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12dee50 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 | 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
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