| import OAI.Geometry.Perimeter.Regularity.RegularSupersolutionC1Test | |
| 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 | |
| namespace RPI.Boundary | |
| section TruncationRegularLimit | |
| variable {n : ℕ} {S Z : Set (Space n)} | |
| variable (hS : IsClosed S) (hZ : IsClosed Z) | |
| variable (hfinite : ∀ R : ℝ, 0 < R → (surfaceMeasure S) (Metric.ball 0 R) < ⊤) | |
| variable (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ 1 f → | |
| AEStronglyMeasurable (surfaceGradient S f) (surfaceMeasure S)) | |
| variable {W : Space n → ℝ} {D : Space n → Space n} | |
| variable (htrace : ∀ x ∈ S \ Z, HasC1TraceJetAt S W D x) | |
| variable (hsup : ∀ φ : Space n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ → | |
| Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) → | |
| 0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient S φ x) ∂surfaceMeasure S) | |
| include hS hZ hfinite hgrad htrace hsup | |
| theorem c1_min_regular_test {φ : Space n → ℝ} (hφ : ContDiff ℝ 1 φ) | |
| (hcφ : HasCompactSupport φ) (hφZ : Disjoint (tsupport φ) Z) (hpφ : ∀ x, 0 ≤ φ x) : | |
| 0 ≤ ∫ x, inner ℝ (truncatedGradient W D x) (surfaceGradient S φ x) ∂surfaceMeasure S := by | |
| let b : Space n → ℝ := fun x ↦ inner ℝ (D x) (surfaceGradient S φ x) | |
| have hb : Integrable b (surfaceMeasure S) := | |
| c1_integrable_regular_test_pair hS hZ hfinite hgrad htrace hφ hcφ hφZ | |
| let q : ℕ → Space n → ℝ := fun j x ↦ | |
| inner ℝ (truncationSlope (truncationParameter j) (W x) • D x) (surfaceGradient S φ x) | |
| have hq : ∀ j, AEStronglyMeasurable (q j) (surfaceMeasure S) := by | |
| intro j | |
| have htt : ∀ x ∈ S \ Z, HasC1TraceJetAt S (smoothTruncation (truncationParameter j) ∘ W) | |
| (fun y ↦ truncationSlope (truncationParameter j) (W y) • D y) x := by | |
| intro x hx | |
| simpa only [smoothTruncation_deriv] using | |
| (htrace x hx).comp hx.1 ((smoothTruncation_contDiff (truncationParameter j)).of_le (by norm_num)) | |
| exact (c1_integrable_regular_test_pair hS hZ hfinite hgrad htt hφ hcφ hφZ).aestronglyMeasurable | |
| have hlim : Tendsto (fun j ↦ ∫ x, q j x ∂surfaceMeasure S) atTop | |
| (𝓝 (∫ x, inner ℝ (truncatedGradient W D x) (surfaceGradient S φ x) ∂surfaceMeasure S)) := by | |
| apply tendsto_integral_of_dominated_convergence (fun x ↦ ‖b x‖) hq hb.norm | |
| · intro j | |
| apply ae_of_all | |
| intro x | |
| simp only [q, real_inner_smul_left, norm_mul] | |
| exact mul_le_of_le_one_left (norm_nonneg _) (truncationSlope_norm_le _ _) | |
| · apply ae_of_all | |
| intro x | |
| have ht := (truncationSlope_tendsto (W x)).mul_const (b x) | |
| by_cases hx : W x < 1 | |
| · simpa only [q, b, truncatedGradient, ite_eq_left hx, one_mul, real_inner_smul_left] using ht | |
| · simpa only [q, b, truncatedGradient, ite_eq_right hx, zero_mul, inner_zero_left, | |
| real_inner_smul_left] using ht | |
| apply ge_of_tendsto hlim | |
| apply Eventually.of_forall | |
| intro j | |
| exact c1_smoothTruncation_regular_test hS hZ hfinite hgrad htrace hsup | |
| (truncationParameter_pos j) hφ hcφ hφZ hpφ | |
| theorem c1_min_regular_energy (hW0 : ∀ x ∈ S \ Z, 0 ≤ W x) | |
| {η : Space n → ℝ} (hη : ContDiff ℝ 1 η) (hcη : HasCompactSupport η) | |
| (hηZ : Disjoint (tsupport η) Z) : | |
| MemLp (fun x ↦ η x • truncatedGradient W D x) 2 (surfaceMeasure S) ∧ | |
| vectorEnergy (surfaceMeasure S) (fun x ↦ η x • truncatedGradient W D x) ≤ | |
| 4 * vectorEnergy (surfaceMeasure S) (surfaceGradient S η) := by | |
| let A : ℕ → Space n → Space n := fun j x ↦ η x • (truncationSlope (truncationParameter j) (W x) • D x) | |
| let H : Space n → Space n := fun x ↦ η x • truncatedGradient W D x | |
| have hA : ∀ j, MemLp (A j) 2 (surfaceMeasure S) := by | |
| intro j | |
| have htt : ∀ x ∈ S \ Z, HasC1TraceJetAt S (smoothTruncation (truncationParameter j) ∘ W) | |
| (fun y ↦ truncationSlope (truncationParameter j) (W y) • D y) x := by | |
| intro x hx | |
| simpa only [smoothTruncation_deriv] using | |
| (htrace x hx).comp hx.1 ((smoothTruncation_contDiff (truncationParameter j)).of_le (by norm_num)) | |
| exact c1_regular_gradient_memLp_test hS hZ hfinite hgrad htt hη hcη hηZ | |
| have hB : MemLp (surfaceGradient S η) 2 (surfaceMeasure S) := | |
| c1_surfaceGradient_memLp_of_compact hfinite hgrad hη hcη | |
| have he : ∀ j, vectorEnergy (surfaceMeasure S) (A j) ≤ | |
| 4 * vectorEnergy (surfaceMeasure S) (surfaceGradient S η) := by | |
| intro j | |
| have h := ENNReal.ofReal_le_ofReal (c1_smoothTruncation_regular_energy hS hZ hfinite hgrad | |
| htrace hW0 hsup (truncationParameter_pos j) hη hcη hηZ) | |
| rw [ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 4)] at h | |
| simpa only [A, vectorEnergy, ofReal_integral_eq_lintegral_ofReal | |
| ((memLp_two_iff_integrable_sq_norm (hA j).aestronglyMeasurable).mp (hA j)) | |
| (ae_of_all _ (fun x ↦ sq_nonneg ‖A j x‖)), | |
| ofReal_integral_eq_lintegral_ofReal | |
| ((memLp_two_iff_integrable_sq_norm hB.aestronglyMeasurable).mp hB) | |
| (ae_of_all _ (fun x ↦ sq_nonneg ‖surfaceGradient S η x‖)), ENNReal.ofReal_ofNat] using h | |
| have hlim : ∀ᵐ x ∂surfaceMeasure S, Tendsto (fun j ↦ A j x) atTop (𝓝 (H x)) := by | |
| apply ae_of_all | |
| intro x | |
| have ht := (tendsto_const_nhds (x := η x)).smul | |
| ((truncationSlope_tendsto (W x)).smul_const (D x)) | |
| by_cases hx : W x < 1 | |
| · simpa only [A, H, truncatedGradient, ite_eq_left hx, one_smul] using ht | |
| · simpa only [A, H, truncatedGradient, ite_eq_right hx, zero_smul] using ht | |
| have hHm : AEStronglyMeasurable H (surfaceMeasure S) := | |
| aestronglyMeasurable_of_tendsto_ae atTop (fun j ↦ (hA j).aestronglyMeasurable) hlim | |
| have hHE := vectorEnergy_le_of_ae_tendsto (fun j ↦ (hA j).aestronglyMeasurable) hlim | |
| (Eventually.of_forall he) tendsto_const_nhds | |
| refine ⟨(memLp_two_iff_vectorEnergy hHm).mpr (hHE.trans_lt ?_), hHE⟩ | |
| exact ENNReal.mul_lt_top (by norm_num) ((memLp_two_iff_vectorEnergy hB.aestronglyMeasurable).mp hB) | |
| end TruncationRegularLimit | |
| end RPI.Boundary | |
| 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 | |
| namespace RPI.Boundary | |
| theorem c1_min_local_energy {n : ℕ} (hn : 8 ≤ n) | |
| {S Z : Set (Space n)} (hS : IsClosed S) (hZ : IsClosed Z) | |
| (hZd : dimH Z ≤ (n - 8 : ℕ)) | |
| (hfinite : ∀ R : ℝ, 0 < R → (surfaceMeasure S) (Metric.ball 0 R) < ⊤) | |
| (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ 1 f → | |
| AEStronglyMeasurable (surfaceGradient S f) (surfaceMeasure S)) | |
| (hgrowth : ∀ R : ℝ, 0 < R → ∃ A r₀ : ℝ, 0 < A ∧ 0 < r₀ ∧ | |
| ∀ c ∈ Z ∩ Metric.closedBall 0 R, ∀ r : ℝ, 0 < r → r < r₀ → | |
| (surfaceMeasure S).restrict (Metric.ball 0 R) (Metric.ball c r) ≤ | |
| ENNReal.ofReal (A * r ^ (n - 1))) | |
| {W : Space n → ℝ} {D : Space n → Space n} | |
| (htrace : ∀ x ∈ S \ Z, HasC1TraceJetAt S W D x) | |
| (hW0 : ∀ x ∈ S \ Z, 0 ≤ W x) | |
| (hsup : ∀ φ : Space n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ → | |
| Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) → | |
| 0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient S φ x) ∂surfaceMeasure S) | |
| {R : ℝ} (hR : 0 < R) : | |
| MemLp (truncatedGradient W D) 2 ((surfaceMeasure S).restrict (Metric.ball 0 R)) := by | |
| let b : ContDiffBump (0 : Space n) := ⟨R + 1, R + 2, by linarith, by linarith⟩ | |
| let μ := (surfaceMeasure S).restrict (Metric.ball 0 (R + 3)) | |
| have hR3 : 0 < R + 3 := by linarith | |
| have : IsFiniteMeasure μ := ⟨by simpa only [μ, Measure.restrict_apply_univ] using hfinite _ hR3⟩ | |
| let K := Z ∩ tsupport b | |
| have hKc : IsCompact K := b.hasCompactSupport.of_isClosed_subset | |
| (hZ.inter (isClosed_tsupport b)) inter_subset_right | |
| have hKd : dimH K ≤ (n - 8 : ℕ) := (dimH_mono inter_subset_left).trans hZd | |
| have hbs : tsupport b ⊆ Metric.ball (0 : Space n) (R + 3) := by | |
| rw [b.tsupport_eq] | |
| exact Metric.closedBall_subset_ball (by dsimp [b]; linarith) | |
| obtain ⟨A, r₀, hA, hr₀, hgr⟩ := hgrowth _ hR3 | |
| obtain ⟨χ, hχ, hχlim, hχE⟩ := exists_memLp_capacity_sequence hn S | |
| (Metric.ball 0 (R + 3)) K hKc hKd | |
| (fun f hf ↦ (hgrad f (hf.of_le (by norm_num))).mono_measure Measure.restrict_le_self) hA hr₀ | |
| (fun c hc s hs hsr ↦ hgr c ⟨hc.1, Metric.ball_subset_closedBall (hbs hc.2)⟩ s hs hsr) | |
| have hb : MemLp (fun x ↦ b x) 2 μ := MemLp.of_bound b.continuous.aestronglyMeasurable 1 | |
| (ae_of_all _ (fun x ↦ by simpa only [Real.norm_eq_abs, abs_of_nonneg b.nonneg] using b.le_one)) | |
| have hDb : MemLp (surfaceGradient S b) 2 μ := surfaceGradient_memLp_on_ball b.contDiff | |
| ((hgrad b b.contDiff).mono_measure Measure.restrict_le_self) | |
| obtain ⟨_, hB, hBt⟩ := capacity_product_pair_tendsto hb hDb χ | |
| (fun i ↦ (hχ i).1.continuous.aestronglyMeasurable) | |
| (fun i ↦ ae_of_all _ (hχ i).2.1) hχlim zero_le_one | |
| (ae_of_all _ (fun x ↦ by simpa only [Real.norm_eq_abs, abs_of_nonneg b.nonneg] using b.le_one)) | |
| (fun i ↦ surfaceGradient S (χ i)) (fun i ↦ (hχ i).2.2.2) hχE | |
| let η : ℕ → Space n → ℝ := fun i x ↦ b x * χ i x | |
| let F : ℕ → Space n → Space n := fun i x ↦ η i x • truncatedGradient W D x | |
| let H : Space n → Space n := fun x ↦ b x • truncatedGradient W D x | |
| have hη : ∀ i, ContDiff ℝ ∞ (η i) := fun i ↦ b.contDiff.mul (hχ i).1 | |
| have hcη : ∀ i, HasCompactSupport (η i) := fun _ ↦ b.hasCompactSupport.mul_right | |
| have hηZ : ∀ i, Disjoint (tsupport (η i)) Z := fun i ↦ | |
| tsupport_mul_disjoint_Singular (hχ i).2.2.1 | |
| have hFE : ∀ i, MemLp (F i) 2 (surfaceMeasure S) ∧ | |
| vectorEnergy (surfaceMeasure S) (F i) ≤ | |
| 4 * vectorEnergy (surfaceMeasure S) (surfaceGradient S (η i)) := by | |
| intro i | |
| exact c1_min_regular_energy hS hZ hfinite hgrad htrace hsup hW0 ((hη i).of_le (by norm_num)) (hcη i) (hηZ i) | |
| have hFm : ∀ i, MemLp (F i) 2 μ := fun i ↦ (hFE i).1.mono_measure Measure.restrict_le_self | |
| have hB_eq : ∀ i, (fun x ↦ χ i x • surfaceGradient S b x + b x • surfaceGradient S (χ i) x) = | |
| surfaceGradient S (η i) := by | |
| intro i | |
| funext x | |
| change _ = surfaceGradient S (fun y ↦ b y * χ i y) x | |
| have hbDiff : ContDiff ℝ ∞ (fun y ↦ b y) := b.contDiff | |
| rw [surfaceGradient_mul (hbDiff.differentiable (by simp) x) | |
| ((hχ i).1.differentiable (by simp) x), add_comm] | |
| have hBE := vectorEnergy_tendsto_of_toLp_tendsto hB hDb hBt.snd_nhds | |
| simp_rw [hB_eq] at hBE | |
| have hbound : ∀ i, vectorEnergy μ (F i) ≤ 4 * vectorEnergy μ (surfaceGradient S (η i)) := by | |
| intro i | |
| rw [vectorEnergy_restrict_of_vanish measurableSet_ball, vectorEnergy_restrict_of_vanish measurableSet_ball] | |
| · exact (hFE i).2 | |
| · intro x hx | |
| exact surfaceGradient_eq_zero_of_notMem_tsupport S | |
| (fun h ↦ hx (hbs (tsupport_mul_subset_left h))) | |
| · intro x hx | |
| have hb0 : b x = 0 := image_eq_zero_of_notMem_tsupport (fun h ↦ hx (hbs h)) | |
| simp only [F, η, hb0, zero_mul, zero_smul] | |
| have hlim : ∀ᵐ x ∂μ, Tendsto (fun i ↦ F i x) atTop (𝓝 (H x)) := by | |
| filter_upwards [hχlim] with x hx | |
| simpa only [F, H, η, mul_one] using | |
| (tendsto_const_nhds.mul hx).smul_const (truncatedGradient W D x) | |
| have hHm : AEStronglyMeasurable H μ := | |
| aestronglyMeasurable_of_tendsto_ae atTop (fun i ↦ (hFm i).aestronglyMeasurable) hlim | |
| have hHE := vectorEnergy_le_of_ae_tendsto (fun i ↦ (hFm i).aestronglyMeasurable) hlim | |
| (Eventually.of_forall hbound) (ENNReal.Tendsto.const_mul hBE (Or.inr (by norm_num))) | |
| have hH : MemLp H 2 μ := (memLp_two_iff_vectorEnergy hHm).mpr (hHE.trans_lt | |
| (ENNReal.mul_lt_top (by norm_num) ((memLp_two_iff_vectorEnergy hDb.aestronglyMeasurable).mp hDb))) | |
| have hHR : MemLp H 2 ((surfaceMeasure S).restrict (Metric.ball 0 R)) := | |
| hH.mono_measure (Measure.restrict_mono_set _ (Metric.ball_subset_ball (by linarith))) | |
| apply hHR.ae_eq | |
| filter_upwards [ae_restrict_mem measurableSet_ball] with x hx | |
| have hbx : b x = 1 := b.one_of_mem_closedBall (Metric.mem_closedBall.mpr | |
| ((Metric.mem_ball.mp hx).le.trans (by dsimp [b]; linarith))) | |
| simp only [H, hbx, one_smul] | |
| end RPI.Boundary | |
| 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 | |
| namespace RPISingularHarnack | |
| open RPI RPI.Boundary | |
| theorem c1_min_wholeBoundarySobolev_and_supersolution_of_regular_geometry | |
| {n : ℕ} (hn : 8 ≤ n) {E Z : Set (Euc n)} (hE : MeasurableSet E) | |
| (hZ : IsClosed Z) (hZd : dimH Z ≤ (n - 8 : ℕ)) | |
| (hfinite : ∀ R : ℝ, 0 < R → surfaceMeasure (perimeterSupport E) (Metric.ball 0 R) < ⊤) | |
| (hgrowth : ∀ R : ℝ, 0 < R → ∃ A r₀ : ℝ, 0 < A ∧ 0 < r₀ ∧ | |
| ∀ c ∈ Z ∩ Metric.closedBall 0 R, ∀ r : ℝ, 0 < r → r < r₀ → | |
| (surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R) (Metric.ball c r) ≤ | |
| ENNReal.ofReal (A * r ^ (n - 1))) | |
| (hreg : ∀ x ∈ perimeterSupport E \ Z, ∃ U : Set (Euc n), IsOpen U ∧ x ∈ U ∧ | |
| ∃ f : Euc n → ℝ, ContDiffOn ℝ ∞ f U ∧ | |
| (∀ y ∈ U, fderiv ℝ f y ≠ 0) ∧ | |
| (∀ y ∈ U, y ∈ perimeterSupport E ↔ f y = 0) ∧ | |
| (∀ y ∈ U, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0))) | |
| {W : Euc n → ℝ} {D : Euc n → Euc n} | |
| (htrace : ∀ x ∈ perimeterSupport E \ Z, HasC1TraceJetAt (perimeterSupport E) W D x) | |
| (hW0 : ∀ x ∈ perimeterSupport E \ Z, 0 ≤ W x) | |
| (hsup : ∀ φ : Euc n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ → | |
| Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) → | |
| 0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient (perimeterSupport E) φ x) | |
| ∂surfaceMeasure (perimeterSupport E)) : | |
| WholeBoundarySobolev E (fun x ↦ min (W x) 1) (truncatedGradient W D) ∧ | |
| WholeBoundarySupersolution E (truncatedGradient W D) := by | |
| have hS := isClosed_perimeterSupport E | |
| have hlevel : ∀ x ∈ perimeterSupport E \ Z, ∃ U : Set (Euc n), IsOpen U ∧ x ∈ U ∧ | |
| ∃ f : Euc n → ℝ, ContDiffOn ℝ 1 f U ∧ | |
| (∀ y ∈ U, fderiv ℝ f y ≠ 0) ∧ | |
| (∀ y ∈ U, y ∈ perimeterSupport E ↔ f y = 0) := by | |
| intro x hx | |
| obtain ⟨U, hU, hxU, f, hf, hdf, hs, _⟩ := hreg x hx | |
| exact ⟨U, hU, hxU, f, hf.of_le (by norm_num), hdf, hs⟩ | |
| have hfilled : ∀ x ∈ perimeterSupport E \ Z, ∃ f : Euc n → ℝ, | |
| ContDiffAt ℝ 1 f x ∧ f x = 0 ∧ fderiv ℝ f x ≠ 0 ∧ | |
| (∀ᶠ y in 𝓝 x, y ∈ perimeterSupport E ↔ f y = 0) ∧ | |
| (∀ᶠ y in 𝓝 x, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0)) := by | |
| intro x hx | |
| obtain ⟨U, hU, hxU, f, hf, hdf, hs, he⟩ := hreg x hx | |
| refine ⟨f, (hf.of_le (by norm_num)).contDiffAt (hU.mem_nhds hxU), | |
| (hs x hxU).mp hx.1, hdf x hxU, ?_, ?_⟩ | |
| · filter_upwards [hU.mem_nhds hxU] with y hy using hs y hy | |
| · filter_upwards [hU.mem_nhds hxU] with y hy using he y hy | |
| have hgrad : ∀ f : Euc n → ℝ, ContDiff ℝ 1 f → | |
| AEStronglyMeasurable (surfaceGradient (perimeterSupport E) f) | |
| (surfaceMeasure (perimeterSupport E)) := by | |
| intro f hf | |
| exact surfaceGradient_aestronglyMeasurable_of_regular_locus hn hS.measurableSet | |
| hZ.measurableSet hZd hlevel hf | |
| have hloc : ∀ R : ℝ, 0 < R → MemLp (truncatedGradient W D) 2 | |
| ((surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R)) := by | |
| intro R hR | |
| exact c1_min_local_energy hn hS hZ hZd hfinite hgrad hgrowth htrace hW0 hsup hR | |
| have haereg : ∀ᵐ x ∂surfaceMeasure (perimeterSupport E), x ∈ perimeterSupport E \ Z := by | |
| have hz : ∀ᵐ x ∂surfaceMeasure (perimeterSupport E), x ∉ Z := by | |
| simpa only [ae_iff, not_not, Set.ofPred_mem_eq] using | |
| surfaceMeasure_zero_of_singular_dim hn (perimeterSupport E) Z hZd | |
| filter_upwards [ae_restrict_mem hS.measurableSet, hz] with x hx hzx | |
| exact ⟨hx, hzx⟩ | |
| constructor | |
| · apply wholeBoundarySobolev_of_local_sobolevGraph_filled hn hE hS.measurableSet hZd hfilled | |
| intro R hR | |
| have : IsFiniteMeasure ((surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R)) := | |
| ⟨by simpa only [Measure.restrict_apply_univ] using hfinite R hR⟩ | |
| have hw : ∀ᵐ x ∂(surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R), 0 ≤ W x := by | |
| filter_upwards [haereg.filter_mono (ae_mono Measure.restrict_le_self)] with x hx | |
| exact hW0 x hx | |
| obtain ⟨A, r₀, hA, hr₀, hgr⟩ := hgrowth (R + 2) (by linarith) | |
| obtain ⟨hh, hpair⟩ := c1_min_one_mem_sobolevGraph_ball hn hS hZ hZd hR htrace hw (hloc R hR) | |
| (fun f hf ↦ (hgrad f (hf.of_le (by norm_num))).mono_measure Measure.restrict_le_self) hA hr₀ (by | |
| intro c hc r hr hrr | |
| apply le_trans _ (hgr c hc r hr hrr) | |
| exact Measure.le_iff'.mp (Measure.restrict_mono_set _ | |
| (Metric.ball_subset_ball (by linarith : R ≤ R + 2))) _) | |
| exact ⟨hh, hloc R hR, hpair⟩ | |
| · intro φ hφ hcφ hpφ | |
| have hh := full_test_of_regular_tests hn hZ hZd hloc hfinite (fun f hf ↦ hgrad f (hf.of_le (by norm_num))) hgrowth | |
| (fun ψ hψ hcψ hψZ hpψ ↦ c1_min_regular_test hS hZ hfinite hgrad htrace hsup (hψ.of_le (by norm_num)) hcψ hψZ hpψ) | |
| hφ hcφ hpφ | |
| have heq := surfaceGradient_ae_eq_tangentialGradient_filled hn hE hS.measurableSet hZd hfilled φ | |
| rw [boundaryArea_eq_smul_surfaceMeasure, integral_smul_measure] | |
| have hi : (∫ x, inner ℝ (truncatedGradient W D x) (tangentialGradient E φ x) | |
| ∂surfaceMeasure (perimeterSupport E)) = | |
| ∫ x, inner ℝ (truncatedGradient W D x) (surfaceGradient (perimeterSupport E) φ x) | |
| ∂surfaceMeasure (perimeterSupport E) := by | |
| apply integral_congr_ae | |
| filter_upwards [heq] with x hx | |
| rw [hx] | |
| rw [hi] | |
| exact mul_nonneg ENNReal.toReal_nonneg hh | |
| 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 | |
| namespace RPISingularHarnack | |
| open RPI RPI.Boundary | |
| theorem c1_min_wholeBoundarySobolev_and_supersolution_of_minimizing_regular_geometry | |
| {n : ℕ} (hn : 8 ≤ n) {E Z : Set (Euc n)} | |
| (hmin : IsLocallyPerimeterMinimizing E) | |
| (hZ : IsClosed Z) (hZd : dimH Z ≤ (n - 8 : ℕ)) | |
| (hreg : ∀ x ∈ perimeterSupport E \ Z, ∃ U : Set (Euc n), IsOpen U ∧ x ∈ U ∧ | |
| ∃ f : Euc n → ℝ, ContDiffOn ℝ ∞ f U ∧ | |
| (∀ y ∈ U, fderiv ℝ f y ≠ 0) ∧ | |
| (∀ y ∈ U, y ∈ perimeterSupport E ↔ f y = 0) ∧ | |
| (∀ y ∈ U, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0))) | |
| {W : Euc n → ℝ} {D : Euc n → Euc n} | |
| (htrace : ∀ x ∈ perimeterSupport E \ Z, HasC1TraceJetAt (perimeterSupport E) W D x) | |
| (hW0 : ∀ x ∈ perimeterSupport E \ Z, 0 ≤ W x) | |
| (hsup : ∀ φ : Euc n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ → | |
| Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) → | |
| 0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient (perimeterSupport E) φ x) | |
| ∂surfaceMeasure (perimeterSupport E)) : | |
| WholeBoundarySobolev E (fun x ↦ min (W x) 1) (truncatedGradient W D) ∧ | |
| WholeBoundarySupersolution E (truncatedGradient W D) := by | |
| have hhalf : ∀ x ∈ perimeterSupport E \ Z, ∃ v, HasHalfspaceBlowup E x v := by | |
| intro x hx | |
| obtain ⟨U, hU, hxU, f, hf, hdf, hs, he⟩ := hreg x hx | |
| refine ⟨_, hasHalfspaceBlowup_gradient_filled hmin.1 | |
| ((hf.contDiffAt (hU.mem_nhds hxU)).differentiableAt (by norm_num)) | |
| ((hs x hxU).mp hx.1) (hdf x hxU) ?_⟩ | |
| filter_upwards [hU.mem_nhds hxU] with y hy using he y hy | |
| obtain ⟨A, hA, harea⟩ := exists_minimizing_surfaceMeasure_growth_of_halfspace hn hmin hZd hhalf | |
| apply c1_min_wholeBoundarySobolev_and_supersolution_of_regular_geometry hn hmin.1 hZ hZd | |
| (fun R hR ↦ (harea 0 R hR).trans_lt ENNReal.ofReal_lt_top) _ hreg htrace hW0 hsup | |
| intro R hR | |
| refine ⟨A, 1, hA, by norm_num, ?_⟩ | |
| intro c hc r hr hrr | |
| exact (Measure.restrict_apply_le (Metric.ball 0 R) (Metric.ball c r)).trans (harea c r hr) | |
| 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 | |
| namespace RPISingularHarnack | |
| open RPI.Boundary | |
| open ContinuousLinearMap | |
| open scoped Convolution | |
| theorem divergence_comp_add_right {n : ℕ} {V : Euc n → Euc n} | |
| (hV : Differentiable ℝ V) (t x : Euc n) : | |
| divergence (fun y ↦ V (y + t)) x = divergence V (x + t) := by | |
| have hd : fderiv ℝ (fun y ↦ V (y + t)) x = fderiv ℝ V (x + t) := by | |
| simpa only [Function.comp_def, id_eq, ContinuousLinearMap.comp_id] using ((hV _).hasFDerivAt.comp x ((hasFDerivAt_id x).add_const t)).fderiv | |
| simp only [divergence, hd] | |
| theorem integral_divergence_le_perimeter_real {n : ℕ} (E U : Set (Euc n)) | |
| (hP : perimeter E U ≠ ⊤) {V : Euc n → Euc n} (hV : PerimeterAdmissible U V) : | |
| ∫ x in E, divergence V x ≤ (perimeter E U).toReal := by | |
| apply (ENNReal.ofReal_le_iff_le_toReal hP).mp | |
| rw [perimeter_eq_iSup_tests] | |
| exact le_iSup_of_le V (le_iSup_of_le hV le_rfl) | |
| theorem indicator_convolution_variation_le {n : ℕ} {E : Set (Euc n)} | |
| (hE : MeasurableSet E) (hP : perimeter E univ ≠ ⊤) | |
| {k : Euc n → ℝ} (hkc : Continuous k) (hk : Integrable k volume) | |
| (hk0 : ∀ x, 0 ≤ k x) (hk1 : ∫ x, k x = 1) | |
| {V : Euc n → Euc n} (hV : PerimeterAdmissible univ V) : | |
| (∫ x, (k ⋆[lsmul ℝ ℝ, volume] E.indicator (fun _ ↦ (1 : ℝ))) x * divergence V x) ≤ | |
| (perimeter E univ).toReal := by | |
| let χ : Euc n → ℝ := E.indicator (fun _ ↦ 1) | |
| have hmχ : Measurable χ := measurable_const.indicator hE | |
| have hbχ : ∀ x, ‖χ x‖ ≤ 1 := by | |
| intro x | |
| by_cases hx : x ∈ E <;> simp [χ, hx] | |
| have hd := integrable_divergence hV.1 hV.2.1 | |
| let F : Euc n × Euc n → ℝ := fun p ↦ k p.1 * χ (p.2 - p.1) * divergence V p.2 | |
| have hFi : Integrable F (volume.prod volume) := by | |
| apply (hk.norm.mul_prod hd.norm).mono' | |
| · exact ((hkc.measurable.comp measurable_fst).mul | |
| (hmχ.comp (measurable_snd.sub measurable_fst))).mul | |
| ((continuous_divergence hV.1).measurable.comp measurable_snd) |>.aestronglyMeasurable | |
| · apply ae_of_all | |
| intro p | |
| simp only [F, norm_mul] | |
| exact mul_le_mul_of_nonneg_right | |
| (by nlinarith [hbχ (p.2 - p.1), norm_nonneg (k p.1)]) (norm_nonneg _) | |
| have hinner : ∀ t, (∫ x, F (t, x)) = k t * (∫ y in E, divergence V (y + t)) := by | |
| intro t | |
| simp only [F, mul_assoc, integral_const_mul] | |
| congr 1 | |
| calc | |
| (∫ x, χ (x - t) * divergence V x) = | |
| ∫ y, χ y * divergence V (y + t) := by | |
| simpa only [add_sub_cancel_right] using | |
| (integral_add_right_eq_self (fun x ↦ χ (x - t) * divergence V x) t).symm | |
| _ = ∫ y in E, divergence V (y + t) := by | |
| rw [← integral_indicator hE] | |
| apply integral_congr_ae | |
| apply ae_of_all | |
| intro y | |
| by_cases hy : y ∈ E <;> simp [χ, hy] | |
| have hInt : Integrable (fun t ↦ k t * (∫ y in E, divergence V (y + t))) volume := by | |
| simpa only [hinner] using hFi.integral_prod_left | |
| have heq : (∫ x, (k ⋆[lsmul ℝ ℝ, volume] χ) x * divergence V x) = | |
| ∫ t, k t * (∫ y in E, divergence V (y + t)) := by | |
| calc | |
| _ = ∫ x, ∫ t, F (t, x) := by | |
| apply integral_congr_ae | |
| apply ae_of_all | |
| intro x | |
| simp only [convolution_def, lsmul_apply, smul_eq_mul, F, integral_mul_const] | |
| _ = ∫ t, ∫ x, F (t, x) := integral_integral_swap hFi.swap | |
| _ = _ := by simp only [hinner] | |
| have hbound : ∀ t, ∫ y in E, divergence V (y + t) ≤ (perimeter E univ).toReal := by | |
| intro t | |
| let Vt : Euc n → Euc n := fun y ↦ V (y + t) | |
| have hVt : PerimeterAdmissible univ Vt := ⟨hV.1.comp (contDiff_id.add contDiff_const), | |
| hV.2.1.comp_homeomorph (Homeomorph.addRight t), subset_univ _, fun y ↦ hV.2.2.2 _⟩ | |
| simpa only [Vt, divergence_comp_add_right (hV.1.differentiable (by simp))] using | |
| integral_divergence_le_perimeter_real E univ hP hVt | |
| rw [heq] | |
| calc | |
| (∫ t, k t * (∫ y in E, divergence V (y + t))) ≤ | |
| ∫ t, k t * (perimeter E univ).toReal := by | |
| exact integral_mono hInt (hk.mul_const _) fun t ↦ | |
| mul_le_mul_of_nonneg_left (hbound t) (hk0 t) | |
| _ = (perimeter E univ).toReal := by rw [integral_mul_const, hk1, one_mul] | |
| end RPISingularHarnack | |
| end | |
| end OAI | |