import OAI.Analysis.HotSpots.VectorEnergy namespace OAI section LipschitzMollifier noncomputable section open Set MeasureTheory Filter Metric ContinuousLinearMap open scoped ContDiff InnerProductSpace ENNReal NNReal Convolution namespace StrictHotSpots def ambientSmooth (φ : ContDiffBump (0 : Plane)) (b : Plane → ℝ) : Plane → ℝ := φ.normed volume ⋆[lsmul ℝ ℝ,volume] b lemma ambientSmooth_smooth (φ : ContDiffBump (0 : Plane)) {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) : ContDiff ℝ ∞ (ambientSmooth φ b) := φ.hasCompactSupport_normed.contDiff_convolution_left _ φ.contDiff_normed hb.continuous.locallyIntegrable lemma ambientSmooth_integrable (φ : ContDiffBump (0 : Plane)) {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) (x : Plane) : Integrable (fun t : Plane => φ.normed volume t * b (x-t)) := φ.hasCompactSupport_normed.convolutionExists_left_of_continuous_right (lsmul ℝ ℝ) φ.integrable_normed.locallyIntegrable hb.continuous x lemma ambientSmooth_lipschitz (φ : ContDiffBump (0 : Plane)) {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) : LipschitzWith L (ambientSmooth φ b) := by rw [lipschitzWith_iff_norm_sub_le] intro x y have hi := (ambientSmooth_integrable φ hb x).sub (ambientSmooth_integrable φ hb y) have hp (t : Plane) : ‖φ.normed volume t*b (x-t)-φ.normed volume t*b (y-t)‖ ≤ φ.normed volume t*((L:ℝ)*‖x-y‖) := by rw [← mul_sub,norm_mul,Real.norm_eq_abs,abs_of_nonneg (φ.nonneg_normed t)] apply mul_le_mul_of_nonneg_left _ (φ.nonneg_normed t) simpa only [sub_sub_sub_cancel_right] using hb.norm_sub_le (x-t) (y-t) change ‖(∫ t : Plane, φ.normed volume t*b (x-t))-(∫ t : Plane, φ.normed volume t*b (y-t))‖ ≤ _ rw [← integral_sub (ambientSmooth_integrable φ hb x) (ambientSmooth_integrable φ hb y)] calc _ ≤ ∫ t : Plane, ‖φ.normed volume t*b (x-t)-φ.normed volume t*b (y-t)‖ := norm_integral_le_integral_norm _ _ ≤ ∫ t : Plane, φ.normed volume t*((L:ℝ)*‖x-y‖) := integral_mono hi.norm (φ.integrable_normed.mul_const _) hp _ = _ := by rw [integral_mul_const,φ.integral_normed,one_mul] lemma ambientSmooth_tendsto {φ : ℕ → ContDiffBump (0 : Plane)} (hφ : Tendsto (fun n => (φ n).rOut) atTop (nhds 0)) {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) (x : Plane) : Tendsto (fun n => ambientSmooth (φ n) b x) atTop (nhds (b x)) := ContDiffBump.convolution_tendsto_right_of_continuous hφ hb.continuous x lemma ambientSmooth_fderiv (φ : ContDiffBump (0 : Plane)) {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) (x : Plane) : fderiv ℝ (ambientSmooth φ b) x = ∫ t : Plane, φ.normed volume t • fderiv ℝ b (x-t) := by have hm : AEStronglyMeasurable (fun t : Plane => φ.normed volume t • fderiv ℝ b (x-t)) volume := (φ.continuous_normed.measurable.smul ((measurable_fderiv ℝ b).comp (measurable_const.sub measurable_id))).aestronglyMeasurable have hl : ∀ᵐ t : Plane ∂volume, LipschitzOnWith (Real.nnabs (φ.normed volume t*(L:ℝ))) (fun y => φ.normed volume t*b (y-t)) univ := by filter_upwards [] with t apply LipschitzWith.lipschitzOnWith apply LipschitzWith.of_dist_le_mul intro y z simp only [dist_eq_norm,← mul_sub,norm_mul,Real.norm_eq_abs,Real.coe_nnabs, abs_of_nonneg (mul_nonneg (φ.nonneg_normed t) L.coe_nonneg), abs_of_nonneg (φ.nonneg_normed t)] calc _ ≤ φ.normed volume t*((L:ℝ)*‖(y-t)-(z-t)‖) := mul_le_mul_of_nonneg_left (hb.norm_sub_le _ _) (φ.nonneg_normed t) _ = _ := by rw [sub_sub_sub_cancel_right]; ring have hd : ∀ᵐ t : Plane ∂volume, HasFDerivAt (fun y => φ.normed volume t*b (y-t)) (φ.normed volume t • fderiv ℝ b (x-t)) x := by filter_upwards [(volume.measurePreserving_sub_left x).quasiMeasurePreserving.ae (hb.ae_differentiableAt (μ := volume))] with t ht have h := ht.hasFDerivAt.comp x ((hasFDerivAt_id x).sub (hasFDerivAt_const t x)) simpa [Function.comp_def,Pi.smul_def,smul_eq_mul] using h.const_smul (φ.normed volume t) have hh := hasFDerivAt_integral_of_dominated_loc_of_lip (s := univ) (bound := fun t => φ.normed volume t*(L:ℝ)) (by simp : univ ∈ nhds x) (Filter.Eventually.of_forall (fun y => (ambientSmooth_integrable φ hb y).aestronglyMeasurable)) (ambientSmooth_integrable φ hb x) hm hl (φ.integrable_normed.mul_const _) hd exact hh.2.fderiv lemma ambientSmooth_gradient (φ : ContDiffBump (0 : Plane)) {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) (x : Plane) : gradient (ambientSmooth φ b) x = (φ.normed volume ⋆[lsmul ℝ ℝ,volume] gradient b) x := by unfold gradient rw [ambientSmooth_fderiv φ hb x] apply ((InnerProductSpace.toDual ℝ Plane).symm.toContinuousLinearEquiv.integral_comp_comm (fun t : Plane => φ.normed volume t • fderiv ℝ b (x-t))).symm.trans apply integral_congr_ae filter_upwards [] with t exact map_smul _ _ _ lemma lipschitz_gradient_locallyIntegrable {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) : LocallyIntegrable (gradient b) volume := by have hm : Measurable (gradient b) := (InnerProductSpace.toDual ℝ Plane).symm.continuous.measurable.comp (measurable_fderiv ℝ b) apply (memLp_top_of_bound hm.aestronglyMeasurable (L:ℝ) ?_).locallyIntegrable (by simp) filter_upwards [] with x change ‖(InnerProductSpace.toDual ℝ Plane).symm (fderiv ℝ b x)‖ ≤ L rw [LinearIsometryEquiv.norm_map] exact norm_fderiv_le_of_lipschitz ℝ hb lemma ambientSmooth_gradient_tendsto {φ : ℕ → ContDiffBump (0 : Plane)} (hφ : Tendsto (fun n => (φ n).rOut) atTop (nhds 0)) {K : ℝ} (hK : ∀ᶠ n in atTop, (φ n).rOut ≤ K*(φ n).rIn) {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) : ∀ᵐ x : Plane ∂volume, Tendsto (fun n => gradient (ambientSmooth (φ n) b) x) atTop (nhds (gradient b x)) := by simp_rw [ambientSmooth_gradient _ hb] exact ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable hφ hK (lipschitz_gradient_locallyIntegrable hb) end StrictHotSpots end end LipschitzMollifier section LipschitzH1 noncomputable section open Set MeasureTheory Filter Metric open scoped ContDiff InnerProductSpace ENNReal NNReal Topology namespace StrictHotSpots lemma HasH1Gradient.of_tendsto_L2 {Ω : Set Plane} {f : ℕ → Plane → ℝ} {g : ℕ → Plane → Plane} (hf : ∀ n, HasH1Gradient Ω (f n) (g n)) {u : Plane → ℝ} {v : Plane → Plane} (hu : MemLp u 2 (volume.restrict Ω)) (hv : MemLp v 2 (volume.restrict Ω)) (hfu : Tendsto (fun n => (hf n).1.toLp (f n)) atTop (𝓝 (hu.toLp u))) (hgv : Tendsto (fun n => (hf n).2.1.toLp (g n)) atTop (𝓝 (hv.toLp v))) : HasH1Gradient Ω u v := by have ht : Tendsto (fun n => (hf n).toH1.val) atTop (𝓝 (WithLp.toLp 2 (hu.toLp u,hv.toLp v))) := (WithLp.prod_continuous_toLp 2 _ _).continuousAt.tendsto.comp (hfu.prodMk_nhds hgv) have hg := (h1Graph_mem_iff Ω (WithLp.toLp 2 (hu.toLp u,hv.toLp v))).mp ((h1Graph_closed Ω).mem_of_tendsto ht (Eventually.of_forall fun n => (hf n).toH1.property)) exact hg.congr hu.coeFn_toLp hv.coeFn_toLp def planeApproxBump (n : ℕ) : ContDiffBump (0 : Plane) where rIn := (2*((n:ℝ)+1))⁻¹ rOut := ((n:ℝ)+1)⁻¹ rIn_pos := by positivity rIn_lt_rOut := by apply inv_strictAnti₀ (by positivity) nlinarith [Nat.cast_nonneg (α:=ℝ) n] lemma planeApproxBump_rOut_tendsto : Tendsto (fun n => (planeApproxBump n).rOut) atTop (𝓝 0) := tendsto_inv_atTop_zero.comp (tendsto_atTop_add_const_right atTop 1 tendsto_natCast_atTop_atTop) lemma planeApproxBump_ratio (n : ℕ) : (planeApproxBump n).rOut ≤ 2*(planeApproxBump n).rIn := by dsimp only [planeApproxBump] have h : (n:ℝ)+1 ≠ 0 := by positivity field_simp norm_num lemma planeApproxBump_rOut_le_one (n : ℕ) : (planeApproxBump n).rOut ≤ 1 := by change ((n:ℝ)+1)⁻¹ ≤ 1 apply inv_le_one_of_one_le₀ linarith [Nat.cast_nonneg (α:=ℝ) n] lemma ambientSmooth_error (φ : ContDiffBump (0 : Plane)) {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) (x : Plane) : ‖ambientSmooth φ b x-b x‖ ≤ (L:ℝ)*φ.rOut := by rw [← dist_eq_norm] apply φ.dist_normed_convolution_le hb.continuous.aestronglyMeasurable intro y hy exact (hb.dist_le_mul y x).trans (mul_le_mul_of_nonneg_left hy.le L.coe_nonneg) lemma lipschitz_gradient_bound {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) (x : Plane) : ‖gradient b x‖ ≤ L := by change ‖(InnerProductSpace.toDual ℝ Plane).symm (fderiv ℝ b x)‖ ≤ L rw [LinearIsometryEquiv.norm_map] exact norm_fderiv_le_of_lipschitz ℝ hb lemma lipschitz_gradient_memLp_top {Ω : Set Plane} {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) : MemLp (gradient b) ∞ (volume.restrict Ω) := by apply memLp_top_of_bound ((InnerProductSpace.toDual ℝ Plane).symm.continuous.measurable.comp (measurable_fderiv ℝ b)).aestronglyMeasurable _ exact Eventually.of_forall (lipschitz_gradient_bound hb) lemma continuous_closure_memLp_top {Ω : Set Plane} {F : Type*} [NormedAddCommGroup F] {f : Plane → F} (ho : IsOpen Ω) (hΩ : Bornology.IsBounded Ω) (hf : ContinuousOn f (closure Ω)) : MemLp f ∞ (volume.restrict Ω) := by obtain ⟨C,hC⟩ := hΩ.isCompact_closure.exists_bound_of_continuousOn hf apply memLp_top_of_bound ((hf.mono subset_closure).aestronglyMeasurable ho.measurableSet) C filter_upwards [ae_restrict_mem ho.measurableSet] with x hx exact hC x (subset_closure hx) lemma ambientSmooth_uniform_bound {Ω : Set Plane} (hΩ : Bornology.IsBounded Ω) {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) : ∃ C : ℝ, 0 ≤ C ∧ ∀ n, ∀ x ∈ Ω, ‖ambientSmooth (planeApproxBump n) b x‖ ≤ C := by obtain ⟨M,hM⟩ := hΩ.isCompact_closure.exists_bound_of_continuousOn hb.continuous.continuousOn refine ⟨max M 0+L,by positivity,fun n x hx => ?_⟩ have he := ambientSmooth_error (planeApproxBump n) hb x have hφ := mul_le_mul_of_nonneg_left (planeApproxBump_rOut_le_one n) L.coe_nonneg have hx' := hM x (subset_closure hx) calc _ ≤ ‖ambientSmooth (planeApproxBump n) b x-b x‖+‖b x‖ := norm_le_norm_sub_add _ _ _ ≤ (L:ℝ)+max M 0 := by linarith [le_max_left M 0] _ = _ := by ring lemma HasH1Gradient.mul_lipschitz {Ω : Set Plane} (ho : IsOpen Ω) (hΩ : Bornology.IsBounded Ω) {u : Plane → ℝ} {g : Plane → Plane} (hu : HasH1Gradient Ω u g) {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) : HasH1Gradient Ω (fun x => b x*u x) (fun x => b x • g x+u x • gradient b x) := by let bn := fun n => ambientSmooth (planeApproxBump n) b have hbn (n : ℕ) : ContDiff ℝ ∞ (bn n) := ambientSmooth_smooth _ hb have hbl (n : ℕ) : LipschitzWith L (bn n) := ambientSmooth_lipschitz _ hb have hb0 := continuous_closure_memLp_top ho hΩ hb.continuous.continuousOn have hb1 := lipschitz_gradient_memLp_top (Ω:=Ω) hb have hn0 (n : ℕ) := continuous_closure_memLp_top ho hΩ (hbl n).continuous.continuousOn have hn1 (n : ℕ) := lipschitz_gradient_memLp_top (Ω:=Ω) (hbl n) have hf (n : ℕ) := hu.mul_smooth (hbn n) (hn0 n) (hn1 n) have hv : MemLp (fun x => b x*u x) 2 (volume.restrict Ω) := hb0.fun_mul hu.1 have hg : MemLp (fun x => b x • g x+u x • gradient b x) 2 (volume.restrict Ω) := (hb0.smul hu.2.1).add (hu.1.smul hb1) obtain ⟨C,hC,hbd⟩ := ambientSmooth_uniform_bound hΩ hb have ht (x : Plane) : Tendsto (fun n => bn n x) atTop (𝓝 (b x)) := ambientSmooth_tendsto planeApproxBump_rOut_tendsto hb x have hgt : ∀ᵐ x ∂volume.restrict Ω, Tendsto (fun n => gradient (bn n) x) atTop (𝓝 (gradient b x)) := ae_restrict_of_ae (ambientSmooth_gradient_tendsto planeApproxBump_rOut_tendsto (Eventually.of_forall planeApproxBump_ratio) hb) apply HasH1Gradient.of_tendsto_L2 hf hv hg · apply L2VectorLimits.tendsto_toLp_of_dominated (fun n => (hf n).1) hv (hu.1.norm.const_mul C) · filter_upwards [] with n filter_upwards [ae_restrict_mem ho.measurableSet] with x hx rw [norm_mul] exact mul_le_mul_of_nonneg_right (hbd n x hx) (norm_nonneg _) · exact Eventually.of_forall fun x => (ht x).mul_const (u x) · apply L2VectorLimits.tendsto_toLp_of_dominated (fun n => (hf n).2.1) hg ((hu.2.1.norm.const_mul C).add (hu.1.norm.const_mul (L:ℝ))) · filter_upwards [] with n filter_upwards [ae_restrict_mem ho.measurableSet] with x hx calc _ ≤ ‖bn n x • g x‖+‖u x • gradient (bn n) x‖ := norm_add_le _ _ _ = ‖bn n x‖*‖g x‖+‖u x‖*‖gradient (bn n) x‖ := by rw [norm_smul,norm_smul] _ ≤ C*‖g x‖+‖u x‖*(L:ℝ) := add_le_add (mul_le_mul_of_nonneg_right (hbd n x hx) (norm_nonneg _)) (mul_le_mul_of_nonneg_left (lipschitz_gradient_bound (hbl n) x) (norm_nonneg _)) _ = _ := by simp only [Pi.add_apply]; ring · filter_upwards [hgt] with x hx exact ((ht x).smul_const (g x)).add (tendsto_const_nhds.smul hx) namespace H1 variable {Ω : Set Plane} (ho : IsOpen Ω) (hΩ : Bornology.IsBounded Ω) {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) def lipschitzMul (u : H1 Ω) : H1 Ω := ((H1.hasH1Gradient u).mul_lipschitz ho hΩ hb).toH1 lemma lipschitzMul_value_ae (u : H1 Ω) : H1.value (lipschitzMul ho hΩ hb u) =ᵐ[volume.restrict Ω] fun x => b x*H1.value u x := ((H1.hasH1Gradient u).mul_lipschitz ho hΩ hb).1.coeFn_toLp lemma lipschitzMul_grad_ae (u : H1 Ω) : H1.grad (lipschitzMul ho hΩ hb u) =ᵐ[volume.restrict Ω] fun x => b x • H1.grad u x+H1.value u x • gradient b x := ((H1.hasH1Gradient u).mul_lipschitz ho hΩ hb).2.1.coeFn_toLp end H1 end StrictHotSpots end end LipschitzH1 section PhysicalHelmholtz noncomputable section open Set MeasureTheory Filter open scoped ContDiff InnerProductSpace ENNReal NNReal namespace StrictHotSpots.PhysicalHelmholtz variable {Ω : Set Plane} (ho : IsOpen Ω) local instance : NormedAddCommGroup (H10 ho) := H10.normedAddCommGroup ho local instance : InnerProductSpace ℝ (H10 ho) := H10.innerProductSpace ho local instance : CompleteSpace (H10 ho) := H10.completeSpace ho local instance : NormedSpace ℝ (H10 ho) := (H10.innerProductSpace ho).toNormedSpace local instance : NormedSpace ℝ (H1 Ω) := (h1Graph Ω).normedSpace def pair (μ : ℝ) : H1 Ω →L[ℝ] H1 Ω →L[ℝ] ℝ := ((innerSL ℝ).comp H1.grad).flip.comp H1.grad - μ • ((innerSL ℝ).comp H1.value).flip.comp H1.value lemma pair_apply (μ : ℝ) (u v : H1 Ω) : pair μ u v = inner ℝ (H1.grad v) (H1.grad u)-μ*inner ℝ (H1.value v) (H1.value u) := rfl lemma pair_comm (μ : ℝ) (u v : H1 Ω) : pair μ u v=pair μ v u := by rw [pair_apply,pair_apply] exact congrArg₂ (fun a b : ℝ => a-μ*b) (real_inner_comm (H1.grad u) (H1.grad v)) (real_inner_comm (H1.value u) (H1.value v)) def incl : H10 ho →L[ℝ] H1 Ω := (h10Submodule ho).subtypeL def zeroPair (μ : ℝ) : H10 ho →L[ℝ] H10 ho →L[ℝ] ℝ := by let a : H10 ho →L[ℝ] H1 Ω →L[ℝ] ℝ := (pair μ).comp (incl ho) let b : H1 Ω →L[ℝ] H10 ho →L[ℝ] ℝ := ContinuousLinearMap.flip (E := H10 ho) (F := H1 Ω) a exact b.comp (incl ho) lemma zeroPair_apply (μ : ℝ) (u v : H10 ho) : zeroPair ho μ u v = inner ℝ (H10.grad ho u) (H10.grad ho v)-μ*inner ℝ (H10.value ho u) (H10.value ho v) := rfl lemma zeroPair_pair (μ : ℝ) (u v : H10 ho) : zeroPair ho μ u v=pair μ v.val u.val := rfl lemma zeroPair_self (μ : ℝ) (v : H10 ho) : zeroPair ho μ v v= ‖H10.grad ho v‖^2-μ*‖H10.value ho v‖^2 := (zeroPair_apply ho μ v v).trans (congrArg₂ (fun a b : ℝ => a-μ*b) (real_inner_self_eq_norm_sq (H10.grad ho v)) (real_inner_self_eq_norm_sq (H10.value ho v))) lemma subcritical_numeric {μ L C x y z : ℝ} (hμ : 0 ≤ μ) (hl : μ < L) (hv : C*x*x ≤ z^2) (hd : L*y^2 ≤ z^2) : ((1-μ/L)*C)*x*x ≤ z^2-μ*y^2 := by have hL : 0 < L := lt_of_le_of_lt hμ hl have hρ : 0 < 1-μ/L := sub_pos.mpr ((div_lt_one hL).mpr hl) have hmul := mul_le_mul_of_nonneg_left hd (div_nonneg hμ hL.le) have he : μ/L*(L*y^2)=μ*y^2 := by field_simp [ne_of_gt hL] rw [he] at hmul have hh := mul_le_mul_of_nonneg_left hv hρ.le nlinarith lemma zeroPair_coercive (hb : Bornology.IsBounded Ω) {μ L : ℝ} (hμ : 0 ≤ μ) (hl : μ < L) (hD : ∀ j : H10 ho, L*‖H10.value ho j‖^2 ≤ ‖H10.grad ho j‖^2) : @IsCoercive (H10 ho) (H10.normedAddCommGroup ho).toSeminormedAddCommGroup (H10.innerProductSpace ho).toNormedSpace (zeroPair ho μ) := by obtain ⟨C,hC,hE⟩ := H10.energy_coercive ho hb have hL : 0 < L := lt_of_le_of_lt hμ hl have hρ : 0 < 1-μ/L := sub_pos.mpr ((div_lt_one hL).mpr hl) refine ⟨(1-μ/L)*C,mul_pos hρ hC,fun v => ?_⟩ have hv : C*‖v‖*‖v‖ ≤ ‖H10.grad ho v‖^2 := (hE v).trans_eq (by rw [H10.energy_apply,real_inner_self_eq_norm_sq]) exact (subcritical_numeric hμ hl hv (hD v)).trans_eq (zeroPair_self ho μ v).symm def forcing (μ : ℝ) : H1 Ω →L[ℝ] H10 ho →L[ℝ] ℝ := by let a : H10 ho →L[ℝ] H1 Ω →L[ℝ] ℝ := (pair μ).flip.comp (incl ho) let b : H1 Ω →L[ℝ] H10 ho →L[ℝ] ℝ := ContinuousLinearMap.flip (E := H10 ho) (F := H1 Ω) a exact -b def correction (hb : Bornology.IsBounded Ω) {μ L : ℝ} (hμ : 0 ≤ μ) (hl : μ < L) (hD : ∀ j : H10 ho, L*‖H10.value ho j‖^2 ≤ ‖H10.grad ho j‖^2) : H1 Ω →L[ℝ] H10 ho := (zeroPair_coercive ho hb hμ hl hD).continuousLinearEquivOfBilin.symm.toContinuousLinearMap.comp ((InnerProductSpace.toDual ℝ (H10 ho)).symm.toContinuousLinearEquiv.toContinuousLinearMap.comp (forcing ho μ)) def extension (hb : Bornology.IsBounded Ω) {μ L : ℝ} (hμ : 0 ≤ μ) (hl : μ < L) (hD : ∀ j : H10 ho, L*‖H10.value ho j‖^2 ≤ ‖H10.grad ho j‖^2) : H1 Ω →L[ℝ] H1 Ω := ContinuousLinearMap.id ℝ (H1 Ω)+(incl ho).comp (correction ho hb hμ hl hD) lemma correction_spec (hb : Bornology.IsBounded Ω) {μ L : ℝ} (hμ : 0 ≤ μ) (hl : μ < L) (hD : ∀ j : H10 ho, L*‖H10.value ho j‖^2 ≤ ‖H10.grad ho j‖^2) (a : H1 Ω) (v : H10 ho) : zeroPair ho μ (correction ho hb hμ hl hD a) v = -pair μ a v.val := by rw [← (zeroPair_coercive ho hb hμ hl hD).continuousLinearEquivOfBilin_apply] simp only [correction,ContinuousLinearMap.comp_apply,ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.apply_symm_apply] exact InnerProductSpace.toDual_symm_apply lemma extension_class (hb : Bornology.IsBounded Ω) {μ L : ℝ} (hμ : 0 ≤ μ) (hl : μ < L) (hD : ∀ j : H10 ho, L*‖H10.value ho j‖^2 ≤ ‖H10.grad ho j‖^2) (a : H1 Ω) : ∃ w : H10 ho, extension ho hb hμ hl hD a-a=w.val := by exact ⟨correction ho hb hμ hl hD a,add_sub_cancel_left _ _⟩ lemma extension_weak (hb : Bornology.IsBounded Ω) {μ L : ℝ} (hμ : 0 ≤ μ) (hl : μ < L) (hD : ∀ j : H10 ho, L*‖H10.value ho j‖^2 ≤ ‖H10.grad ho j‖^2) (a : H1 Ω) (v : H10 ho) : pair μ (extension ho hb hμ hl hD a) v.val=0 := by change pair μ (a+(correction ho hb hμ hl hD a).val) v.val=0 rw [map_add,add_apply] have he := correction_spec ho hb hμ hl hD a v have hh : pair μ (correction ho hb hμ hl hD a).val v.val= zeroPair ho μ (correction ho hb hμ hl hD a) v := by exact (pair_comm μ _ _).trans (zeroPair_pair ho μ _ _).symm rw [hh,he,add_neg_cancel] lemma norm_zero_numeric {C x : ℝ} (hC : 0 < C) (hx : 0 ≤ x) (h : C*x*x ≤ 0) : x=0 := by by_contra hn have hp : 0 < x := lt_of_le_of_ne hx (Ne.symm hn) exact (not_le_of_gt (mul_pos (mul_pos hC hp) hp)) h lemma self_zero (hb : Bornology.IsBounded Ω) {μ L : ℝ} (hμ : 0 ≤ μ) (hl : μ < L) (hD : ∀ j : H10 ho, L*‖H10.value ho j‖^2 ≤ ‖H10.grad ho j‖^2) (w : H10 ho) (hz : zeroPair ho μ w w=0) : w=0 := by obtain ⟨C,hC,hB⟩ := zeroPair_coercive ho hb hμ hl hD have hn := hB w rw [hz] at hn exact norm_eq_zero.mp (norm_zero_numeric hC (norm_nonneg w) hn) lemma pair_sub (μ : ℝ) (a b v : H1 Ω) : pair μ (a-b) v = pair μ a v-pair μ b v := by exact congrArg (fun t : H1 Ω →L[ℝ] ℝ => t v) ((pair μ).map_sub a b) lemma unique (hb : Bornology.IsBounded Ω) {μ L : ℝ} (hμ : 0 ≤ μ) (hl : μ < L) (hD : ∀ j : H10 ho, L*‖H10.value ho j‖^2 ≤ ‖H10.grad ho j‖^2) {a b : H1 Ω} (ha : ∀ v : H10 ho, pair μ a v.val=0) (hb' : ∀ v : H10 ho, pair μ b v.val=0) (hab : ∃ w : H10 ho, a-b=w.val) : a=b := by obtain ⟨w,hw⟩ := hab have h0 : pair μ w.val w.val=0 := by calc _ = pair μ (a-b) w.val := congrArg (fun z : H1 Ω => pair μ z w.val) hw.symm _ = pair μ a w.val-pair μ b w.val := pair_sub μ a b w.val _ = 0 := by rw [ha,hb',sub_self] have hz : zeroPair ho μ w w=0 := h0 have hw0 : w=0 := self_zero ho hb hμ hl hD w hz apply sub_eq_zero.mp exact hw.trans (congrArg (fun z : H10 ho => z.val) hw0) end StrictHotSpots.PhysicalHelmholtz end end PhysicalHelmholtz section PhysicalSolutionWeak noncomputable section open Set MeasureTheory Filter open scoped ContDiff InnerProductSpace ENNReal NNReal namespace StrictHotSpots.Conformal variable (e : OpenPartialHomeomorph Plane Plane) (he : ContDiffOn ℝ ∞ e.symm e.target) (d : Plane → ℂ) (hd : ∀ x ∈ e.source, HasFDerivAt e (planeMul (d x)) x) (hb : Bornology.IsBounded e.target) lemma inner_H1grad_add_single {S : Set Plane} (a b z : H1 S) : inner ℝ (H1.grad (a+b)) (H1.grad z) = inner ℝ (H1.grad a) (H1.grad z)+inner ℝ (H1.grad b) (H1.grad z) := by rw [map_add,inner_add_left] include d hd hb in lemma affine_H10_grad_pair {w : Plane → ℝ} (hw : DifferentiableOn ℝ w e.target) (hW : HasH1Gradient e.target w (gradient w)) (hV : HasH1Gradient e.source (w ∘ e) (gradient (w ∘ e))) (u v : H10 e.open_source) : inner ℝ (H1.grad (hW.toH1+(transportH10 e he u).val)) (H10.grad e.open_target (transportH10 e he v)) = inner ℝ (H1.grad (hV.toH1+u.val)) (H10.grad e.open_source v) := by exact (inner_H1grad_add_single hW.toH1 (transportH10 e he u).val (transportH10 e he v).val).trans ((congrArg₂ (fun a b : ℝ => a+b) (transportH10_mixed_grad e he d hd hb hw hW hV v) (transportH10_inner_grad e he d hd hb u v)).trans (inner_H1grad_add_single hV.toH1 u.val v.val).symm) include hd hb in lemma affine_H10_mass_pair {w : Plane → ℝ} (hW : HasH1Gradient e.target w (gradient w)) (hV : HasH1Gradient e.source (w ∘ e) (gradient (w ∘ e))) (u v : H10 e.open_source) : inner ℝ (H1.value (hW.toH1+(transportH10 e he u).val)) (H10.value e.open_target (transportH10 e he v)) = ∫ x, H1.value (hV.toH1+u.val) x*H10.value e.open_source v x ∂chartMeasure e d := isometry_value_pair (chartPull e d hd) _ _ (affine_value_ae e he d hd hb hW hV u) (weightedValue_ae e he d hd hb v).symm namespace ClosedDiskChart open DiskH10 PlaneGreen variable {Ω : Set Plane} (c : ClosedDiskChart Ω) variable (hb : Bornology.IsBounded Ω) (hs : SmoothBoundary Ω) variable (hμ : 0 ≤ firstPositiveNeumannValue Ω) lemma physicalSolution_mixed_grad {w : Plane → ℝ} (hw : ContDiffOn ℝ ∞ w (closure Ω)) {B : ℝ≥0} (hl : LipschitzWith B (c.datumTrace w hw.continuousOn)) (v : H10 diskOpen) : inner ℝ (H1.grad (c.physicalSolution hb hs hμ hw hl)) (H10.grad c.diskMap.open_target (transportH10 c.diskMap c.diskMap_inverse_smooth v)) = inner ℝ (H1.grad (boundarySolution c.densityBound_ne_top (c.density_bound hμ) (c.datumTrace w hw.continuousOn) hl)) (H10.grad diskOpen v) := by have hh := affine_H10_grad_pair c.diskMap c.diskMap_inverse_smooth c.d c.diskMap_deriv hb ((hw.mono subset_closure).differentiableOn (by simp)) (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) v apply hh.trans apply congrArg (fun U : H1 disk => inner ℝ (H1.grad U) (H10.grad diskOpen v)) exact (congrArg (fun U : H1 disk => U+(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 lemma physicalSolution_mixed_mass {w : Plane → ℝ} (hw : ContDiffOn ℝ ∞ w (closure Ω)) {B : ℝ≥0} (hl : LipschitzWith B (c.datumTrace w hw.continuousOn)) (v : H10 diskOpen) : firstPositiveNeumannValue Ω * inner ℝ (H1.value (c.physicalSolution hb hs hμ hw hl)) (H10.value c.diskMap.open_target (transportH10 c.diskMap c.diskMap_inverse_smooth v)) = inner ℝ (SubcriticalExtension.observe c.densityBound_ne_top (c.density_bound hμ) (boundarySolution c.densityBound_ne_top (c.density_bound hμ) (c.datumTrace w hw.continuousOn) hl)) (H10.weightedValue diskOpen c.densityBound_ne_top (c.density_bound hμ) v) := by have hh := affine_H10_mass_pair 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) v 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 U : H1 disk => U+(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 let S := boundarySolution c.densityBound_ne_top (c.density_bound hμ) (c.datumTrace w hw.continuousOn) hl have hr := congrArg (fun U : H1 disk => ∫ x, H1.value U x*H10.value diskOpen v x ∂chartMeasure c.diskMap c.d) he have hp := potentialMeasure_integral c.diskMap c.d hμ (fun x => H1.value S x*H10.value diskOpen v x) have hq := congrArg (fun ν : Measure Plane => ∫ x, H1.value S x*H10.value diskOpen v x ∂ν) c.potential_diskMap exact (congrArg (fun t : ℝ => firstPositiveNeumannValue Ω*t) (hh.trans hr)).trans (hp.symm.trans (hq.trans (observe_inner c.densityBound_ne_top (c.density_bound hμ) S v.val).symm)) lemma pair_forward {S : Set Plane} (μ : ℝ) (u v : H1 S) : PhysicalHelmholtz.pair μ u v = inner ℝ (H1.grad u) (H1.grad v)- μ*inner ℝ (H1.value u) (H1.value v) := (PhysicalHelmholtz.pair_comm μ u v).trans (PhysicalHelmholtz.pair_apply μ v u) lemma physicalSolution_transported_weak (hne : Ω.Nonempty) (hpos : 0 < firstPositiveNeumannValue Ω) {w : Plane → ℝ} (hw : ContDiffOn ℝ ∞ w (closure Ω)) {B : ℝ≥0} (hl : LipschitzWith B (c.datumTrace w hw.continuousOn)) (t : H10 diskOpen) : PhysicalHelmholtz.pair (firstPositiveNeumannValue Ω) (c.physicalSolution hb hs hμ hw hl) (show H1 Ω from (transportH10 c.diskMap c.diskMap_inverse_smooth t).val)=0 := by have hn := c.operator_norm_lt_one hb hne hpos have hwk := boundarySolution_weak c.densityBound_ne_top (c.density_bound hμ) hn (c.datumTrace w hw.continuousOn) hl t have hg := c.physicalSolution_mixed_grad hb hs hμ hw hl t have hm := c.physicalSolution_mixed_mass hb hs hμ hw hl t exact (pair_forward _ _ _).trans (sub_eq_zero.mpr (hg.trans (hwk.trans hm.symm))) lemma physicalSolution_weak (hne : Ω.Nonempty) (hpos : 0 < firstPositiveNeumannValue Ω) {w : Plane → ℝ} (hw : ContDiffOn ℝ ∞ w (closure Ω)) {B : ℝ≥0} (hl : LipschitzWith B (c.datumTrace w hw.continuousOn)) (v : H10 (show IsOpen Ω from c.diskMap.open_target)) : PhysicalHelmholtz.pair (firstPositiveNeumannValue Ω) (c.physicalSolution hb hs hμ hw hl) v.val=0 := by let t : H10 diskOpen := transportH10 c.diskMap.symm c.diskMap_smooth v have ht : transportH10 c.diskMap c.diskMap_inverse_smooth t=v := transportH10_symm c.diskMap.symm c.diskMap_smooth c.diskMap_inverse_smooth c.d' c.d c.diskMap_inverse_deriv c.diskMap_deriv diskBounded hb v have he := c.physicalSolution_transported_weak hb hs hμ hne hpos hw hl t exact (congrArg (fun V : H10 (show IsOpen Ω from c.diskMap.open_target) => PhysicalHelmholtz.pair (firstPositiveNeumannValue Ω) (c.physicalSolution hb hs hμ hw hl) V.val) ht).symm.trans he end ClosedDiskChart end StrictHotSpots.Conformal end end PhysicalSolutionWeak section LipschitzH1Convergence noncomputable section open Set MeasureTheory Filter Metric open scoped ContDiff InnerProductSpace ENNReal NNReal Topology namespace StrictHotSpots lemma H1.tendsto_of_components {Ω : Set Plane} {v : ℕ → H1 Ω} {u : H1 Ω} (hv : Tendsto (fun n => H1.value (v n)) atTop (𝓝 (H1.value u))) (hg : Tendsto (fun n => H1.grad (v n)) atTop (𝓝 (H1.grad u))) : Tendsto v atTop (𝓝 u) := by apply tendsto_subtype_rng.mpr exact (WithLp.prod_continuous_toLp 2 _ _).continuousAt.tendsto.comp (hv.prodMk_nhds hg) lemma smoothMul_tendsto_lipschitzMul {Ω : Set Plane} (ho : IsOpen Ω) (hΩ : Bornology.IsBounded Ω) {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) (u : H1 Ω) : Tendsto (fun n => H1.smoothMul (ambientSmooth_smooth (planeApproxBump n) hb) (continuous_closure_memLp_top ho hΩ (ambientSmooth_lipschitz (planeApproxBump n) hb).continuous.continuousOn) (lipschitz_gradient_memLp_top (ambientSmooth_lipschitz (planeApproxBump n) hb)) u) atTop (𝓝 (H1.lipschitzMul ho hΩ hb u)) := by let bn := fun n => ambientSmooth (planeApproxBump n) b have hbn (n : ℕ) : ContDiff ℝ ∞ (bn n) := ambientSmooth_smooth _ hb have hbl (n : ℕ) : LipschitzWith L (bn n) := ambientSmooth_lipschitz _ hb have hn0 (n : ℕ) := continuous_closure_memLp_top ho hΩ (hbl n).continuous.continuousOn have hn1 (n : ℕ) := lipschitz_gradient_memLp_top (Ω:=Ω) (hbl n) have hu := H1.hasH1Gradient u have hf (n : ℕ) := hu.mul_smooth (hbn n) (hn0 n) (hn1 n) have hl := hu.mul_lipschitz ho hΩ hb obtain ⟨C,hC,hbd⟩ := ambientSmooth_uniform_bound hΩ hb have ht (x : Plane) : Tendsto (fun n => bn n x) atTop (𝓝 (b x)) := ambientSmooth_tendsto planeApproxBump_rOut_tendsto hb x have hgt : ∀ᵐ x ∂volume.restrict Ω, Tendsto (fun n => gradient (bn n) x) atTop (𝓝 (gradient b x)) := ae_restrict_of_ae (ambientSmooth_gradient_tendsto planeApproxBump_rOut_tendsto (Eventually.of_forall planeApproxBump_ratio) hb) apply H1.tendsto_of_components · apply L2VectorLimits.tendsto_toLp_of_dominated (fun n => (hf n).1) hl.1 (hu.1.norm.const_mul C) · filter_upwards [] with n filter_upwards [ae_restrict_mem ho.measurableSet] with x hx rw [norm_mul] exact mul_le_mul_of_nonneg_right (hbd n x hx) (norm_nonneg _) · exact Eventually.of_forall fun x => (ht x).mul_const (H1.value u x) · apply L2VectorLimits.tendsto_toLp_of_dominated (fun n => (hf n).2.1) hl.2.1 ((hu.2.1.norm.const_mul C).add (hu.1.norm.const_mul (L:ℝ))) · filter_upwards [] with n filter_upwards [ae_restrict_mem ho.measurableSet] with x hx calc _ ≤ ‖bn n x • H1.grad u x‖+‖H1.value u x • gradient (bn n) x‖ := norm_add_le _ _ _ = ‖bn n x‖*‖H1.grad u x‖+‖H1.value u x‖*‖gradient (bn n) x‖ := by rw [norm_smul,norm_smul] _ ≤ C*‖H1.grad u x‖+‖H1.value u x‖*(L:ℝ) := add_le_add (mul_le_mul_of_nonneg_right (hbd n x hx) (norm_nonneg _)) (mul_le_mul_of_nonneg_left (lipschitz_gradient_bound (hbl n) x) (norm_nonneg _)) _ = _ := by simp only [Pi.add_apply]; ring · filter_upwards [hgt] with x hx exact ((ht x).smul_const (H1.grad u x)).add (tendsto_const_nhds.smul hx) end StrictHotSpots end end LipschitzH1Convergence section LipschitzVectorMultiplier noncomputable section open Set MeasureTheory Filter open scoped ContDiff InnerProductSpace ENNReal NNReal namespace StrictHotSpots.Hodge variable {Ω : Set Plane} (ho : IsOpen Ω) (hbound : Bornology.IsBounded Ω) variable {b : Plane → ℝ} {L : ℝ≥0} (hb : LipschitzWith L b) theorem boundaryDefect_lipschitz_multiplier (X : Fin 2 → H1 Ω) : boundaryDefect (fun i => H1.lipschitzMul ho hbound hb (X i)) (fun i => H1.lipschitzMul ho hbound hb (X i)) = boundaryDefect X (fun i => H1.lipschitzMul ho hbound hb (H1.lipschitzMul ho hbound hb (X i))) := by rw [boundaryDefect_integral,boundaryDefect_integral] apply integral_congr_ae let M := H1.lipschitzMul ho hbound hb filter_upwards [H1.lipschitzMul_grad_ae ho hbound hb (X 0), H1.lipschitzMul_grad_ae ho hbound hb (X 1), H1.lipschitzMul_value_ae ho hbound hb (X 0), H1.lipschitzMul_value_ae ho hbound hb (X 1), H1.lipschitzMul_grad_ae ho hbound hb (M (X 0)), H1.lipschitzMul_grad_ae ho hbound hb (M (X 1))] with x h0 h1 hv0 hv1 hh0 hh1 change H1.grad (M (X 0)) x = _ at h0 change H1.grad (M (X 1)) x = _ at h1 change H1.value (M (X 0)) x = _ at hv0 change H1.value (M (X 1)) x = _ at hv1 change -inner ℝ (H1.grad (M (X 0)) x) (rot (H1.grad (M (X 1)) x))+ inner ℝ (H1.grad (M (X 1)) x) (rot (H1.grad (M (X 0)) x)) = _ exact defect_multiplier_substitute (b x) (H1.value (X 0) x) (H1.value (X 1) x) (H1.value (M (X 0)) x) (H1.value (M (X 1)) x) (H1.grad (X 0) x) (H1.grad (X 1) x) (gradient b x) (H1.grad (M (X 0)) x) (H1.grad (M (X 1)) x) (H1.grad (M (M (X 0))) x) (H1.grad (M (M (X 1))) x) h0 h1 hv0 hv1 hh0 hh1 lemma vectorValue_lipschitzMul_ae (X : Fin 2 → H1 Ω) : vectorValue (fun i => H1.lipschitzMul ho hbound hb (X i)) =ᵐ[volume.restrict Ω] fun x => b x • vectorValue X x := by let M := H1.lipschitzMul ho hbound hb filter_upwards [vectorValue_ae (fun i => M (X i)),vectorValue_ae X, H1.lipschitzMul_value_ae ho hbound hb (X 0),H1.lipschitzMul_value_ae ho hbound hb (X 1)] with x hV hX h0 h1 apply hV.trans exact (congrArg₂ (fun a d : ℝ => a • e 0+d • e 1) h0 h1).trans (by rw [hX]; module) lemma divergenceL_lipschitzMul_ae (X : Fin 2 → H1 Ω) : divergenceL (fun i => H1.lipschitzMul ho hbound hb (X i)) =ᵐ[volume.restrict Ω] fun x => b x*divergenceL X x+inner ℝ (vectorValue X x) (gradient b x) := by let M := H1.lipschitzMul ho hbound hb filter_upwards [divergenceL_ae (fun i => M (X i)),divergenceL_ae X,vectorValue_ae X, H1.lipschitzMul_grad_ae ho hbound hb (X 0),H1.lipschitzMul_grad_ae ho hbound hb (X 1)] with x hD hX hV h0 h1 apply hD.trans exact (congrArg₂ (fun A B : Plane => c 0 A+c 1 B) h0 h1).trans ((divergence_multiplier_algebra _ _ _ _ _ _).trans (congrArg₂ (fun d : ℝ => fun V : Plane => b x*d+inner ℝ V (gradient b x)) hX hV).symm) lemma normal_pair_lipschitzMul (X : Fin 2 → H1 Ω) (u : H1 Ω) : inner ℝ (vectorValue (fun i => H1.lipschitzMul ho hbound hb (X i))) (H1.grad u)+ inner ℝ (divergenceL (fun i => H1.lipschitzMul ho hbound hb (X i))) (H1.value u) = inner ℝ (vectorValue X) (H1.grad (H1.lipschitzMul ho hbound hb u))+ inner ℝ (divergenceL X) (H1.value (H1.lipschitzMul ho hbound hb u)) := by refine Eq.trans (normal_pair_integral _ _) (Eq.trans ?_ (normal_pair_integral _ _).symm) apply integral_congr_ae filter_upwards [vectorValue_lipschitzMul_ae ho hbound hb X,divergenceL_lipschitzMul_ae ho hbound hb X, H1.lipschitzMul_grad_ae ho hbound hb u,H1.lipschitzMul_value_ae ho hbound hb u] with x hV hD hG hu exact (congrArg₂ (fun V : Plane => fun F : ℝ => inner ℝ V (H1.grad u x)+F*H1.value u x) hV hD).trans ((normal_multiplier_algebra _ _ _ _ _ _).trans (congrArg₂ (fun G : Plane => fun v : ℝ => inner ℝ (vectorValue X x) G+divergenceL X x*v) hG hu).symm) theorem TangentWeak.lipschitzMul {X : Fin 2 → H1 Ω} (hX : TangentWeak X) : TangentWeak (fun i => H1.lipschitzMul ho hbound hb (X i)) := by intro u have he := normal_pair_lipschitzMul ho hbound hb X u have hz := hX (H1.lipschitzMul ho hbound hb u) linarith only [he,hz] theorem boundaryClass_lipschitz_multiplier_identity (μ : ℝ) (X Y Z : Fin 2 → H1 Ω) (hX : TangentWeak X) (hY : ∀ i, ∃ w : H10 ho, Y i-H1.lipschitzMul ho hbound hb (X i)=w.val) (hZ : ∀ i, ∃ w : H10 ho, Z i-H1.lipschitzMul ho hbound hb (H1.lipschitzMul ho hbound hb (X i))=w.val) (hpos : ∀ V : Fin 2 → H1 Ω, TangentWeak V → 0 ≤ vectorForm μ V V) (hnull : vectorForm μ X X=0) : vectorForm μ Y Y=cartesianForm μ Y Y-cartesianForm μ X Z := by let M := H1.lipschitzMul ho hbound hb have htZ : TangentWeak Z := TangentWeak.of_class ho ((hX.lipschitzMul ho hbound hb).lipschitzMul ho hbound hb) hZ have hz : vectorForm μ X Z=0 := vectorForm_null_pair μ X Z hX htZ hpos hnull have hYY : boundaryDefect Y Y = boundaryDefect (fun i => M (X i)) (fun i => M (X i)) := boundaryDefect_class ho Y Y _ _ hY hY have hZZ : boundaryDefect X Z = boundaryDefect X (fun i => M (M (X i))) := boundaryDefect_class_right ho X Z _ hZ have he : boundaryDefect Y Y = boundaryDefect X Z := hYY.trans ((boundaryDefect_lipschitz_multiplier ho hbound hb X).trans hZZ.symm) have hqY := vectorForm_eq_cartesian_add_defect μ Y Y have hqZ := vectorForm_eq_cartesian_add_defect μ X Z linarith only [hqY,hqZ,he,hz] end StrictHotSpots.Hodge end end LipschitzVectorMultiplier section PhysicalMultiplierLimit 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_weak {w : Plane → ℝ} (hw : ContDiffOn ℝ ∞ w (closure Ω)) (v : H10 hΩ.2.1) : PhysicalHelmholtz.pair (firstPositiveNeumannValue Ω) (c.scalarSolution hΩ.2.2.1 hΩ.2.2.2.2 hμ hw) v.val=0 := c.physicalSolution_weak hΩ.2.2.1 hΩ.2.2.2.2 hμ.le hΩ.1 hμ hw _ v lemma class_difference {S : Set Plane} (ho : IsOpen S) {a b f : H1 S} (ha : ∃ v : H10 ho, a-f=v.val) (hb : ∃ v : H10 ho, b-f=v.val) : ∃ v : H10 ho, a-b=v.val := by obtain ⟨v,hv⟩ := ha obtain ⟨w,hw⟩ := hb refine ⟨v-w,?_⟩ change a-b=v.val-w.val exact (show a-b=(a-f)-(b-f) from by abel).trans (congrArg₂ (fun x y : H1 S => x-y) hv hw) lemma scalarSolution_eq_extension {w : Plane → ℝ} (hw : ContDiffOn ℝ ∞ w (closure Ω)) : c.scalarSolution hΩ.2.2.1 hΩ.2.2.2.2 hμ hw = PhysicalHelmholtz.extension hΩ.2.1 hΩ.2.2.1 hμ.le hl hD (smooth_closure_H1 hΩ.2.1 hΩ.2.2.1 hΩ.2.2.2.2 hw).toH1 := by apply PhysicalHelmholtz.unique hΩ.2.1 hΩ.2.2.1 hμ.le hl hD (c.scalarSolution_weak hΩ hμ hw) (PhysicalHelmholtz.extension_weak hΩ.2.1 hΩ.2.2.1 hμ.le hl hD _) exact class_difference hΩ.2.1 (c.scalarSolution_class hΩ.2.2.1 hΩ.2.2.2.2 hμ hw) (PhysicalHelmholtz.extension_class hΩ.2.1 hΩ.2.2.1 hμ.le hl hD _) lemma multipliedGradient_eq_extension {u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u) {b : Plane → ℝ} (hb : ContDiff ℝ ∞ b) (hb0 : MemLp b ∞ (volume.restrict Ω)) (hb1 : MemLp (gradient b) ∞ (volume.restrict Ω)) (i : Fin 2) : c.multipliedGradient hΩ hu hμ hb i = PhysicalHelmholtz.extension hΩ.2.1 hΩ.2.2.1 hμ.le hl hD (H1.smoothMul hb hb0 hb1 (eigenGradientField hΩ hu i)) := by have hh := c.scalarSolution_eq_extension hΩ hμ hl hD (hb.contDiffOn.mul (closure_directional_smooth hΩ.2.1 hΩ.2.2.2.2 hu.1 (Hodge.e i))) have he := smoothMul_smooth_closure hΩ.2.1 hΩ.2.2.1 hΩ.2.2.2.2 hb hb0 hb1 (closure_directional_smooth hΩ.2.1 hΩ.2.2.2.2 hu.1 (Hodge.e i)) exact hh.trans (congrArg (PhysicalHelmholtz.extension hΩ.2.1 hΩ.2.2.1 hμ.le hl hD) he.symm) def lipschitzMultipliedGradient {u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u) {b : Plane → ℝ} {B : ℝ≥0} (hb : LipschitzWith B b) : Fin 2 → H1 Ω := fun i => PhysicalHelmholtz.extension hΩ.2.1 hΩ.2.2.1 hμ.le hl hD (H1.lipschitzMul hΩ.2.1 hΩ.2.2.1 hb (eigenGradientField hΩ hu i)) lemma multipliedGradient_tendsto {u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u) {b : Plane → ℝ} {B : ℝ≥0} (hb : LipschitzWith B b) (i : Fin 2) : Tendsto (fun n => c.multipliedGradient hΩ hu hμ (ambientSmooth_smooth (planeApproxBump n) hb) i) atTop (𝓝 (lipschitzMultipliedGradient hΩ hμ hl hD hu hb i)) := by 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 (eigenGradientField hΩ hu i)) apply ht.congr' exact Eventually.of_forall fun n => (c.multipliedGradient_eq_extension hΩ hμ hl hD hu (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)) i).symm lemma lipschitzMultipliedGradient_class {u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u) {b : Plane → ℝ} {B : ℝ≥0} (hb : LipschitzWith B b) (i : Fin 2) : ∃ w : H10 hΩ.2.1, lipschitzMultipliedGradient hΩ hμ hl hD hu hb i- H1.lipschitzMul hΩ.2.1 hΩ.2.2.1 hb (eigenGradientField hΩ hu i)=w.val := PhysicalHelmholtz.extension_class hΩ.2.1 hΩ.2.2.1 hμ.le hl hD _ lemma lipschitzMultipliedGradient_tangent {u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u) {b : Plane → ℝ} {B : ℝ≥0} (hb : LipschitzWith B b) : TangentWeak (lipschitzMultipliedGradient hΩ hμ hl hD hu hb) := TangentWeak.of_class hΩ.2.1 ((eigenGradientField_null hΩ hu).1.lipschitzMul hΩ.2.1 hΩ.2.2.1 hb) (lipschitzMultipliedGradient_class hΩ hμ hl hD hu hb) end StrictHotSpots.Conformal.ClosedDiskChart namespace StrictHotSpots.Hodge variable {Ω : Set Plane} lemma vectorForm_tendsto {X : ℕ → Fin 2 → H1 Ω} {Y : Fin 2 → H1 Ω} (h : ∀ i, Tendsto (fun n => X n i) atTop (𝓝 (Y i))) (μ : ℝ) : Tendsto (fun n => vectorForm μ (X n) (X n)) atTop (𝓝 (vectorForm μ Y Y)) := by have hg (i : Fin 2) := H1.grad.continuous.continuousAt.tendsto.comp (h i) have hv (i : Fin 2) := H1.value.continuous.continuousAt.tendsto.comp (h i) have hp (i j : Fin 2) := (coordinateL Ω i).continuous.continuousAt.tendsto.comp (hg j) have ha (i : Fin 2) := (axisL Ω i).continuous.continuousAt.tendsto.comp (hv i) have hd := (hp 0 0).add (hp 1 1) have hc := (hp 0 1).sub (hp 1 0) have hb := (ha 0).add (ha 1) exact ((hd.inner hd).add (hc.inner hc)).sub ((hb.inner hb).const_mul μ) end StrictHotSpots.Hodge end end PhysicalMultiplierLimit section MultiplierKernelConvergence noncomputable section open Set MeasureTheory Filter Metric open scoped ENNReal NNReal Topology InnerProductSpace namespace StrictHotSpots.PlaneGreen variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] lemma harmonic_square_bound {b : Boundary → ℝ} {B : ℝ≥0} (hb : LipschitzWith B b) (s t : Boundary) : ‖boundaryInteraction s t*(b s-b t)^2‖ ≤ (B:ℝ)*B/Real.pi := by simpa only [Real.inner_apply,pow_two] using harmonic_boundary_bound hb hb s t lemma multiplier_integrand_bound {R : Boundary → Boundary → ℝ} {g : Boundary → E} {b : Boundary → ℝ} {B : ℝ≥0} (hb : LipschitzWith B b) {M G : ℝ} (_hM : 0 ≤ M) (hG : 0 ≤ G) (hbM : ∀ s, ‖b s‖ ≤ M) (hgG : ∀ s, ‖g s‖ ≤ G) (s t : Boundary) : ‖(boundaryInteraction s t+R s t)*(b s-b t)^2*inner ℝ (g s) (g t)‖ ≤ ((B:ℝ)*B/Real.pi+(M+M)^2*‖R s t‖)*(G*G) := by have hd : ‖b s-b t‖ ≤ M+M := (norm_sub_le _ _).trans (add_le_add (hbM s) (hbM t)) have hi : ‖inner ℝ (g s) (g t)‖ ≤ G*G := (norm_inner_le_norm _ _).trans (mul_le_mul (hgG s) (hgG t) (norm_nonneg _) hG) have hr : ‖R s t*(b s-b t)^2‖ ≤ (M+M)^2*‖R s t‖ := by rw [norm_mul,norm_pow,mul_comm] exact mul_le_mul_of_nonneg_right (pow_le_pow_left₀ (norm_nonneg _) hd 2) (norm_nonneg _) have he : (boundaryInteraction s t+R s t)*(b s-b t)^2 = boundaryInteraction s t*(b s-b t)^2+R s t*(b s-b t)^2 := by ring rw [he,norm_mul] exact mul_le_mul ((norm_add_le _ _).trans (add_le_add (harmonic_square_bound hb s t) hr)) hi (norm_nonneg _) (by positivity) theorem multiplier_kernel_tendsto {R : Boundary → Boundary → ℝ} (hR : Integrable (fun z : Boundary × Boundary => R z.1 z.2) (boundaryMeasure.prod boundaryMeasure)) {g : Boundary → E} (hg : Continuous g) {b : Boundary → ℝ} {bn : ℕ → Boundary → ℝ} {B : ℝ≥0} (hb : ∀ n, LipschitzWith B (bn n)) {M : ℝ} (hM : 0 ≤ M) (hMb : ∀ n s, ‖bn n s‖ ≤ M) (ht : ∀ s, Tendsto (fun n => bn n s) atTop (𝓝 (b s))) : Tendsto (fun n => ∫ z : Boundary × Boundary, (boundaryInteraction z.1 z.2+R z.1 z.2)*(bn n z.1-bn n z.2)^2* inner ℝ (g z.1) (g z.2) ∂boundaryMeasure.prod boundaryMeasure) atTop (𝓝 (∫ z : Boundary × Boundary, (boundaryInteraction z.1 z.2+R z.1 z.2)*(b z.1-b z.2)^2* inner ℝ (g z.1) (g z.2) ∂boundaryMeasure.prod boundaryMeasure)) := by obtain ⟨G,hG,hGb⟩ := boundary_continuous_bound hg apply tendsto_integral_of_dominated_convergence (fun z : Boundary × Boundary => ((B:ℝ)*B/Real.pi+(M+M)^2*‖R z.1 z.2‖)*(G*G)) · intro n exact ((boundaryInteraction_measurable.aestronglyMeasurable.add hR.aestronglyMeasurable).mul ((((hb n).continuous.comp continuous_fst).sub ((hb n).continuous.comp continuous_snd)).pow 2).aestronglyMeasurable).mul ((hg.comp continuous_fst).inner (hg.comp continuous_snd)).aestronglyMeasurable · exact ((integrable_const _).add (hR.norm.const_mul ((M+M)^2))).mul_const (G*G) · intro n exact Eventually.of_forall fun z => multiplier_integrand_bound (hb n) hM hG (hMb n) hGb z.1 z.2 · exact Eventually.of_forall fun z => ((((ht z.1).sub (ht z.2)).pow 2).const_mul _).mul_const _ end StrictHotSpots.PlaneGreen end end MultiplierKernelConvergence section ScalarBoundaryForm noncomputable section open Set MeasureTheory Filter open scoped ContDiff InnerProductSpace ENNReal NNReal namespace StrictHotSpots.Conformal.ClosedDiskChart open DiskH10 PlaneGreen Hodge variable {Ω : Set Plane} (c : ClosedDiskChart Ω) variable (hb : Bornology.IsBounded Ω) (hs : SmoothBoundary Ω) variable (hμ : 0 < firstPositiveNeumannValue Ω) lemma scalarSolution_form (hne : Ω.Nonempty) {w z : Plane → ℝ} (hw : ContDiffOn ℝ ∞ w (closure Ω)) (hz : ContDiffOn ℝ ∞ z (closure Ω)) : let _ : IsFiniteMeasure c.potential := c.potential_finite hb H1.form (firstPositiveNeumannValue Ω) (c.scalarSolution hb hs hμ hw) (c.scalarSolution hb hs hμ hz) = boundaryForm (fullBoundaryR c.densityBound_ne_top (c.density_bound hμ.le)) 0 (fun s : Boundary => w (c.F s)) (fun s : Boundary => z (c.F s)) := by let _ : IsFiniteMeasure c.potential := c.potential_finite hb have he := c.scalarSolution_energy hb hs hμ hne hw hz simp only [boundaryForm,Pi.zero_apply,zero_mul,integral_zero,add_zero, RCLike.inner_apply,conj_trivial] at ⊢ change _ = _ rw [show H1.form (firstPositiveNeumannValue Ω) (c.scalarSolution hb hs hμ hw) (c.scalarSolution hb hs hμ hz) = _ from he] congr 1 congr 1 funext q ring end StrictHotSpots.Conformal.ClosedDiskChart namespace StrictHotSpots.Hodge lemma cartesianForm_eq_sum {Ω : Set Plane} (μ : ℝ) (X Y : Fin 2 → H1 Ω) : cartesianForm μ X Y=H1.form μ (X 0) (Y 0)+H1.form μ (X 1) (Y 1) := by simp only [cartesianForm,cartesianEnergy,H1.form] ring end StrictHotSpots.Hodge end end ScalarBoundaryForm section SmoothMultiplierKernel noncomputable section open Set MeasureTheory Filter open scoped ContDiff InnerProductSpace ENNReal NNReal namespace StrictHotSpots.Conformal.ClosedDiskChart open DiskH10 PlaneGreen Hodge variable {Ω : Set Plane} (c : ClosedDiskChart Ω) (hΩ : AdmissibleDomain Ω) variable {u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u) variable (hμ : 0 < firstPositiveNeumannValue Ω) lemma multipliedGradient_coordinate_difference {b : Plane → ℝ} (hb : ContDiff ℝ ∞ b) (i : Fin 2) : let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1 H1.form (firstPositiveNeumannValue Ω) (c.multipliedGradient hΩ hu hμ hb i) (c.multipliedGradient hΩ hu hμ hb i)- H1.form (firstPositiveNeumannValue Ω) (eigenGradientField hΩ hu i) (c.multipliedGradient hΩ hu hμ (hb.mul hb) i) = (1/2:ℝ)*∫ z : Boundary × Boundary, (boundaryInteraction z.1 z.2+fullBoundaryR c.densityBound_ne_top (c.density_bound hμ.le) z.1 z.2)* (b (c.F z.1)-b (c.F z.2))^2* inner ℝ (fderivWithin ℝ u (closure Ω) (c.F z.1) (Hodge.e i)) (fderivWithin ℝ u (closure Ω) (c.F z.2) (Hodge.e i)) ∂boundaryMeasure.prod boundaryMeasure := by let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1 let hw := closure_directional_smooth hΩ.2.1 hΩ.2.2.2.2 hu.1 (Hodge.e i) have hx : eigenGradientField hΩ hu i = c.scalarSolution hΩ.2.2.1 hΩ.2.2.2.2 hμ hw := (c.scalarSolution_eq_of_helmholtz hΩ.2.2.1 hΩ.2.2.2.2 hμ hΩ.1 hw (fun _ hx => closed_directional_helmholtz hΩ.2.1 hu (Hodge.e i) hx)).symm rw [hx] change H1.form _ (c.scalarSolution _ _ hμ (hb.contDiffOn.mul hw)) (c.scalarSolution _ _ hμ (hb.contDiffOn.mul hw))- H1.form _ (c.scalarSolution _ _ hμ hw) (c.scalarSolution _ _ hμ ((hb.contDiffOn.mul hb.contDiffOn).mul hw)) = _ rw [c.scalarSolution_form _ _ _ hΩ.1,c.scalarSolution_form _ _ _ hΩ.1] obtain ⟨A,hA⟩ := c.datumTrace_lipschitz hΩ.2.2.1 hΩ.2.2.2.2 hw obtain ⟨B,hB⟩ := c.datumTrace_lipschitz hΩ.2.2.1 hΩ.2.2.2.2 hb.contDiffOn change LipschitzWith A (fun s : Boundary => fderivWithin ℝ u (closure Ω) (c.F s) (Hodge.e i)) at hA change LipschitzWith B (fun s : Boundary => b (c.F s)) at hB exact (by simpa only [smul_eq_mul,pow_two] using (boundaryForm_multiplier_difference (c := (0 : Boundary → ℝ)) (fullBoundaryR_integrable c.densityBound_ne_top (c.density_bound hμ.le)) (fullBoundaryR_symm c.densityBound_ne_top (c.density_bound hμ.le) (c.operator_norm_lt_one hΩ.2.2.1 hΩ.1 hμ)) hA hB)) lemma datum_multiplier_integrable {w b : Plane → ℝ} (hw : ContDiffOn ℝ ∞ w (closure Ω)) (hb : ContDiff ℝ ∞ b) : let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1 Integrable (fun z : Boundary × Boundary => (boundaryInteraction z.1 z.2+fullBoundaryR c.densityBound_ne_top (c.density_bound hμ.le) z.1 z.2)* (b (c.F z.1)-b (c.F z.2))^2 * inner ℝ (w (c.F z.1)) (w (c.F z.2))) (boundaryMeasure.prod boundaryMeasure) := by let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1 obtain ⟨A,hA⟩ := c.datumTrace_lipschitz hΩ.2.2.1 hΩ.2.2.2.2 hw obtain ⟨B,hB⟩ := c.datumTrace_lipschitz hΩ.2.2.1 hΩ.2.2.2.2 hb.contDiffOn change LipschitzWith A (fun s : Boundary => w (c.F s)) at hA change LipschitzWith B (fun s : Boundary => b (c.F s)) at hB have hR : Integrable (fun z : Boundary × Boundary => fullBoundaryR c.densityBound_ne_top (c.density_bound hμ.le) z.1 z.2) (boundaryMeasure.prod boundaryMeasure) := fullBoundaryR_integrable c.densityBound_ne_top (c.density_bound hμ.le) exact multiplier_integrand_integrable (E := ℝ) (g := fun s : Boundary => w (c.F s)) (b := fun s : Boundary => b (c.F s)) hR hA hB theorem smooth_multiplier_kernel {b : Plane → ℝ} (hb : ContDiff ℝ ∞ b) (hb0 : MemLp b ∞ (volume.restrict Ω)) (hb1 : MemLp (gradient b) ∞ (volume.restrict Ω)) : let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1 vectorForm (firstPositiveNeumannValue Ω) (c.multipliedGradient hΩ hu hμ hb) (c.multipliedGradient hΩ hu hμ hb) = (1/2:ℝ)*∫ z : Boundary × Boundary, (boundaryInteraction z.1 z.2+fullBoundaryR c.densityBound_ne_top (c.density_bound hμ.le) z.1 z.2)* (b (c.F z.1)-b (c.F z.2))^2* inner ℝ (closedGradient Ω u (c.F z.1)) (closedGradient Ω u (c.F z.2)) ∂boundaryMeasure.prod boundaryMeasure := by let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1 let T (i : Fin 2) (z : Boundary × Boundary) := (boundaryInteraction z.1 z.2+fullBoundaryR c.densityBound_ne_top (c.density_bound hμ.le) z.1 z.2)* (b (c.F z.1)-b (c.F z.2))^2* inner ℝ (fderivWithin ℝ u (closure Ω) (c.F z.1) (Hodge.e i)) (fderivWithin ℝ u (closure Ω) (c.F z.2) (Hodge.e i)) have hi (i : Fin 2) : Integrable (T i) (boundaryMeasure.prod boundaryMeasure) := c.datum_multiplier_integrable hΩ hμ (closure_directional_smooth hΩ.2.1 hΩ.2.2.2.2 hu.1 (Hodge.e i)) hb rw [c.multipliedGradient_identity hΩ hu hμ hb hb0 hb1, cartesianForm_eq_sum,cartesianForm_eq_sum] have h0 := c.multipliedGradient_coordinate_difference hΩ hu hμ hb 0 have h1 := c.multipliedGradient_coordinate_difference hΩ hu hμ hb 1 change _ = (1/2:ℝ)* _ at h0 h1 have ha : (∫ z, T 0 z ∂boundaryMeasure.prod boundaryMeasure)+ (∫ z, T 1 z ∂boundaryMeasure.prod boundaryMeasure) = ∫ z : Boundary × Boundary, (boundaryInteraction z.1 z.2+fullBoundaryR c.densityBound_ne_top (c.density_bound hμ.le) z.1 z.2)* (b (c.F z.1)-b (c.F z.2))^2* inner ℝ (closedGradient Ω u (c.F z.1)) (closedGradient Ω u (c.F z.2)) ∂boundaryMeasure.prod boundaryMeasure := by rw [← integral_add (hi 0) (hi 1)] apply integral_congr_ae filter_upwards [] with z rw [Hodge.inner_two] simp only [T,Hodge.c,closedGradient, InnerProductSpace.toDual_symm_apply,Real.inner_apply] ring change (∫ z, T 0 z ∂boundaryMeasure.prod boundaryMeasure)+ (∫ z, T 1 z ∂boundaryMeasure.prod boundaryMeasure) = _ at ha change _ = (1/2:ℝ)*(∫ z, T 0 z ∂boundaryMeasure.prod boundaryMeasure) at h0 change _ = (1/2:ℝ)*(∫ z, T 1 z ∂boundaryMeasure.prod boundaryMeasure) at h1 linarith include hu in theorem smooth_multiplier_kernel_nonneg {b : Plane → ℝ} (hb : ContDiff ℝ ∞ b) (hb0 : MemLp b ∞ (volume.restrict Ω)) (hb1 : MemLp (gradient b) ∞ (volume.restrict Ω)) : let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1 0 ≤ (1/2:ℝ)*∫ z : Boundary × Boundary, (boundaryInteraction z.1 z.2+fullBoundaryR c.densityBound_ne_top (c.density_bound hμ.le) z.1 z.2)* (b (c.F z.1)-b (c.F z.2))^2* inner ℝ (closedGradient Ω u (c.F z.1)) (closedGradient Ω u (c.F z.2)) ∂boundaryMeasure.prod boundaryMeasure := by let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1 change 0 ≤ (1/2:ℝ) * _ rw [← c.smooth_multiplier_kernel hΩ hu hμ hb hb0 hb1] exact c.multipliedGradient_nonneg hΩ hu hμ hb hb0 hb1 end StrictHotSpots.Conformal.ClosedDiskChart end end SmoothMultiplierKernel section LipschitzMultiplierKernel 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 {u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u) 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) include hΩ hu in lemma closedGradient_trace_continuous : Continuous (fun s : Boundary => closedGradient Ω u (c.F s)) := by apply (closedGradient_continuous hΩ.2.1 hΩ.2.2.2.2 hu.1).comp_continuous (c.lipschitz.continuous.comp continuous_subtype_val) intro s exact c.maps_closed (boundary_mem_closed s) lemma physical_boundary_lipschitz {b : Plane → ℝ} {B : ℝ≥0} (hb : LipschitzWith B b) : LipschitzWith ((B*c.Lip)*1) (fun s : Boundary => b (c.F s)) := (hb.comp c.lipschitz).comp (LipschitzWith.subtype_val _) include hΩ in lemma smooth_physical_boundary_bound {b : Plane → ℝ} {B : ℝ≥0} (hb : LipschitzWith B b) : ∃ M : ℝ, 0 ≤ M ∧ ∀ n (s : Boundary), ‖ambientSmooth (planeApproxBump n) b (c.F s)‖ ≤ M := by have hc : Bornology.IsBounded (closure Ω) := hΩ.2.2.1.closure obtain ⟨M,hM,hMb⟩ := ambientSmooth_uniform_bound hc hb exact ⟨M,hM,fun n s => hMb n (c.F s) (c.maps_closed (boundary_mem_closed s))⟩ include hΩ in lemma physical_multiplier_integral_tendsto {R : Boundary → Boundary → ℝ} (hR : Integrable (fun z : Boundary × Boundary => R z.1 z.2) (boundaryMeasure.prod boundaryMeasure)) {g : Boundary → Plane} (hg : Continuous g) {b : Plane → ℝ} {B : ℝ≥0} (hb : LipschitzWith B b) : Tendsto (fun n => ∫ z : Boundary × Boundary, (boundaryInteraction z.1 z.2+R z.1 z.2)* (ambientSmooth (planeApproxBump n) b (c.F z.1)-ambientSmooth (planeApproxBump n) b (c.F z.2))^2* inner ℝ (g z.1) (g z.2) ∂boundaryMeasure.prod boundaryMeasure) atTop (𝓝 (∫ z : Boundary × Boundary, (boundaryInteraction z.1 z.2+R z.1 z.2)*(b (c.F z.1)-b (c.F z.2))^2* inner ℝ (g z.1) (g z.2) ∂boundaryMeasure.prod boundaryMeasure)) := by let bn : ℕ → Boundary → ℝ := fun n s => ambientSmooth (planeApproxBump n) b (c.F s) have hn (n : ℕ) : LipschitzWith ((B*c.Lip)*1) (bn n) := c.physical_boundary_lipschitz (ambientSmooth_lipschitz (planeApproxBump n) hb) obtain ⟨M,hM,hMb⟩ := c.smooth_physical_boundary_bound hΩ hb have hm (n : ℕ) (s : Boundary) : ‖bn n s‖ ≤ M := hMb n s have ht (s : Boundary) : Tendsto (fun n => bn n s) atTop (𝓝 (b (c.F s))) := ambientSmooth_tendsto planeApproxBump_rOut_tendsto hb (c.F s) exact multiplier_kernel_tendsto (bn:=bn) (B:=(B*c.Lip)*1) hR hg hn hM hm ht theorem lipschitz_multiplier_kernel {b : Plane → ℝ} {B : ℝ≥0} (hb : LipschitzWith B b) : let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1 vectorForm (firstPositiveNeumannValue Ω) (lipschitzMultipliedGradient hΩ hμ hl hD hu hb) (lipschitzMultipliedGradient hΩ hμ hl hD hu hb) = (1/2:ℝ)*∫ z : Boundary × Boundary, (boundaryInteraction z.1 z.2+fullBoundaryR c.densityBound_ne_top (c.density_bound hμ.le) z.1 z.2)* (b (c.F z.1)-b (c.F z.2))^2* inner ℝ (closedGradient Ω u (c.F z.1)) (closedGradient Ω u (c.F z.2)) ∂boundaryMeasure.prod boundaryMeasure := by let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1 have hlhs := vectorForm_tendsto (c.multipliedGradient_tendsto hΩ hμ hl hD hu hb) (firstPositiveNeumannValue Ω) have hrhs := (c.physical_multiplier_integral_tendsto hΩ (fullBoundaryR_integrable c.densityBound_ne_top (c.density_bound hμ.le)) (c.closedGradient_trace_continuous hΩ hu) hb).const_mul (1/2:ℝ) apply tendsto_nhds_unique hlhs apply hrhs.congr' exact Eventually.of_forall fun n => (c.smooth_multiplier_kernel hΩ hu hμ (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))).symm lemma lipschitzMultipliedGradient_nonneg {b : Plane → ℝ} {B : ℝ≥0} (hb : LipschitzWith B b) : 0 ≤ vectorForm (firstPositiveNeumannValue Ω) (lipschitzMultipliedGradient hΩ hμ hl hD hu hb) (lipschitzMultipliedGradient hΩ hμ hl hD hu hb) := vectorForm_nonneg hΩ.2.1 hΩ.1 hΩ.2.2.2.1 hΩ.2.2.1 hμ hl hD _ (lipschitzMultipliedGradient_tangent hΩ hμ hl hD hu hb) end StrictHotSpots.Conformal.ClosedDiskChart end end LipschitzMultiplierKernel section HilbertMultiplierSum noncomputable section open Set MeasureTheory Metric Function open scoped InnerProductSpace Topology namespace StrictHotSpots.HilbertMultipliers lemma orthonormal_countable {H ι : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [TopologicalSpace.SeparableSpace H] {e : ι → H} (he : Orthonormal ℝ e) : Countable ι := by have hp : Pairwise (Disjoint on fun i => ball (e i) (1/2:ℝ)) := by intro i j hij apply disjoint_left.mpr intro x hxi hxj have hd : dist (e i) (e j) < 1 := by have h := dist_triangle (e i) x (e j) have hi : dist (e i) x < 1/2 := by simpa [dist_comm] using hxi have hj : dist x (e j) < 1/2 := hxj linarith have hn : ‖e i-e j‖^2 = 2 := by rw [norm_sub_sq_real,he.1 i,he.1 j,he.2 hij] norm_num rw [dist_eq_norm] at hd nlinarith [norm_nonneg (e i-e j)] exact hp.countable_of_isOpen_disjoint (fun _ => isOpen_ball) (fun i => ⟨e i,mem_ball_self (by norm_num)⟩) lemma coordinate_hasSum {H ι : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (e : HilbertBasis ι ℝ H) (x : H) : HasSum (fun i => (inner ℝ (e i) x)^2) (‖x‖^2) := by have h := e.hasSum_inner_mul_inner x x simpa only [real_inner_comm x,← pow_two,real_inner_self_eq_norm_sq] using h theorem coordinate_energy_sum {X H E ι : Type*} [MeasurableSpace X] [NormedAddCommGroup H] [InnerProductSpace ℝ H] [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [Countable ι] (μ : Measure X) [SFinite μ] (e : HilbertBasis ι ℝ H) (N : X → X → ℝ) (K : X → ℝ) (b : X → H) (g : X → E) (hN : ∀ s t, 0 ≤ N s t) (hidentity : ∀ᵐ z : X × X ∂μ.prod μ, N z.1 z.2 * ‖b z.1-b z.2‖^2 = K z.1*K z.2) (hKg : Integrable (fun s => K s • g s) μ) (hbound : Integrable (fun z : X × X => N z.1 z.2 * ‖b z.1-b z.2‖^2 * (‖g z.1‖*‖g z.2‖)) (μ.prod μ)) (hF : ∀ i, AEStronglyMeasurable (fun z : X × X => N z.1 z.2 * (inner ℝ (e i) (b z.1-b z.2))^2 * inner ℝ (g z.1) (g z.2)) (μ.prod μ)) : HasSum (fun i => (1/2:ℝ) * ∫ z : X × X, N z.1 z.2 * (inner ℝ (e i) (b z.1-b z.2))^2 * inner ℝ (g z.1) (g z.2) ∂μ.prod μ) ((1/2:ℝ)*‖∫ s, K s • g s ∂μ‖^2) := by let B : ι → X × X → ℝ := fun i z => N z.1 z.2 * (inner ℝ (e i) (b z.1-b z.2))^2 * (‖g z.1‖*‖g z.2‖) have hB (z : X × X) : HasSum (fun i => B i z) (N z.1 z.2 * ‖b z.1-b z.2‖^2 * (‖g z.1‖*‖g z.2‖)) := ((coordinate_hasSum e (b z.1-b z.2)).mul_left (N z.1 z.2)).mul_right _ have hsum := hasSum_integral_of_dominated_convergence (μ := μ.prod μ) B hF (fun i => Filter.Eventually.of_forall (fun z => by rw [Real.norm_eq_abs,abs_mul,abs_mul,abs_of_nonneg (hN _ _),abs_of_nonneg (sq_nonneg _)] exact mul_le_mul_of_nonneg_left (abs_real_inner_le_norm _ _) (mul_nonneg (hN _ _) (sq_nonneg _)))) (Filter.Eventually.of_forall (fun z => (hB z).summable)) (hbound.congr (Filter.Eventually.of_forall (fun z => (hB z).tsum_eq.symm))) (Filter.Eventually.of_forall (fun z => ((coordinate_hasSum e (b z.1-b z.2)).mul_left (N z.1 z.2)).mul_right (inner ℝ (g z.1) (g z.2)))) have hint : (∫ z : X × X, N z.1 z.2 * ‖b z.1-b z.2‖^2 * inner ℝ (g z.1) (g z.2) ∂μ.prod μ) = ‖∫ s, K s • g s ∂μ‖^2 := by calc _ = ∫ z : X × X, inner ℝ (K z.1 • g z.1) (K z.2 • g z.2) ∂μ.prod μ := by apply integral_congr_ae filter_upwards [hidentity] with z hz rw [hz,real_inner_smul_left,real_inner_smul_right] ring _ = ∫ s, ∫ t, inner ℝ (K s • g s) (K t • g t) ∂μ ∂μ := by exact integral_prod _ (Integrable.op_fst_snd (op := inner ℝ) (continuous_fst.inner continuous_snd) ⟨1, fun x y => by simpa only [one_mul] using norm_inner_le_norm x y⟩ hKg hKg) _ = ∫ s, inner ℝ (K s • g s) (∫ t, K t • g t ∂μ) ∂μ := by apply integral_congr_ae exact Filter.Eventually.of_forall (fun s => integral_inner hKg _) _ = ∫ s, inner ℝ (∫ t, K t • g t ∂μ) (K s • g s) ∂μ := by apply integral_congr_ae exact Filter.Eventually.of_forall (fun s => real_inner_comm _ _) _ = inner ℝ (∫ s, K s • g s ∂μ) (∫ s, K s • g s ∂μ) := integral_inner hKg _ _ = _ := real_inner_self_eq_norm_sq _ rw [hint] at hsum exact hsum.mul_left (1/2:ℝ) lemma nonnegative_coordinate_energy_zero {ι : Type*} {E : ι → ℝ} (hs : HasSum E 0) (hE : ∀ i, 0 ≤ E i) (i : ι) : E i = 0 := by apply le_antisymm _ (hE i) simpa only [hs.tsum_eq] using hs.summable.le_tsum i (fun j _ => hE j) end StrictHotSpots.HilbertMultipliers end end HilbertMultiplierSum section HilbertKernelCoordinates noncomputable section open Set MeasureTheory Filter Metric Function open scoped ENNReal NNReal Topology InnerProductSpace namespace StrictHotSpots.PlaneGreen open DiskH10 variable {ν : Measure Plane} [IsFiniteMeasure ν] {C : ℝ≥0∞} (hC : C ≠ ∞) (hν : ν ≤ C • volume.restrict disk) lemma fullBoundaryN_nonneg (hT : ‖weightedIntegralOperator hC hν‖ < 1) (s t : Boundary) : 0 ≤ fullBoundaryN hC hν s t := by rw [fullBoundaryN_eq] exact add_nonneg (by unfold boundaryInteraction DiskFormula.boundaryInteraction; positivity) (fullBoundaryR_nonneg hC hν hT s t) lemma boundary_pair_ne : ∀ᵐ z : Boundary × Boundary ∂boundaryMeasure.prod boundaryMeasure, z.1 ≠ z.2 := by apply (Measure.ae_prod_iff_ae_ae (isClosed_eq continuous_fst continuous_snd).isOpen_compl.measurableSet).mpr exact Eventually.of_forall fun s => (boundaryMeasure.ae_ne s).mono fun _ h => h.symm lemma full_embedding_identity {H : Type*} [NormedAddCommGroup H] (p : disk) (hT : ‖weightedIntegralOperator hC hν‖ < 1) {b : Boundary → H} (hb : ∀ s t, ‖b s-b t‖^2 = fullDistanceKernel hC hν p s t) : ∀ᵐ z : Boundary × Boundary ∂boundaryMeasure.prod boundaryMeasure, fullBoundaryN hC hν z.1 z.2 * ‖b z.1-b z.2‖^2 = fullBoundaryK hC hν p z.1 * fullBoundaryK hC hν p z.2 := by filter_upwards [boundary_pair_ne] with z hz rw [hb,fullDistanceKernel,ite_eq_right hz] exact mul_div_cancel₀ _ (fullBoundaryN_pos hC hν hT _ _ hz).ne' theorem full_embedding_coordinate_sum {H E ι : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [Countable ι] (e : HilbertBasis ι ℝ H) (p : disk) (hT : ‖weightedIntegralOperator hC hν‖ < 1) {b : Boundary → H} (hb : Continuous b) (hid : ∀ s t, ‖b s-b t‖^2 = fullDistanceKernel hC hν p s t) {g : Boundary → E} (hg : Continuous g) : HasSum (fun i => (1/2:ℝ)*∫ z : Boundary × Boundary, fullBoundaryN hC hν z.1 z.2 * (inner ℝ (e i) (b z.1-b z.2))^2 * inner ℝ (g z.1) (g z.2) ∂boundaryMeasure.prod boundaryMeasure) ((1/2:ℝ)*‖∫ s, fullBoundaryK hC hν p s • g s ∂boundaryMeasure‖^2) := by have hidentity := full_embedding_identity hC hν p hT hid obtain ⟨B,hB,hBK⟩ := fullBoundaryK_bounded hC hν p obtain ⟨M,hM,hMg⟩ := boundary_continuous_bound hg have hKg : Integrable (fun s => fullBoundaryK hC hν p s • g s) boundaryMeasure := by apply Integrable.of_bound ((fullBoundaryK_measurable hC hν p).aestronglyMeasurable.smul hg.aestronglyMeasurable) (B*M) exact Eventually.of_forall fun s => by change ‖fullBoundaryK hC hν p s • g s‖ ≤ B*M rw [norm_smul,Real.norm_of_nonneg (fullBoundaryK_pos hC hν hT p s).le] exact mul_le_mul (hBK s) (hMg s) (norm_nonneg _) hB have hKn : Integrable (fun s => fullBoundaryK hC hν p s * ‖g s‖) boundaryMeasure := by apply hKg.norm.congr exact Eventually.of_forall fun s => by change ‖fullBoundaryK hC hν p s • g s‖ = _ rw [norm_smul,Real.norm_of_nonneg (fullBoundaryK_pos hC hν hT p s).le] apply HilbertMultipliers.coordinate_energy_sum boundaryMeasure e (fullBoundaryN hC hν) (fullBoundaryK hC hν p) b g (fullBoundaryN_nonneg hC hν hT) hidentity hKg · apply (hKn.mul_prod hKn).congr filter_upwards [hidentity] with z hz rw [hz] ring · intro i exact ((fullBoundaryN_measurable hC hν).mul ((continuous_const.inner ((hb.comp continuous_fst).sub (hb.comp continuous_snd))).measurable.pow_const 2)).mul ((hg.comp continuous_fst).inner (hg.comp continuous_snd)).measurable |>.aestronglyMeasurable end StrictHotSpots.PlaneGreen end end HilbertKernelCoordinates section OriginalGradientVector noncomputable section open Set MeasureTheory Filter Metric Function open scoped ENNReal NNReal Topology InnerProductSpace namespace StrictHotSpots.PlaneGreen open DiskH10 variable {ν : Measure Plane} [IsFiniteMeasure ν] {C : ℝ≥0∞} (hC : C ≠ ∞) (hν : ν ≤ C • volume.restrict disk) lemma boundary_K_vector_integrable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] (p : disk) (hT : ‖weightedIntegralOperator hC hν‖ < 1) {g : Boundary → E} (hg : Continuous g) : Integrable (fun s => fullBoundaryK hC hν p s • g s) boundaryMeasure := by obtain ⟨B,hB,hBK⟩ := fullBoundaryK_bounded hC hν p obtain ⟨M,_,hMg⟩ := boundary_continuous_bound hg apply Integrable.of_bound ((fullBoundaryK_measurable hC hν p).aestronglyMeasurable.smul hg.aestronglyMeasurable) (B*M) exact Eventually.of_forall fun s => by change ‖fullBoundaryK hC hν p s • g s‖ ≤ B*M rw [norm_smul,Real.norm_of_nonneg (fullBoundaryK_pos hC hν hT p s).le] exact mul_le_mul (hBK s) (hMg s) (norm_nonneg _) hB end StrictHotSpots.PlaneGreen namespace StrictHotSpots.Conformal.ClosedDiskChart open PlaneGreen DiskH10 variable {Ω : Set Plane} (c : ClosedDiskChart Ω) lemma boundary_gradient_integral_zero (hΩ : AdmissibleDomain Ω) {u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u) (hne : ∃ x ∈ Ω, u x ≠ 0) (p : disk) (hz : gradient u (c.F p) = 0) : let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1 (∫ s, fullBoundaryK c.densityBound_ne_top (c.density_bound (firstPositiveNeumannValue_pos_of_eigenfunction hΩ hu hne).le) p s • closedGradient Ω u (c.F s) ∂boundaryMeasure) = 0 := by let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1 have hμ := firstPositiveNeumannValue_pos_of_eigenfunction hΩ hu hne have hi := boundary_K_vector_integrable c.densityBound_ne_top (c.density_bound hμ.le) p (c.operator_norm_lt_one hΩ.2.2.1 hΩ.1 hμ) (c.closedGradient_trace_continuous hΩ hu) apply integral_eq_zero_of_forall_integral_inner_eq_zero ℝ _ hi intro e calc _ = ∫ s, fullBoundaryK c.densityBound_ne_top (c.density_bound hμ.le) p s * inner ℝ (closedGradient Ω u (c.F s)) e ∂boundaryMeasure := by apply integral_congr_ae exact Eventually.of_forall fun s => by simp only [real_inner_smul_right,real_inner_comm] _ = inner ℝ (gradient u (c.F p)) e := (c.gradient_K_representation hΩ hu hne p e).symm _ = 0 := by rw [hz,inner_zero_left] end StrictHotSpots.Conformal.ClosedDiskChart end end OriginalGradientVector end OAI