hν hT] apply tendsto_integral_of_dominated_convergence (fun _ : Boundary => A*M) · intro n exact ((fullBoundaryK_measurable hC hν p).mul (fn n).continuous.measurable).aestronglyMeasurable · exact integrable_const _ · intro n exact Eventually.of_forall fun s => (norm_mul _ _).trans_le (mul_le_mul (hn s) (hm n s) (norm_nonneg _) hA) · exact Eventually.of_forall fun s => (ht s).const_mul _ end StrictHotSpots.PlaneGreen end end BoundaryClosedConvergence section PhysicalMultiplierClosed noncomputable section open Set MeasureTheory Filter open scoped ContDiff InnerProductSpace ENNReal NNReal Topology namespace StrictHotSpots.Conformal.ClosedDiskChart open DiskH10 PlaneGreen Hodge variable {Ω : Set Plane} (c : ClosedDiskChart Ω) (hΩ : AdmissibleDomain Ω) variable (hμ : 0 < firstPositiveNeumannValue Ω) variable {L : ℝ} (hl : firstPositiveNeumannValue Ω < L) (hD : ∀ j : H10 hΩ.2.1, L*‖H10.value hΩ.2.1 j‖^2 ≤ ‖H10.grad hΩ.2.1 j‖^2) lemma scalarSolution_closed_ae {w : Plane → ℝ} (hw : ContDiffOn ℝ ∞ w (closure Ω)) {M : ℝ} (hM : ∀ s : Boundary, ‖w (c.F s)‖ ≤ M) : let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1 (H1.value (c.scalarSolution hΩ.2.2.1 hΩ.2.2.2.2 hμ hw) : Plane → ℝ) =ᵐ[volume.restrict Ω] (boundaryClosedSolution c.densityBound_ne_top (c.density_bound hμ.le) (c.datumTrace w hw.continuousOn) hM) ∘ c.G := c.physicalSolution_closed_ae hΩ.2.2.1 hΩ.2.2.2.2 hμ.le hw _ hM lemma scalarSolution_mul_tendsto {w b : Plane → ℝ} (hw : ContDiffOn ℝ ∞ w (closure Ω)) {B : ℝ≥0} (hb : LipschitzWith B b) : Tendsto (fun n => c.scalarSolution hΩ.2.2.1 hΩ.2.2.2.2 hμ ((ambientSmooth_smooth (planeApproxBump n) hb).contDiffOn.mul hw)) atTop (𝓝 (PhysicalHelmholtz.extension hΩ.2.1 hΩ.2.2.1 hμ.le hl hD (H1.lipschitzMul hΩ.2.1 hΩ.2.2.1 hb (smooth_closure_H1 hΩ.2.1 hΩ.2.2.1 hΩ.2.2.2.2 hw).toH1))) := by let U := (smooth_closure_H1 hΩ.2.1 hΩ.2.2.1 hΩ.2.2.2.2 hw).toH1 have ht := (PhysicalHelmholtz.extension hΩ.2.1 hΩ.2.2.1 hμ.le hl hD).continuous.continuousAt.tendsto.comp (smoothMul_tendsto_lipschitzMul hΩ.2.1 hΩ.2.2.1 hb U) apply ht.congr' apply Eventually.of_forall intro n have hen := smoothMul_smooth_closure hΩ.2.1 hΩ.2.2.1 hΩ.2.2.2.2 (ambientSmooth_smooth (planeApproxBump n) hb) (continuous_closure_memLp_top hΩ.2.1 hΩ.2.2.1 (ambientSmooth_lipschitz (planeApproxBump n) hb).continuous.continuousOn) (lipschitz_gradient_memLp_top (ambientSmooth_lipschitz (planeApproxBump n) hb)) hw exact ((c.scalarSolution_eq_extension hΩ hμ hl hD ((ambientSmooth_smooth (planeApproxBump n) hb).contDiffOn.mul hw)).trans (congrArg (PhysicalHelmholtz.extension hΩ.2.1 hΩ.2.2.1 hμ.le hl hD) hen.symm)).symm def multiplierTrace {b : Plane → ℝ} {B : ℝ≥0} (hb : LipschitzWith B b) (w : Plane → ℝ) (hw : ContinuousOn w (closure Ω)) : C(Boundary,ℝ) where toFun s := b (c.F s)*w (c.F s) continuous_toFun := (c.physical_boundary_lipschitz hb).continuous.mul (c.datumTrace w hw).continuous include hΩ in lemma multiplierTrace_lipschitz {b w : Plane → ℝ} {B : ℝ≥0} (hb : LipschitzWith B b) (hw : ContDiffOn ℝ ∞ w (closure Ω)) : ∃ A : ℝ≥0, LipschitzWith A (c.multiplierTrace hb w hw.continuousOn) := by obtain ⟨D,hD⟩ := c.datumTrace_lipschitz hΩ.2.2.1 hΩ.2.2.2.2 hw exact boundary_smul_lipschitz (c.physical_boundary_lipschitz hb) hD lemma physical_lipschitzMul_closed_ae {w b : Plane → ℝ} (hw : ContDiffOn ℝ ∞ w (closure Ω)) {B : ℝ≥0} (hb : LipschitzWith B b) {A : ℝ} (hA : ∀ s, ‖c.multiplierTrace hb w hw.continuousOn s‖ ≤ A) : let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1 (H1.value (PhysicalHelmholtz.extension hΩ.2.1 hΩ.2.2.1 hμ.le hl hD (H1.lipschitzMul hΩ.2.1 hΩ.2.2.1 hb (smooth_closure_H1 hΩ.2.1 hΩ.2.2.1 hΩ.2.2.2.2 hw).toH1)) : Plane → ℝ) =ᵐ[volume.restrict Ω] (boundaryClosedSolution c.densityBound_ne_top (c.density_bound hμ.le) (c.multiplierTrace hb w hw.continuousOn) hA) ∘ c.G := by let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1 let hn (n : ℕ) := (ambientSmooth_smooth (planeApproxBump n) hb).contDiffOn.mul hw let fn (n : ℕ) := c.datumTrace _ (hn n).continuousOn obtain ⟨M,hM,hMb⟩ := c.smooth_physical_boundary_bound hΩ hb obtain ⟨W,hW,hWb⟩ := boundary_continuous_bound (c.datumTrace w hw.continuousOn).continuous have hMW (n : ℕ) (s : Boundary) : ‖fn n s‖ ≤ M*W := by change ‖ambientSmooth (planeApproxBump n) b (c.F s)*w (c.F s)‖ ≤ M*W rw [norm_mul] exact mul_le_mul (hMb n s) (hWb s) (norm_nonneg _) hM have ht (s : Boundary) : Tendsto (fun n => fn n s) atTop (𝓝 (c.multiplierTrace hb w hw.continuousOn s)) := (ambientSmooth_tendsto planeApproxBump_rOut_tendsto hb (c.F s)).mul_const _ have he (n : ℕ) := c.scalarSolution_closed_ae hΩ hμ (hn n) (hMW n) have hv := H1.value.continuous.continuousAt.tendsto.comp (c.scalarSolution_mul_tendsto hΩ hμ hl hD hw hb) obtain ⟨φ,hφ,hφae⟩ := (tendstoInMeasure_of_tendsto_Lp hv).exists_seq_tendsto_ae filter_upwards [hφae,ae_all_iff.mpr he,ae_restrict_mem hΩ.2.1.measurableSet] with x hx hxe hxΩ have hxD : c.G x ∈ disk := by rw [← c.diskMap_inverse_agrees hxΩ] exact c.diskMap.map_target hxΩ have hk := boundaryClosedSolution_tendsto c.densityBound_ne_top (c.density_bound hμ.le) (c.operator_norm_lt_one hΩ.2.2.1 hΩ.1 hμ) fn (c.multiplierTrace hb w hw.continuousOn) hMW hA ht ⟨c.G x,hxD⟩ apply tendsto_nhds_unique hx exact (hk.comp hφ.tendsto_atTop).congr' (Eventually.of_forall fun n => (hxe (φ n)).symm) end StrictHotSpots.Conformal.ClosedDiskChart end end PhysicalMultiplierClosed end OAI