| import OAI.Analysis.AllenCahn.EllipticRegularity |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace DeGiorgi |
| open Filter Set Topology |
| open scoped NNReal |
| open scoped SchwartzMap Laplacian ContDiff |
| open FourierTransform |
| open Filter Topology |
| open scoped ContDiff |
| open scoped ENNReal |
| open MeasureTheory Filter Topology |
|
|
| lemma coordDeriv_continuous_of_C1 {n : ℕ} {f : Space n → ℝ} |
| (hf : ContDiff ℝ 1 f) (i : Fin n) : Continuous (coordDeriv f i) := by |
| exact (hf.continuous_fderiv (by norm_num)).clm_apply continuous_const |
|
|
| lemma coordDeriv_hasCompactSupport {n : ℕ} {f : Space n → ℝ} |
| (hf : HasCompactSupport f) (i : Fin n) : HasCompactSupport (coordDeriv f i) := by |
| exact hf.fderiv_apply ℝ _ |
|
|
| lemma integral_test_laplacian {n : ℕ} {f h : Space n → ℝ} |
| (hf : ContDiff ℝ 1 f) (hc : HasCompactSupport f) (hh : ContDiff ℝ 2 h) : |
| (∫ x, f x * Laplacian.laplacian h x) = |
| -(∫ x, ∑ i : Fin n, coordDeriv f i x * coordDeriv h i x) := by |
| have h1 : ContDiff ℝ 1 h := hh.of_le (by norm_num) |
| have hI (i : Fin n) : Integrable |
| (fun x => f x * coordDeriv (coordDeriv h i) i x) := |
| (hf.continuous.mul (coordDeriv_continuous_of_C1 (coordDeriv_contDiff hh i) i)).integrable_of_hasCompactSupport hc.mul_right |
| have hJ (i : Fin n) : Integrable |
| (fun x => coordDeriv f i x * coordDeriv h i x) := |
| ((coordDeriv_continuous_of_C1 hf i).mul (coordDeriv_continuous_of_C1 h1 i)).integrable_of_hasCompactSupport (coordDeriv_hasCompactSupport hc i).mul_right |
| simp_rw [laplacian_eq_sum_coordDeriv hh, Finset.mul_sum] |
| rw [integral_finsetSum _ (fun i _ => hI i), integral_finsetSum _ (fun i _ => hJ i), |
| ← Finset.sum_neg_distrib] |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable (hJ i) (hI i) |
| ((hf.continuous.mul (coordDeriv_continuous_of_C1 h1 i)).integrable_of_hasCompactSupport hc.mul_right) |
| (fun x _ => hf.differentiable (by norm_num) x) |
| (fun x _ => (coordDeriv_contDiff hh i).differentiable (by norm_num) x) |
|
|
| theorem positive_ground_state_identity {n : ℕ} {h η : Space n → ℝ} |
| (hh : ContDiff ℝ 2 h) (hp : ∀ x, 0 < h x) |
| (hη : ContDiff ℝ 1 η) (hηc : HasCompactSupport η) : |
| (∫ x, (‖gradient η x‖ ^ 2 + (Laplacian.laplacian h x / h x) * η x ^ 2)) = |
| ∫ x, h x ^ 2 * ‖gradient (fun y => η y / h y) x‖ ^ 2 := by |
| have hn : ∀ x, h x ≠ 0 := fun x => (hp x).ne' |
| have h1 : ContDiff ℝ 1 h := hh.of_le (by norm_num) |
| have hd : Differentiable ℝ h := hh.differentiable (by norm_num) |
| have hηd : Differentiable ℝ η := hη.differentiable (by norm_num) |
| let f : Space n → ℝ := fun x => η x ^ 2 / h x |
| have hf : ContDiff ℝ 1 f := (hη.pow 2).div h1 hn |
| have hfc : HasCompactSupport f := by |
| simpa only [f, pow_two, div_eq_mul_inv, Pi.mul_def] using |
| (hηc.mul_right (f' := η)).mul_right (f' := fun x => (h x)⁻¹) |
| have hI (i : Fin n) : Integrable |
| (fun x => f x * coordDeriv (coordDeriv h i) i x) := |
| (hf.continuous.mul (coordDeriv_continuous_of_C1 (coordDeriv_contDiff hh i) i)).integrable_of_hasCompactSupport hfc.mul_right |
| have hJ (i : Fin n) : Integrable |
| (fun x => coordDeriv f i x * coordDeriv h i x) := |
| ((coordDeriv_continuous_of_C1 hf i).mul (coordDeriv_continuous_of_C1 h1 i)).integrable_of_hasCompactSupport (coordDeriv_hasCompactSupport hfc i).mul_right |
| have hK (i : Fin n) : Integrable (fun x => (coordDeriv η i x) ^ 2) := by |
| apply ((coordDeriv_continuous_of_C1 hη i).pow 2).integrable_of_hasCompactSupport |
| simpa only [pow_two, Pi.mul_apply] using |
| (coordDeriv_hasCompactSupport hηc i).mul_right (f' := coordDeriv η i) |
| have hIl : Integrable (fun x => f x * Laplacian.laplacian h x) := by |
| simp_rw [laplacian_eq_sum_coordDeriv hh, Finset.mul_sum] |
| exact integrable_finsetSum _ (fun i _ => hI i) |
| have hJs : Integrable (fun x => ∑ i : Fin n, coordDeriv f i x * coordDeriv h i x) := |
| integrable_finsetSum _ (fun i _ => hJ i) |
| have hKg : Integrable (fun x => ‖gradient η x‖ ^ 2) := by |
| simp_rw [gradient_norm_sq] |
| exact integrable_finsetSum _ (fun i _ => hK i) |
| have hid (x : Space n) : |
| ‖gradient η x‖ ^ 2 - (∑ i : Fin n, coordDeriv f i x * coordDeriv h i x) = |
| h x ^ 2 * ‖gradient (fun y => η y / h y) x‖ ^ 2 := by |
| simp_rw [gradient_norm_sq, ← Finset.sum_sub_distrib, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| dsimp only [f] |
| rw [coordDeriv_div (hηd.fun_pow 2) hd hn, |
| coordDeriv_square hηd, coordDeriv_div hηd hd hn] |
| field_simp [hn x] |
| ring |
| calc |
| (∫ x, (‖gradient η x‖ ^ 2 + (Laplacian.laplacian h x / h x) * η x ^ 2)) = |
| ∫ x, (‖gradient η x‖ ^ 2 + f x * Laplacian.laplacian h x) := by |
| refine integral_congr_ae (Filter.Eventually.of_forall (fun x => ?_)) |
| dsimp only [f] |
| ring |
| _ = (∫ x, ‖gradient η x‖ ^ 2) + ∫ x, f x * Laplacian.laplacian h x := |
| integral_add hKg hIl |
| _ = (∫ x, ‖gradient η x‖ ^ 2) - |
| ∫ x, ∑ i : Fin n, coordDeriv f i x * coordDeriv h i x := by |
| rw [integral_test_laplacian hf hfc hh, sub_eq_add_neg] |
| _ = ∫ x, (‖gradient η x‖ ^ 2 - |
| ∑ i : Fin n, coordDeriv f i x * coordDeriv h i x) := |
| (integral_sub hKg hJs).symm |
| _ = ∫ x, h x ^ 2 * ‖gradient (fun y => η y / h y) x‖ ^ 2 := |
| integral_congr_ae (Filter.Eventually.of_forall hid) |
|
|
| lemma stable_of_positive_jacobi {n : ℕ} {u h : Space n → ℝ} |
| (hh : ContDiff ℝ 2 h) (hp : ∀ x, 0 < h x) |
| (hEq : ∀ x, Laplacian.laplacian h x = (3 * u x ^ 2 - 1) * h x) : Stable u := by |
| intro η hη hηc |
| have hQ := positive_ground_state_identity hh hp hη hηc |
| have hn : ∀ x, h x ≠ 0 := fun x => (hp x).ne' |
| simp_rw [hEq, mul_div_cancel_right₀ _ (hn _)] at hQ |
| change 0 ≤ ∫ x, (‖gradient η x‖ ^ 2 + (3 * u x ^ 2 - 1) * η x ^ 2) |
| rw [hQ] |
| exact integral_nonneg (fun x => mul_nonneg (sq_nonneg _) (sq_nonneg _)) |
| |
| lemma coordDeriv_allenCahn_jacobi {n : ℕ} {u : Space n → ℝ} |
| (hu : ContDiff ℝ ∞ u) (hAC : AllenCahn u) (i : Fin n) (x : Space n) : |
| Laplacian.laplacian (coordDeriv u i) x = |
| (3 * u x ^ 2 - 1) * coordDeriv u i x := by |
| rw [laplacian_coordDeriv hu] |
| have he : Laplacian.laplacian u = fun x => u x ^ 3 - u x := funext hAC |
| rw [he] |
| have hd := hu.differentiable (by simp) |
| rw [coordDeriv_sub (hd.fun_pow 3) hd] |
| change fderiv ℝ (fun y => u y ^ 3) x _ - _ = _ |
| rw [fderiv_fun_pow 3 (hd x)] |
| simp only [smul_apply, smul_eq_mul, nsmul_eq_mul, Nat.cast_ofNat, Nat.reduceSub] |
| change 3 * u x ^ 2 * coordDeriv u i x - coordDeriv u i x = _ |
| ring |
| |
| theorem stable_of_coordinate_monotone {n : ℕ} {u : Space n → ℝ} |
| (hu : ContDiff ℝ 2 u) (hb : ∀ x, u x ∈ Set.Icc (-1 : ℝ) 1) |
| (hAC : AllenCahn u) (j : Fin n) |
| (hmon : ∀ x, 0 < fderiv ℝ u x (EuclideanSpace.single j 1)) : Stable u := by |
| have hs : ContDiff ℝ ∞ u := RegularitySupport.DeGiorgi.allenCahn_smooth hu hb hAC |
| apply stable_of_positive_jacobi (smooth_C2 (coordDeriv_smooth hs j)) |
| · intro x |
| change 0 < fderiv ℝ u x (EuclideanSpace.basisFun (Fin n) ℝ j) |
| simpa only [EuclideanSpace.basisFun_apply] using hmon x |
| · exact coordDeriv_allenCahn_jacobi hs hAC j |
| |
| end DeGiorgi |
| end |
| |
| end OAI |
| |
