| import OAI.Analysis.AllenCahn.DyadicDerivatives | |
| namespace OAI | |
| noncomputable section | |
| namespace RegularitySupport.DeGiorgi | |
| open Filter Set Topology | |
| open scoped NNReal | |
| open scoped SchwartzMap Laplacian ContDiff | |
| open FourierTransform | |
| open Filter Topology | |
| open scoped ContDiff | |
| open MeasureTheory | |
| open scoped ENNReal | |
| lemma iteratedFDeriv_apply_bound_of_difference_bound {n k : ℕ} {f : Space n → ℝ} {M : ℝ} | |
| (hf : ContDiff ℝ ∞ f) (hM : 0 ≤ M) (hb : DifferenceBound (2 * k) f M) | |
| (x : Space n) (v : Fin k → Space n) : | |
| |iteratedFDeriv ℝ k f x v| ≤ M * ∏ i : Fin k, ‖v i‖ := by | |
| induction k generalizing f M with | |
| | zero => | |
| have h := hb [] (by simp) x | |
| simpa [iteratedFDeriv_zero_apply, iterDifference, differenceScale] using h | |
| | succ k ih => | |
| let h : Space n := v (Fin.last k) | |
| have hder : ContDiff ℝ ∞ (fderiv ℝ f) := hf.fderiv_right (by simp) | |
| have hg : ContDiff ℝ ∞ (fun y => fderiv ℝ f y h) := hder.clm_apply contDiff_const | |
| have hbd : DifferenceBound (2 * k + 2) f M := by convert hb using 1 | |
| have hh := differenceBound_fderiv_apply hbd hM h | |
| have hi := ih hg (mul_nonneg hM (norm_nonneg h)) hh (Fin.init v) | |
| rw [iteratedFDeriv_succ_apply_right, ← iteratedFDeriv_clm_apply_const_apply hder (i := k) (by exact WithTop.coe_le_coe.mpr le_top)] | |
| calc | |
| _ ≤ M * ‖h‖ * ∏ i : Fin k, ‖Fin.init v i‖ := hi | |
| _ = M * ∏ i : Fin (k + 1), ‖v i‖ := by | |
| rw [Fin.prod_univ_castSucc] | |
| simp only [Fin.init, h] | |
| ring | |
| lemma iteratedFDeriv_bound_of_difference_bound {n k : ℕ} {f : Space n → ℝ} {M : ℝ} | |
| (hf : ContDiff ℝ ∞ f) (hM : 0 ≤ M) (hb : DifferenceBound (2 * k) f M) | |
| (x : Space n) : ‖iteratedFDeriv ℝ k f x‖ ≤ M := by | |
| apply ContinuousMultilinearMap.opNorm_le_bound hM | |
| intro v | |
| simpa only [Real.norm_eq_abs] using iteratedFDeriv_apply_bound_of_difference_bound hf hM hb x v | |
| lemma allenCahn_smooth {n : ℕ} {v : Space n → ℝ} | |
| (hv : ContDiff ℝ 2 v) (hrange : ∀ x, v x ∈ Set.Icc (-1 : ℝ) 1) (hpde : AllenCahn v) : | |
| ContDiff ℝ ∞ v := | |
| contDiff_infty_of_difference_bounds (allenCahn_difference_bounds hv hrange hpde) | |
| lemma allenCahn_uniform_derivatives {n : ℕ} {v : Space n → ℝ} | |
| (hv : ContDiff ℝ 2 v) (hrange : ∀ x, v x ∈ Set.Icc (-1 : ℝ) 1) (hpde : AllenCahn v) | |
| (k : ℕ) : ∃ C : ℝ, 0 ≤ C ∧ ∀ x, ‖iteratedFDeriv ℝ k v x‖ ≤ C := by | |
| obtain ⟨M, hM, hb⟩ := allenCahn_difference_bounds hv hrange hpde (2 * k) | |
| exact ⟨M, hM, iteratedFDeriv_bound_of_difference_bound (allenCahn_smooth hv hrange hpde) hM hb⟩ | |
| lemma allenCahn_uniform_derivatives_dim (n k : ℕ) : | |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ v : Space n → ℝ, ContDiff ℝ 2 v → | |
| (∀ x, v x ∈ Set.Icc (-1 : ℝ) 1) → AllenCahn v → | |
| ∀ x, ‖iteratedFDeriv ℝ k v x‖ ≤ C := by | |
| obtain ⟨M, hM, hb⟩ := allenCahn_difference_bounds_uniform n (2 * k) | |
| refine ⟨M, hM, ?_⟩ | |
| intro v hv hrange hpde | |
| exact iteratedFDeriv_bound_of_difference_bound (allenCahn_smooth hv hrange hpde) hM (hb v hv hrange hpde) | |
| end RegularitySupport.DeGiorgi | |
| end | |
| end OAI | |