| import OAI.Geometry.Perimeter.Regularity.SmulMemPhaseSupport |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal ContDiff |
| open scoped Topology ENNReal NNReal |
| open scoped ENNReal Topology ContDiff Pointwise |
| open scoped Topology ENNReal ContDiff |
| open scoped BoundedContinuousFunction |
| open scoped ENNReal Topology |
| open scoped ENNReal Topology ContDiff |
| open scoped Topology ENNReal |
| namespace RPISingularHarnack |
|
|
| def coneRadialCap {n : ℕ} (R : ℝ) (x : Euc n) : ℝ := |
| expNegInvGlue (R ^ 2 - ‖x‖ ^ 2) |
|
|
| theorem coneRadialCap_contDiff {n : ℕ} (R : ℝ) : |
| ContDiff ℝ ∞ (coneRadialCap (n := n) R) := |
| expNegInvGlue.contDiff.comp (contDiff_const.sub (contDiff_id.norm_sq ℝ)) |
|
|
| theorem coneRadialCap_tsupport {n : ℕ} {R : ℝ} (hR : 0 < R) : |
| tsupport (coneRadialCap (n := n) R) ⊆ Metric.closedBall (0 : Euc n) R := by |
| apply closure_minimal _ Metric.isClosed_closedBall |
| intro x hx |
| have hh : 0 < R ^ 2 - ‖x‖ ^ 2 := by |
| by_contra hh |
| exact hx (expNegInvGlue.zero_of_nonpos (le_of_not_gt hh)) |
| rw [Metric.mem_closedBall, dist_zero_right] |
| nlinarith [norm_nonneg x] |
|
|
| theorem gradient_coneRadialCap {n : ℕ} (R : ℝ) (x : Euc n) : |
| gradient (coneRadialCap R) x = |
| (-2 * ((R ^ 2 - ‖x‖ ^ 2)⁻¹ ^ 2 * expNegInvGlue (R ^ 2 - ‖x‖ ^ 2))) • x := by |
| have hg (s : ℝ) : HasDerivAt expNegInvGlue (s⁻¹ ^ 2 * expNegInvGlue s) s := by |
| simpa using expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul 1 s |
| have hq := ((hasFDerivAt_id (𝕜 := ℝ) x).norm_sq).const_sub (R ^ 2) |
| have hh := (hg (R ^ 2 - ‖x‖ ^ 2)).comp_hasFDerivAt x hq |
| change HasFDerivAt (coneRadialCap R) _ x at hh |
| apply (InnerProductSpace.toDual ℝ (Euc n)).injective |
| rw [toDual_gradient, hh.fderiv] |
| ext y |
| simp only [smul_apply, neg_apply, |
| ContinuousLinearMap.comp_apply, ContinuousLinearMap.id_apply, id_eq, |
| innerSL_apply_apply, map_smul, InnerProductSpace.toDual_apply_apply, |
| smul_eq_mul] |
| ring |
|
|
| theorem minimizing_cone_support_in_halfspace_plane {n : ℕ} (hn : 1 < n) |
| {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) (hcone : IsCone E) |
| (v : Euc n) (hside : ∀ x ∈ perimeterSupport E, 0 ≤ inner ℝ x v) : |
| ∀ x ∈ perimeterSupport E, inner ℝ x v = 0 := by |
| have hs : ∀ᵐ x ∂perimeterMeasure E, x ∈ perimeterSupport E := by |
| rw [← perimeterMeasure_support] |
| exact (perimeterMeasure E).support_mem_ae |
| have h0 : ∀ᵐ x ∂perimeterMeasure E, 0 ≤ inner ℝ x v := hs.mono fun x hx ↦ hside x hx |
| have hball (R : ℝ) (hR : 0 < R) : |
| ∀ᵐ x ∂(perimeterMeasure E).restrict (Metric.ball 0 R), inner ℝ x v = 0 := by |
| let φ : Euc n → ℝ := coneRadialCap R |
| let b : Euc n → ℝ := fun x ↦ (R ^ 2 - ‖x‖ ^ 2)⁻¹ ^ 2 * expNegInvGlue (R ^ 2 - ‖x‖ ^ 2) |
| let V : Euc n → Euc n := fun x ↦ φ x • v |
| have hφ : ContDiff ℝ ∞ φ := coneRadialCap_contDiff R |
| have htφ : HasCompactSupport φ := (isCompact_closedBall (0 : Euc n) R).of_isClosed_subset |
| (isClosed_tsupport _) (coneRadialCap_tsupport hR) |
| have hV : ContDiff ℝ ∞ V := hφ.smul contDiff_const |
| have htV : HasCompactSupport V := htφ.smul_right |
| have hVs : tsupport V ⊆ Metric.ball (0 : Euc n) (R + 1) := |
| (tsupport_smul_subset_left _ _).trans |
| ((coneRadialCap_tsupport hR).trans (Metric.closedBall_subset_ball (by linarith))) |
| have hdiv : ∀ᵐ x ∂perimeterMeasure E, |
| tangentialDivergence E V x = -2 * (b x * inner ℝ x v) := by |
| filter_upwards [minimizing_cone_radial_tangent_ae hn hmin hcone] with x hx |
| rw [tangentialDivergence_smul E (hφ.differentiable (by simp)) (differentiable_const v)] |
| have hc : tangentialDivergence E (fun _ : Euc n ↦ v) x = 0 := by |
| simp [tangentialDivergence, divergence] |
| rw [hc, mul_zero, zero_add, tangentialGradient, gradient_coneRadialCap] |
| simp only [real_inner_smul_left, hx, mul_zero, zero_smul, sub_zero] |
| dsimp [b] |
| ring |
| have hdf : Integrable (fun x ↦ b x * inner ℝ x v) |
| ((perimeterMeasure E).restrict (Metric.ball 0 (R + 1))) := by |
| have hh := (tangentialDivergence_integrable_ball_perimeter hmin 0 (by linarith : 0 < R + 1) hV).const_mul (-1/2 : ℝ) |
| apply hh.congr |
| filter_upwards [ae_restrict_of_ae hdiv] with x hx |
| rw [hx] |
| ring |
| have hnn : ∀ᵐ x ∂(perimeterMeasure E).restrict (Metric.ball 0 (R + 1)), |
| 0 ≤ b x * inner ℝ x v := by |
| filter_upwards [ae_restrict_of_ae h0] with x hx |
| exact mul_nonneg (mul_nonneg (sq_nonneg _) (expNegInvGlue.nonneg _)) hx |
| have hint : (∫ x in Metric.ball (0 : Euc n) (R + 1), b x * inner ℝ x v ∂perimeterMeasure E) = 0 := by |
| have hh := minimizing_tangentialDivergence_integral_eq_zero hn hmin 0 |
| (by linarith : 0 < R + 1) hV htV hVs |
| rw [integral_congr_ae (ae_restrict_of_ae hdiv), integral_const_mul] at hh |
| linarith |
| have he := (integral_eq_zero_iff_of_nonneg_ae hnn hdf).mp hint |
| have her : ∀ᵐ x ∂(perimeterMeasure E).restrict (Metric.ball 0 R), b x * inner ℝ x v = 0 := |
| ae_mono (Measure.restrict_mono (Metric.ball_subset_ball (by linarith)) le_rfl) he |
| filter_upwards [her, ae_restrict_mem measurableSet_ball] with x hx hxr |
| have hp : 0 < R ^ 2 - ‖x‖ ^ 2 := by |
| have hh : ‖x‖ < R := by simpa using hxr |
| nlinarith [norm_nonneg x] |
| have hb : 0 < b x := mul_pos (sq_pos_of_pos (inv_pos.mpr hp)) (expNegInvGlue.pos_of_pos hp) |
| exact (mul_eq_zero.mp hx).resolve_left hb.ne' |
| have hae : ∀ᵐ x ∂perimeterMeasure E, inner ℝ x v = 0 := by |
| let B (j : ℕ) := Metric.ball (0 : Euc n) (j + 1 : ℝ) |
| have hcover : (⋃ j, B j) = univ := by |
| apply eq_univ_of_forall |
| intro x |
| obtain ⟨j, hj⟩ := exists_nat_gt ‖x‖ |
| exact mem_iUnion.mpr ⟨j, by simpa [B] using (show ‖x‖ < (j : ℝ) + 1 by linarith)⟩ |
| have hh := (ae_restrict_iUnion_iff B (fun x ↦ inner ℝ x v = 0)).mpr |
| (fun j ↦ hball (j + 1) (by positivity)) |
| simpa only [hcover, Measure.restrict_univ] using hh |
| have hsup := Measure.support_subset_of_isClosed |
| (isClosed_eq (continuous_id.inner continuous_const) continuous_const) hae |
| rw [perimeterMeasure_support] at hsup |
| exact hsup |
| |
| end RPISingularHarnack |
| |
| end |
| |
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal ContDiff |
| open scoped Topology ENNReal NNReal |
| open scoped ENNReal Topology ContDiff Pointwise |
| open scoped Topology ENNReal ContDiff |
| open scoped BoundedContinuousFunction |
| open scoped ENNReal Topology |
| open scoped ENNReal Topology ContDiff |
| open scoped Topology ENNReal |
| namespace RPISingularHarnack |
| |
| theorem minimizing_cone_containing_halfspace_ae_eq {n : ℕ} (hn : 1 < n) |
| {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) (hcone : IsCone E) |
| (hvertex : (0 : Euc n) ∈ perimeterSupport E) |
| {v : Euc n} (hv : ‖v‖ = 1) (hinside : {x | inner ℝ x v < 0} ⊆ E) : |
| E =ᵐ[volume] {x | inner ℝ x v < 0} := by |
| let K := (volume.restrict E).support |
| have hK : IsClosed K := Measure.isClosed_support |
| have hP : IsOpen {x : Euc n | inner ℝ x v < 0} := |
| isOpen_lt (continuous_id.inner continuous_const) continuous_const |
| have hPK : {x : Euc n | inner ℝ x v < 0} ⊆ K := by |
| intro x hx |
| apply Measure.interior_inter_support |
| refine ⟨?_, ?_⟩ |
| · exact (interior_mono hinside) (hP.interior_eq.symm ▸ hx) |
| · rw [Measure.support_eq_univ]; trivial |
| have hPint : {x : Euc n | inner ℝ x v < 0} ⊆ interior K := by |
| exact hP.interior_eq ▸ interior_mono hPK |
| have hfK : frontier K = perimeterSupport E := |
| minimizing_frontier_phaseSupport |
| (Real.HolderConjugate.conjExponent (by exact_mod_cast hn : 1 < (n : ℝ))) hmin |
| have hside : ∀ x ∈ perimeterSupport E, 0 ≤ inner ℝ x v := by |
| intro x hx |
| by_contra hh |
| have hi := hPint (lt_of_not_ge hh) |
| rw [← hfK, frontier] at hx |
| exact hx.2 hi |
| have hplane := minimizing_cone_support_in_halfspace_plane hn hmin hcone v hside |
| have hcover : {x : Euc n | 0 < inner ℝ x v} ⊆ interior K ∪ Kᶜ := by |
| intro x hx |
| by_cases hk : x ∈ K |
| · left |
| by_contra hi |
| have hs : x ∈ perimeterSupport E := by |
| rw [← hfK, frontier, hK.closure_eq] |
| exact ⟨hk, hi⟩ |
| exact (ne_of_gt hx) (hplane x hs) |
| · exact Or.inr hk |
| have hc : IsPreconnected {x : Euc n | 0 < inner ℝ x v} := by |
| have hh := (convex_halfSpace_gt (innerSL ℝ v).toLinearMap.isLinear (0 : ℝ)).isPreconnected |
| change IsPreconnected {x : Euc n | 0 < inner ℝ v x} at hh |
| simpa only [real_inner_comm v] using hh |
| have hor := hc.subset_or_subset isOpen_interior hK.isOpen_compl |
| (disjoint_compl_right.mono_left interior_subset) hcover |
| have hEK : E =ᵐ[volume] K := minimizing_ae_eq_phaseSupport |
| (Real.HolderConjugate.conjExponent (by exact_mod_cast hn : 1 < (n : ℝ))) hmin |
| rcases hor with hall | hnone |
| · have hae : ∀ᵐ x ∂(volume : Measure (Euc n)), x ∈ K := by |
| filter_upwards [ae_inner_ne_value hv 0] with x hx |
| rcases lt_or_gt_of_ne hx with hn | hp |
| · exact hPK hn |
| · exact interior_subset (hall hp) |
| have hKU : K = univ := by |
| have hh := Measure.support_subset_of_isClosed hK hae |
| rw [Measure.support_eq_univ] at hh |
| exact eq_univ_iff_forall.mpr fun x ↦ hh trivial |
| rw [← hfK, hKU, frontier_univ] at hvertex |
| exact False.elim hvertex |
| · apply hEK.trans |
| apply Filter.eventuallyEqSet_iff.mpr |
| filter_upwards [ae_inner_ne_value hv 0] with x hx |
| rcases lt_or_gt_of_ne hx with hn | hp |
| · exact iff_of_true (hPK hn) hn |
| · exact iff_of_false (hnone hp) (not_lt.mpr hp.le) |
| |
| theorem minimizing_cone_containing_halfspace_support_eq {n : ℕ} (hn : 1 < n) |
| {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) (hcone : IsCone E) |
| (hvertex : (0 : Euc n) ∈ perimeterSupport E) |
| {v : Euc n} (hv : ‖v‖ = 1) (hinside : {x | inner ℝ x v < 0} ⊆ E) : |
| perimeterSupport E = perimeterSupport {x | inner ℝ x v < 0} := |
| perimeterSupport_eq_of_ae |
| (minimizing_cone_containing_halfspace_ae_eq hn hmin hcone hvertex hv hinside) |
| |
| end RPISingularHarnack |
| |
| end |
| |
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal ContDiff |
| open scoped Topology ENNReal NNReal |
| open scoped ENNReal Topology ContDiff Pointwise |
| open scoped Topology ENNReal ContDiff |
| open scoped BoundedContinuousFunction |
| open scoped ENNReal Topology |
| open scoped ENNReal Topology ContDiff |
| open scoped Topology ENNReal |
| namespace RPISingularHarnack |
| |
| theorem measure_eq_of_Iio_finite {μ ν : Measure ℝ} |
| (hfin : ∀ a : ℝ, μ (Iio a) ≠ ⊤) (heq : ∀ a : ℝ, μ (Iio a) = ν (Iio a)) : |
| μ = ν := by |
| apply Measure.ext_of_Ico' μ ν |
| · intro a b _ |
| exact ne_top_of_le_ne_top (hfin b) (measure_mono Ico_subset_Iio_self) |
| · intro a b hab |
| rw [← Iio_sdiff_Iio, measure_sdiff (Iio_subset_Iio hab.le) |
| measurableSet_Iio.nullMeasurableSet (hfin a), |
| measure_sdiff (Iio_subset_Iio hab.le) measurableSet_Iio.nullMeasurableSet |
| (by rw [← heq a]; exact hfin a), heq a, heq b] |
|
|
| noncomputable def radialScaledMeasure {n : ℕ} (μ : Measure (Euc n)) (k : ℕ) (t : ℝ) : Measure ℝ := |
| ENNReal.ofReal (t^k) • μ.map (fun x ↦ t * ‖x‖) |
|
|
| theorem radialScaledMeasure_Iio {n k : ℕ} (μ : Measure (Euc n)) |
| {a t : ℝ} (ht : 0 < t) : |
| radialScaledMeasure μ k t (Iio a) = |
| ENNReal.ofReal (t^k) * μ (Metric.ball 0 (a/t)) := by |
| have hm : Measurable (fun x : Euc n ↦ t * ‖x‖) := by fun_prop |
| rw [radialScaledMeasure, Measure.smul_apply, smul_eq_mul, |
| Measure.map_apply hm measurableSet_Iio] |
| congr 2 |
| ext x |
| simp only [mem_preimage, mem_Iio, Metric.mem_ball, dist_zero_right] |
| rw [lt_div_iff₀ ht, mul_comm] |
|
|
| theorem radialScaledMeasure_eq_of_constant_density {n k : ℕ} |
| (μ : Measure (Euc n)) (hfin : ∀ r : ℝ, 0 < r → μ (Metric.ball 0 r) ≠ ⊤) |
| {a : ℝ} (ha : 0 ≤ a) (hdensity : ∀ r : ℝ, 0 < r → μ.real (Metric.ball 0 r) = a * r^k) |
| {s t : ℝ} (hs : 0 < s) (ht : 0 < t) : |
| radialScaledMeasure μ k s = radialScaledMeasure μ k t := by |
| have hval (t : ℝ) (ht : 0 < t) (r : ℝ) : |
| radialScaledMeasure μ k t (Iio r) = ENNReal.ofReal (a * (max r 0)^k) * |
| (if 0 < r then 1 else 0) := by |
| rw [radialScaledMeasure_Iio μ ht] |
| by_cases hr : 0 < r |
| · rw [ite_eq_left hr, mul_one, max_eq_left hr.le] |
| have hrt : 0 < r/t := div_pos hr ht |
| have hp := hdensity (r/t) hrt |
| have he : μ (Metric.ball 0 (r/t)) = ENNReal.ofReal (a * (r/t)^k) := by |
| rw [← hp, Measure.real, ENNReal.ofReal_toReal (hfin _ hrt)] |
| rw [he, ← ENNReal.ofReal_mul' (mul_nonneg ha (pow_nonneg hrt.le _))] |
| congr 1 |
| rw [div_pow] |
| field_simp |
| · rw [ite_eq_right hr, mul_zero, Metric.ball_eq_empty.mpr (div_nonpos_of_nonpos_of_nonneg |
| (le_of_not_gt hr) ht.le), measure_empty, mul_zero] |
| apply measure_eq_of_Iio_finite |
| · intro r |
| rw [hval s hs r] |
| split_ifs <;> finiteness |
| · intro r |
| rw [hval s hs r, hval t ht r] |
| |
| theorem radialDensity_integral_eq_pushforward {n k : ℕ} (μ : Measure (Euc n)) |
| {ψ : ℝ → ℝ} (hψ : Continuous ψ) {t : ℝ} (ht : 0 < t) : |
| (∫ x, radialDensityIntegrand 0 k ψ t x ∂μ) = |
| ∫ s, ψ (s^2) ∂radialScaledMeasure μ k t := by |
| have hm : Measurable (fun x : Euc n ↦ t * ‖x‖) := by fun_prop |
| have hc : Continuous (fun s : ℝ ↦ ψ (s^2)) := by fun_prop |
| rw [radialScaledMeasure, integral_smul_measure, |
| integral_map_of_stronglyMeasurable hm hc.stronglyMeasurable, |
| ENNReal.toReal_ofReal (pow_nonneg ht.le _)] |
| simp only [radialDensityIntegrand, sub_zero, smul_eq_mul, mul_pow, integral_const_mul] |
| |
| theorem radialDensity_integral_ball_eq {n k : ℕ} (μ : Measure (Euc n)) |
| {ψ : ℝ → ℝ} (hψz : ∀ s, 1 ≤ s → ψ s = 0) |
| {M t : ℝ} (hM : 0 < M) (ht : 1/M < t) : |
| (∫ x in Metric.ball (0 : Euc n) M, radialDensityIntegrand 0 k ψ t x ∂μ) = |
| ∫ x, radialDensityIntegrand 0 k ψ t x ∂μ := by |
| apply setIntegral_eq_integral_of_forall_compl_eq_zero |
| intro x hx |
| have ht0 : 0 < t := (one_div_pos.mpr hM).trans ht |
| have htM : 1 < t*M := (div_lt_iff₀ hM).mp ht |
| have hxM : M ≤ ‖x‖ := by simpa only [Metric.mem_ball, dist_zero_right, not_lt] using hx |
| have h1 : 1 ≤ t*‖x‖ := by nlinarith |
| have h2 := pow_le_pow_left₀ (by norm_num : (0:ℝ) ≤ 1) h1 2 |
| simp only [one_pow, mul_pow] at h2 |
| simp only [radialDensityIntegrand, sub_zero, hψz _ h2, mul_zero] |
| |
| theorem minimizing_radialDensityDerivative_eq_zero_of_constant_density |
| {n : ℕ} {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) |
| {a : ℝ} (ha : 0 ≤ a) |
| (hdensity : ∀ r : ℝ, 0 < r → (perimeterMeasure E).real (Metric.ball 0 r) = a*r^(n-1)) |
| {M : ℝ} (hM : 0 < M) {ψ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ) |
| (hψz : ∀ s, 1 ≤ s → ψ s = 0) {t : ℝ} (ht : 1/M < t) : |
| (∫ x in Metric.ball 0 M, radialDensityDerivative 0 (n-1) ψ t x ∂perimeterMeasure E) = 0 := by |
| let μ := (perimeterMeasure E).restrict (Metric.ball 0 M) |
| have : IsFiniteMeasure μ := ⟨by |
| rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball] |
| exact hmin.2.1 0 M hM⟩ |
| have hμ : ∀ᵐ x ∂μ, x ∈ Metric.closedBall 0 M := |
| (ae_restrict_mem measurableSet_ball).mono fun _ hx ↦ Metric.ball_subset_closedBall hx |
| have hd := (radialDensity_integral_hasDerivAt 0 M (n-1) hψ μ hμ t).2 |
| have ht0 : 0 < t := (one_div_pos.mpr hM).trans ht |
| have hfin (r : ℝ) (hr : 0 < r) : perimeterMeasure E (Metric.ball 0 r) ≠ ⊤ := by |
| rw [perimeterMeasure_open E Metric.isOpen_ball] |
| exact (hmin.2.1 0 r hr).ne |
| have he (s : ℝ) (hs : 1/M < s) : |
| (∫ x, radialDensityIntegrand 0 (n-1) ψ s x ∂μ) = |
| ∫ x, radialDensityIntegrand 0 (n-1) ψ t x ∂μ := by |
| have hs0 := (one_div_pos.mpr hM).trans hs |
| change (∫ x in Metric.ball 0 M, _ ∂perimeterMeasure E) = |
| ∫ x in Metric.ball 0 M, _ ∂perimeterMeasure E |
| rw [radialDensity_integral_ball_eq _ hψz hM hs, |
| radialDensity_integral_ball_eq _ hψz hM ht, |
| radialDensity_integral_eq_pushforward _ hψ.continuous hs0, |
| radialDensity_integral_eq_pushforward _ hψ.continuous ht0, |
| radialScaledMeasure_eq_of_constant_density _ hfin ha hdensity hs0 ht0] |
| have hdz : HasDerivAt (fun s ↦ ∫ x, radialDensityIntegrand 0 (n-1) ψ s x ∂μ) 0 t := by |
| apply (hasDerivAt_const t (∫ x, radialDensityIntegrand 0 (n-1) ψ t x ∂μ)).congr_of_eventuallyEq |
| filter_upwards [lt_mem_nhds ht] with s hs |
| exact he s hs |
| exact hd.unique hdz |
| |
| theorem minimizing_radial_tangent_of_constant_density {n : ℕ} (hn : 1 < n) |
| {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) |
| {a : ℝ} (ha : 0 ≤ a) |
| (hdensity : ∀ r : ℝ, 0 < r → (perimeterMeasure E).real (Metric.ball 0 r) = a*r^(n-1)) : |
| ∀ᵐ x ∂perimeterMeasure E, inner ℝ (approximateNormal E x) x = 0 := by |
| let ψ : ℝ → ℝ := fun s ↦ expNegInvGlue (1-s) |
| have hψ : ContDiff ℝ ∞ ψ := expNegInvGlue.contDiff.comp (contDiff_const.sub contDiff_id) |
| have hψz (s : ℝ) (hs : 1 ≤ s) : ψ s = 0 := |
| expNegInvGlue.zero_of_nonpos (by linarith) |
| have hψd (s : ℝ) : deriv ψ s = -((1-s)⁻¹^2 * expNegInvGlue (1-s)) := by |
| have hg : HasDerivAt expNegInvGlue ((1-s)⁻¹^2 * expNegInvGlue (1-s)) (1-s) := by |
| simpa using expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul 1 (1-s) |
| simpa [ψ, Function.comp_def] using (hg.comp s ((hasDerivAt_id s).const_sub 1)).deriv |
| have hlocal (R : ℝ) (hR : 0 < R) : |
| ∀ᵐ x ∂(perimeterMeasure E).restrict (Metric.ball 0 R), |
| inner ℝ (approximateNormal E x) x = 0 := by |
| let M := 2*R |
| let t := 1/R |
| have hM : 0 < M := by dsimp [M]; positivity |
| have ht0 : 0 < t := one_div_pos.mpr hR |
| have ht : 1/M < t := one_div_lt_one_div_of_lt hR (by dsimp [M]; linarith) |
| have htM : 1/t < M := by dsimp [t, M]; rw [one_div_one_div]; linarith |
| let μ := (perimeterMeasure E).restrict (Metric.ball 0 M) |
| have : IsFiniteMeasure μ := ⟨by |
| rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball] |
| exact hmin.2.1 0 M hM⟩ |
| have hμ : ∀ᵐ x ∂μ, x ∈ Metric.closedBall 0 M := |
| (ae_restrict_mem measurableSet_ball).mono fun _ hx ↦ Metric.ball_subset_closedBall hx |
| let V : Euc n → Euc n := fun x ↦ ψ (t^2*‖x‖^2) • x |
| have hV : ContDiff ℝ ∞ V := |
| (hψ.comp (contDiff_const.mul (contDiff_id.norm_sq (𝕜 := ℝ)))).smul contDiff_id |
| have hp : tsupport (fun x : Euc n ↦ ψ (t^2*‖x‖^2)) ⊆ Metric.closedBall 0 (1/t) := by |
| simpa only [sub_zero] using (radialProfile_tsupport (n := n) 0 hψz ht0) |
| have hs : tsupport V ⊆ Metric.closedBall 0 (1/t) := |
| (tsupport_smul_subset_left (fun x : Euc n ↦ ψ (t^2*‖x‖^2)) |
| (id : Euc n → Euc n)).trans hp |
| have hc : HasCompactSupport V := |
| (isCompact_closedBall (0 : Euc n) (1/t)).of_isClosed_subset (isClosed_tsupport _) hs |
| have hdiv := minimizing_tangentialDivergence_integral_eq_zero hn hmin 0 hM hV hc |
| (hs.trans (Metric.closedBall_subset_ball htM)) |
| have hD0 := minimizing_radialDensityDerivative_eq_zero_of_constant_density |
| hmin ha hdensity hM hψ hψz ht |
| have hDi := (radialDensity_integral_hasDerivAt 0 M (n-1) hψ μ hμ t).1 |
| have hVi := (tangentialDivergence_integrable_ball_perimeter hmin 0 hM hV).const_mul |
| (t^((n-1)-1)) |
| let q : Euc n → ℝ := fun x ↦ t^((n-1)-1)*tangentialDivergence E V x - |
| radialDensityDerivative 0 (n-1) ψ t x |
| have hqi : Integrable q μ := hVi.sub hDi |
| have hqformula : ∀ᵐ x ∂μ, q x = |
| (2*t^((n-1)-1)*t^2*(inner ℝ (approximateNormal E x) x)^2) * |
| ((1-t^2*‖x‖^2)⁻¹^2 * expNegInvGlue (1-t^2*‖x‖^2)) := by |
| filter_upwards [ae_restrict_of_ae (minimizing_halfspaceBlowup_ae hn hmin)] with x hx |
| have hpow : t^(n-1) = t^((n-1)-1)*t := by |
| rw [← pow_succ, Nat.sub_add_cancel (by omega : 1 ≤ n-1)] |
| have hvf := tangentialDivergence_radialProfile (by omega : 0<n) E 0 hψ t x hx.1 |
| simp only [sub_zero] at hvf |
| dsimp [q, radialDensityDerivative] |
| change t^((n-1)-1)*tangentialDivergence E V x - _ = _ |
| rw [show tangentialDivergence E V x = _ from hvf, hpow] |
| simp only [sub_zero, hψd] |
| ring |
| have hqnonneg : ∀ᵐ x ∂μ, 0 ≤ q x := hqformula.mono fun x hx ↦ by |
| rw [hx] |
| exact mul_nonneg (by positivity) |
| (mul_nonneg (sq_nonneg _) (expNegInvGlue.nonneg _)) |
| have hqint : (∫ x, q x ∂μ) = 0 := by |
| rw [integral_sub hVi hDi, integral_const_mul] |
| change _ * (∫ x in Metric.ball 0 M, tangentialDivergence E V x ∂perimeterMeasure E) - _ = 0 |
| rw [hdiv, hD0] |
| ring |
| have hqzero := (integral_eq_zero_iff_of_nonneg_ae hqnonneg hqi).mp hqint |
| have hqz := (ae_restrict_iff' measurableSet_ball).mp hqzero |
| have hqf := (ae_restrict_iff' measurableSet_ball).mp hqformula |
| apply (ae_restrict_iff' measurableSet_ball).mpr |
| filter_upwards [hqz, hqf] with x hxz hxf |
| intro hx |
| have hxM : x ∈ Metric.ball 0 M := Metric.ball_subset_ball (by dsimp [M]; linarith) hx |
| have htx : t*‖x‖ < 1 := by |
| have hxR : ‖x‖ < R := by simpa only [Metric.mem_ball, dist_zero_right] using hx |
| simpa only [t, one_div, div_eq_mul_inv, mul_comm, mul_one, one_mul] using (div_lt_one hR).mpr hxR |
| have htx0 : 0 ≤ t*‖x‖ := mul_nonneg ht0.le (norm_nonneg x) |
| have hspos : 0 < 1-t^2*‖x‖^2 := by |
| nlinarith [mul_pos (sub_pos.mpr htx) (show 0 < 1+t*‖x‖ by linarith)] |
| have hbpos : 0 < (1-t^2*‖x‖^2)⁻¹^2 * expNegInvGlue (1-t^2*‖x‖^2) := |
| mul_pos (sq_pos_of_pos (inv_pos.mpr hspos)) (expNegInvGlue.pos_of_pos hspos) |
| have he := hxf hxM |
| rw [hxz hxM] at he |
| have hz : (inner ℝ (approximateNormal E x) x)^2 = 0 := by |
| have hp : 0 < 2*t^((n-1)-1)*t^2 := by positivity |
| exact (mul_eq_zero.mp ((mul_eq_zero.mp he.symm).resolve_right hbpos.ne')).resolve_left hp.ne' |
| exact sq_eq_zero_iff.mp hz |
| let U : ℕ → Set (Euc n) := fun j ↦ Metric.ball 0 (j+1 : ℝ) |
| have hcover : (⋃ j, U j) = univ := by |
| apply eq_univ_of_forall |
| intro x |
| obtain ⟨j, hj⟩ := exists_nat_gt ‖x‖ |
| exact mem_iUnion.mpr ⟨j, by simpa [U, Metric.mem_ball, dist_zero_right] using |
| (show ‖x‖ < (j : ℝ)+1 by linarith)⟩ |
| have hall := (ae_restrict_iUnion_iff U |
| (fun x ↦ inner ℝ (approximateNormal E x) x = 0)).mpr |
| (fun j ↦ hlocal (j+1) (by positivity)) |
| simpa only [hcover, Measure.restrict_univ] using hall |
|
|
| end RPISingularHarnack |
|
|
| end |
|
|
| end OAI |
|
|
