| import OAI.Analysis.HenonEmden.OriginBounds |
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| namespace OAI |
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| universe uE uE2 |
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| section Distribution |
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| open Set Filter Topology MeasureTheory |
| open scoped ContDiff |
| open Set Filter Topology |
| open scoped RealInnerProductSpace ContDiff |
| open Set Filter Topology |
| open scoped ContDiff |
| open Set MeasureTheory MeasureTheory.Measure |
| open scoped ENNReal |
| open Set Filter Topology MeasureTheory |
| open Set Filter Topology |
| open MeasureTheory |
| open scoped Convolution |
| open Set Filter Topology |
| open MeasureTheory Set Filter Topology |
| open scoped ContDiff |
| noncomputable section |
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| namespace HenonLaneEmden.WeakPDE |
| variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] |
| [MeasurableSpace E] [BorelSpace E] |
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| lemma cutoff_laplacian_integral_scale {u φ ξ : E → ℝ} |
| (hφ : ContDiff ℝ 2 φ) (hξ : ContDiff ℝ 2 ξ) {R : ℝ} (hR : 0 < R) : |
| (∫ x : E, u x*Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x) = |
| R^(Module.finrank ℝ E)*(R⁻¹)^2* |
| ∫ y : E, u (R • y)*Laplacian.laplacian (fun z => φ (R • z)*ξ z) y := by |
| have hF : ContDiff ℝ 2 (fun z : E => φ (R • z)*ξ z) := |
| (hφ.comp (contDiff_const.smul contDiff_id)).mul hξ |
| have hid : (fun y : E => φ y*ξ (R⁻¹ • y)) = |
| (fun y : E => (fun z => φ (R • z)*ξ z) (R⁻¹ • y)) := by |
| ext y |
| simp [smul_smul,ne_of_gt hR] |
| have hh (x : E) : Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) (R • x) = |
| (R⁻¹)^2*Laplacian.laplacian (fun z => φ (R • z)*ξ z) x := by |
| rw [hid,laplacian_dilation hF] |
| simp [smul_smul,ne_of_gt hR] |
| have hc := Measure.integral_comp_smul_of_nonneg volume |
| (fun x => u x*Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x) R (hR := hR.le) |
| simp only [hh,smul_eq_mul] at hc |
| have hC : (∫ x, u (R • x)*((R⁻¹)^2*Laplacian.laplacian (fun z => φ (R • z)*ξ z) x)) = |
| (R⁻¹)^2*(∫ x, u (R • x)*Laplacian.laplacian (fun z => φ (R • z)*ξ z) x) := by |
| simp_rw [← mul_assoc,mul_comm (u (R • _)) ((R⁻¹)^2),mul_assoc] |
| exact integral_const_mul _ _ |
| rw [hC] at hc |
| have hn : R^(Module.finrank ℝ E) ≠ 0 := pow_ne_zero _ (ne_of_gt hR) |
| calc |
| _ = R^Module.finrank ℝ E * ((R^Module.finrank ℝ E)⁻¹ * |
| ∫ x, u x*Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x) := by field_simp |
| _ = _ := by rw [← hc]; ring |
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| lemma cutoff_laplacian_integral_tendsto_zero {u φ ξ : E → ℝ} |
| (hn : 3 ≤ Module.finrank ℝ E) (hu : Continuous u) |
| (hφ : ContDiff ℝ 2 φ) (hξ : ContDiff ℝ 2 ξ) (hξc : HasCompactSupport ξ) : |
| Tendsto (fun R : ℝ => ∫ x : E, u x*Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x) |
| (𝓝[>] 0) (𝓝 0) := by |
| have heq : ∀ᶠ R : ℝ in 𝓝[>] 0, (∫ x : E, u x*Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x) = |
| R^(Module.finrank ℝ E-2)*∫ y : E, u (R • y)*Laplacian.laplacian (fun z => φ (R • z)*ξ z) y := by |
| filter_upwards [self_mem_nhdsWithin] with R hR |
| rw [cutoff_laplacian_integral_scale hφ hξ hR] |
| congr 1 |
| rw [inv_pow] |
| have hp := pow_sub₀ R (ne_of_gt hR) (show 2 ≤ Module.finrank ℝ E by omega) |
| exact hp.symm |
| apply Tendsto.congr' (Filter.EventuallyEq.symm heq) |
| have hpow : Tendsto (fun R : ℝ => R^(Module.finrank ℝ E-2)) (𝓝[>] 0) (𝓝 0) := by |
| have := (continuous_pow (Module.finrank ℝ E-2) : Continuous (fun R : ℝ => R^(Module.finrank ℝ E-2))).tendsto 0 |
| simpa [show Module.finrank ℝ E-2 ≠ 0 by omega] using this.mono_left nhdsWithin_le_nhds |
| simpa using hpow.mul ((continuous_cutoff_error hu hφ hξ hξc).tendsto 0 |>.mono_left nhdsWithin_le_nhds) |
| end HenonLaneEmden.WeakPDE |
| |
| namespace HenonLaneEmden.WeakPDE |
| variable {E : Type uE2} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] |
| [MeasurableSpace E] [BorelSpace E] |
| |
| omit [MeasurableSpace E] [BorelSpace E] in |
| lemma bump_dilation_eventually_zero (ξ : ContDiffBump (0 : E)) {x : E} (hx : x ≠ 0) : |
| (fun R : ℝ => ξ (R⁻¹ • x)) =ᶠ[𝓝[>] 0] 0 := by |
| have he : 0 < ‖x‖ / ξ.rOut := div_pos (norm_pos_iff.mpr hx) ξ.rOut_pos |
| filter_upwards [self_mem_nhdsWithin, (eventually_lt_nhds he).filter_mono nhdsWithin_le_nhds] with R hR hlt |
| apply ξ.zero_of_le_dist |
| rw [dist_zero_right,norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hR),← div_eq_inv_mul] |
| apply (le_div_iff₀ hR).mpr |
| simpa only [mul_comm] using (le_of_lt ((lt_div_iff₀ ξ.rOut_pos).mp hlt)) |
| |
| lemma cutoff_source_integral_tendsto_zero [Nontrivial E] {g φ : E → ℝ} |
| (hg : LocallyIntegrable g) (hφ : Continuous φ) (hφc : HasCompactSupport φ) |
| (ξ : ContDiffBump (0 : E)) : |
| Tendsto (fun R : ℝ => ∫ x : E, g x*(φ x*ξ (R⁻¹ • x))) (𝓝[>] 0) (𝓝 0) := by |
| have hi : Integrable (fun x => g x*φ x) := |
| hg.integrable_smul_right_of_hasCompactSupport hφ hφc |
| have hc (R : ℝ) : Continuous (fun x : E => ξ (R⁻¹ • x)) := |
| ξ.continuous.comp (continuous_const.smul continuous_id) |
| have lim := tendsto_integral_filter_of_dominated_convergence |
| (μ := volume) (l := 𝓝[>] (0 : ℝ)) (F := fun R x => g x*(φ x*ξ (R⁻¹ • x))) |
| (f := fun _ => (0 : ℝ)) (fun x => ‖g x*φ x‖) |
| (Eventually.of_forall fun R => by |
| rw [show (fun x => g x*(φ x*ξ (R⁻¹ • x))) = |
| ((fun x => g x*φ x)*(fun x => ξ (R⁻¹ • x))) by ext x; simp only [Pi.mul_apply]; ring] |
| exact hi.aestronglyMeasurable.mul (hc R).aestronglyMeasurable) |
| (Eventually.of_forall fun R => Eventually.of_forall fun x => by |
| rw [← mul_assoc,norm_mul] |
| exact mul_le_of_le_one_right (norm_nonneg _) (by |
| rw [Real.norm_eq_abs,abs_of_nonneg ξ.nonneg]; exact ξ.le_one)) |
| hi.norm ?_ |
| · simpa using lim |
| filter_upwards [ae_iff.mpr (show volume {x : E | ¬ x ≠ 0} = 0 by simp)] with x hx |
| apply tendsto_const_nhds.congr' |
| filter_upwards [bump_dilation_eventually_zero ξ hx] with R hR |
| simp [hR] |
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| omit [MeasurableSpace E] [BorelSpace E] in |
| lemma cutoff_test_off_origin {φ : E → ℝ} (hφ : ContDiff ℝ 2 φ) |
| (hφc : HasCompactSupport φ) (ξ : ContDiffBump (0 : E)) {R : ℝ} (_hR : 0 < R) : |
| ContDiff ℝ 2 (fun x => φ x-φ x*ξ (R⁻¹ • x)) ∧ |
| HasCompactSupport (fun x => φ x-φ x*ξ (R⁻¹ • x)) ∧ |
| 0 ∉ tsupport (fun x => φ x-φ x*ξ (R⁻¹ • x)) := by |
| refine ⟨hφ.sub (hφ.mul (ξ.contDiff.comp (contDiff_const.smul contDiff_id))), |
| hφc.sub (hφc.mul_right), ?_⟩ |
| apply notMem_tsupport_iff_eventuallyEq.mpr |
| have ht : Tendsto (fun x : E => R⁻¹ • x) (𝓝 0) (𝓝 0) := by |
| have h : Continuous (fun x : E => R⁻¹ • x) := continuous_const.smul continuous_id |
| have ht := h.tendsto 0 |
| change Tendsto (fun x : E => R⁻¹ • x) (𝓝 0) (𝓝 (R⁻¹ • (0:E))) at ht |
| simpa only [smul_zero] using ht |
| have he := ξ.eventuallyEq_one.comp_tendsto ht |
| filter_upwards [he] with x hx |
| change ξ (R⁻¹ • x) = 1 at hx |
| simp [hx] |
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| lemma distributional_extension (hn : 3 ≤ Module.finrank ℝ E) {u g : E → ℝ} |
| (hu : Continuous u) (hg : LocallyIntegrable g) |
| (heq : ∀ (φ : E → ℝ), ContDiff ℝ 2 φ → HasCompactSupport φ → 0 ∉ tsupport φ → |
| (∫ x, u x*Laplacian.laplacian φ x) = -(∫ x, g x*φ x)) |
| {φ : E → ℝ} (hφ : ContDiff ℝ 2 φ) (hφc : HasCompactSupport φ) : |
| (∫ x, u x*Laplacian.laplacian φ x) = -(∫ x, g x*φ x) := by |
| have : Nontrivial E := (Module.finrank_pos_iff (R := ℝ)).mp (by omega) |
| let ξ : ContDiffBump (0 : E) := ⟨1,2,by norm_num,by norm_num⟩ |
| have hi : Integrable (fun x => g x*φ x) := |
| hg.integrable_smul_right_of_hasCompactSupport hφ.continuous hφc |
| have hui : Integrable (fun x => u x*Laplacian.laplacian φ x) := |
| integrable_mul_compact (compactSupport_laplacian hφc) hu.continuousOn (continuous_laplacian hφ).continuousOn |
| have hevent : ∀ᶠ R : ℝ in 𝓝[>] 0, |
| (∫ x, u x*Laplacian.laplacian φ x) - |
| (∫ x, u x*Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x) = |
| -(∫ x, g x*φ x) + (∫ x, g x*(φ x*ξ (R⁻¹ • x))) := by |
| filter_upwards [self_mem_nhdsWithin] with R hR |
| have hχ : ContDiff ℝ 2 (fun x => φ x*ξ (R⁻¹ • x)) := |
| hφ.mul (ξ.contDiff.comp (contDiff_const.smul contDiff_id)) |
| have hχc : HasCompactSupport (fun x => φ x*ξ (R⁻¹ • x)) := hφc.mul_right |
| have hχi : Integrable (fun x => u x*Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x) := |
| integrable_mul_compact (compactSupport_laplacian hχc) hu.continuousOn (continuous_laplacian hχ).continuousOn |
| have hgi : Integrable (fun x => g x*(φ x*ξ (R⁻¹ • x))) := |
| hg.integrable_smul_right_of_hasCompactSupport hχ.continuous hχc |
| obtain ⟨hψ,hψc,hψ0⟩ := cutoff_test_off_origin hφ hφc ξ hR |
| have HH := heq _ hψ hψc hψ0 |
| have hid (x : E) : Laplacian.laplacian (fun y => φ y-φ y*ξ (R⁻¹ • y)) x = |
| Laplacian.laplacian φ x-Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x := |
| hφ.contDiffAt.laplacian_sub hχ.contDiffAt |
| simp_rw [hid,mul_sub] at HH |
| rw [integral_sub hui hχi,integral_sub hi hgi] at HH |
| linarith |
| have hlim1 := (tendsto_const_nhds (x := ∫ x, u x*Laplacian.laplacian φ x)).sub (cutoff_laplacian_integral_tendsto_zero hn hu hφ ξ.contDiff ξ.hasCompactSupport) |
| have hlim2 := (tendsto_const_nhds (x := -(∫ x, g x*φ x))).add (cutoff_source_integral_tendsto_zero hg hφ.continuous hφc ξ) |
| have HH := tendsto_nhds_unique hlim1 (hlim2.congr' (Filter.EventuallyEq.symm hevent)) |
| simpa only [sub_zero,add_zero] using HH |
| end HenonLaneEmden.WeakPDE |
| |
| end |
| end Distribution |
| |
| end OAI |
| |
