| import OAI.Analysis.AllenCahn.Modica |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace DeGiorgi |
| open MeasureTheory |
| open scoped ENNReal |
| open Filter Topology |
|
|
| lemma regularized_inverse_lipschitz_gt_one {n : ℕ} {v : Space n → ℝ} |
| (hv : ContDiff ℝ 2 v) (hb : ∀ x, v x ∈ Set.Icc (-1 : ℝ) 1) |
| (hAC : AllenCahn v) {r L : ℝ} (hr : 0 < r) (hr1 : r < 1) (hL : 1 < L) : |
| ∀ x y, inverseProfile (fun z => r * v z) y - inverseProfile (fun z => r * v z) x ≤ |
| L * ‖y - x‖ := by |
| have hreg : ContDiff ℝ 2 (fun y => r * v y) := contDiff_const.mul hv |
| have hic (y : Space n) : r * v y ∈ Set.Icc (-r) r := by |
| have H := mul_le_mul_of_nonneg_left (hb y).1 hr.le |
| have H' := mul_le_mul_of_nonneg_left (hb y).2 hr.le |
| constructor <;> nlinarith |
| have hi (y : Space n) : r * v y ∈ Set.Ioo (-1 : ℝ) 1 := |
| ⟨by linarith [(hic y).1], by linarith [(hic y).2]⟩ |
| have hs : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num) |
| have hqr : 0 < 1 - r ^ 2 := weight_pos ⟨by linarith, hr1⟩ |
| have hL2 : 0 < L ^ 2 - 1 := by nlinarith |
| let B := Real.sqrt 2 * r |
| let M := Real.sqrt 2 * max |Real.artanh (-r)| |Real.artanh r| |
| let κ := (L ^ 2 - 1) * (1 - r ^ 2) |
| let b : Space n → ℝ := fun x => Real.sqrt 2 * (r * v x) |
| let h : Space n → ℝ := fun x => -Real.sqrt 2 * (r * v x) + |
| Real.sqrt 2 * (1 - r ^ 2) / r ^ 2 * ((r * v x) ^ 3 / (1 - (r * v x) ^ 2)) |
| apply lipschitz_of_laplacian_monotonicity |
| (b := b) (h := h) (M := M) (B := B) (κ := κ) |
| (inverseProfile_contDiff hreg hi) (by dsimp [M]; positivity) (by dsimp [B]; positivity) |
| (by linarith) (mul_pos hL2 hqr) |
| · intro x |
| exact inverseProfile_scalar_bound hr hr1 (hic x) |
| · intro x |
| dsimp [b, B] |
| rw [abs_mul, abs_of_pos hs] |
| exact mul_le_mul_of_nonneg_left (abs_le.mpr (hic x)) hs.le |
| · intro x |
| exact laplacian_regularized_inverse hv hb hAC hr hr1 x |
| · intro x y hxy |
| have ha : Real.artanh (r * v x) ≤ Real.artanh (r * v y) := |
| by |
| change Real.sqrt 2 * Real.artanh (r * v x) ≤ Real.sqrt 2 * Real.artanh (r * v y) at hxy |
| nlinarith only [hxy, hs] |
| have ht : r * v x ≤ r * v y := (Real.strictMonoOn_artanh.le_iff_le (hi x) (hi y)).mp ha |
| have H := inverseProfile_scalar_quantitative hr hr1 (hic x) (hic y) ht |
| have H' := mul_le_mul_of_nonneg_left H hL2.le |
| have HH := cube_div_weight_monotone (hi x) (hi y) ht |
| have hc : 0 ≤ Real.sqrt 2 * (1 - r ^ 2) / r ^ 2 := by positivity |
| have HH' := mul_le_mul_of_nonneg_left HH hc |
| dsimp [κ, b, h, inverseProfile] at * |
| nlinarith |
|
|
| lemma regularized_inverse_lipschitz {n : ℕ} {v : Space n → ℝ} |
| (hv : ContDiff ℝ 2 v) (hb : ∀ x, v x ∈ Set.Icc (-1 : ℝ) 1) |
| (hAC : AllenCahn v) {r : ℝ} (hr : 0 < r) (hr1 : r < 1) (x y : Space n) : |
| |inverseProfile (fun z => r * v z) y - inverseProfile (fun z => r * v z) x| ≤ ‖y - x‖ := by |
| have H (a c : Space n) : inverseProfile (fun z => r * v z) c - |
| inverseProfile (fun z => r * v z) a ≤ ‖c - a‖ := by |
| apply le_of_forall_pos_le_add |
| intro ε hε |
| have hh := regularized_inverse_lipschitz_gt_one hv hb hAC hr hr1 |
| (show 1 < 1 + ε / (‖c - a‖ + 1) from lt_add_of_pos_right 1 (div_pos hε (by positivity))) a c |
| have hn := norm_nonneg (c - a) |
| have hratio : ε / (‖c - a‖ + 1) * ‖c - a‖ ≤ ε := by |
| rw [div_mul_eq_mul_div, div_le_iff₀ (by positivity)] |
| nlinarith |
| nlinarith |
| apply abs_le.mpr |
| constructor |
| · have hh := H y x |
| rw [norm_sub_rev x y] at hh |
| linarith |
| · exact H x y |
|
|
| lemma regularized_inverse_gradient_bound {n : ℕ} {v : Space n → ℝ} |
| (hv : ContDiff ℝ 2 v) (hb : ∀ x, v x ∈ Set.Icc (-1 : ℝ) 1) |
| (hAC : AllenCahn v) {r : ℝ} (hr : 0 < r) (hr1 : r < 1) (x : Space n) : |
| ‖gradient (inverseProfile (fun z => r * v z)) x‖ ≤ 1 := by |
| have H := norm_fderiv_le_of_lip' ℝ (f := inverseProfile (fun z => r * v z)) |
| (x₀ := x) (show (0 : ℝ) ≤ 1 by norm_num) |
| (Filter.Eventually.of_forall (fun y => by |
| simpa only [Real.norm_eq_abs, one_mul] using regularized_inverse_lipschitz hv hb hAC hr hr1 x y)) |
| simpa only [gradient, LinearIsometryEquiv.norm_map] using H |
|
|
| theorem modica_linear {n : ℕ} {v : Space n → ℝ} |
| (hv : ContDiff ℝ 2 v) (hb : ∀ x, v x ∈ Set.Icc (-1 : ℝ) 1) |
| (hAC : AllenCahn v) (x : Space n) : |
| Real.sqrt 2 * ‖gradient v x‖ ≤ 1 - v x ^ 2 := by |
| have H (r : ℝ) (hr : r ∈ Set.Ioo (0 : ℝ) 1) : |
| Real.sqrt 2 * r * ‖gradient v x‖ ≤ 1 - (r * v x) ^ 2 := by |
| have hi : r * v x ∈ Set.Ioo (-1 : ℝ) 1 := by |
| have ha := mul_le_mul_of_nonneg_left (hb x).1 hr.1.le |
| have hb' := mul_le_mul_of_nonneg_left (hb x).2 hr.1.le |
| constructor <;> nlinarith [hr.2] |
| have hq := weight_pos hi |
| have hd : DifferentiableAt ℝ (fun z => r * v z) x := |
| (hv.differentiable (by norm_num) x).const_mul r |
| have hg : gradient (fun z => r * v z) x = r • gradient v x := by |
| simp [gradient, fderiv_const_mul (hv.differentiable (by norm_num) x)] |
| have hh := regularized_inverse_gradient_bound hv hb hAC hr.1 hr.2 x |
| rw [gradient_inverseProfile hd hi, hg, norm_smul, norm_smul, |
| Real.norm_eq_abs, Real.norm_eq_abs, abs_of_pos hr.1] at hh |
| have hcoeff : 0 ≤ Real.sqrt 2 / defectWeight (fun z => r * v z) x := by |
| exact div_nonneg (Real.sqrt_nonneg _) hq.le |
| rw [abs_of_nonneg hcoeff] at hh |
| dsimp [defectWeight] at hh |
| rw [div_mul_eq_mul_div, div_le_iff₀ hq] at hh |
| nlinarith |
| have hc : IsClosed {r : ℝ | Real.sqrt 2 * r * ‖gradient v x‖ ≤ 1 - (r * v x) ^ 2} := |
| isClosed_le (by fun_prop) (by fun_prop) |
| have hsub : closure (Set.Ioo (0 : ℝ) 1) ⊆ |
| {r : ℝ | Real.sqrt 2 * r * ‖gradient v x‖ ≤ 1 - (r * v x) ^ 2} := |
| closure_minimal H hc |
| have hmem : (1 : ℝ) ∈ closure (Set.Ioo (0 : ℝ) 1) := by |
| rw [closure_Ioo (by norm_num : (0 : ℝ) ≠ 1)] |
| exact ⟨zero_le_one, le_rfl⟩ |
| simpa using hsub hmem |
|
|
| theorem modica {n : ℕ} {v : Space n → ℝ} |
| (hv : ContDiff ℝ 2 v) (hb : ∀ x, v x ∈ Set.Icc (-1 : ℝ) 1) |
| (hAC : AllenCahn v) (x : Space n) : |
| ‖gradient v x‖ ^ 2 / 2 ≤ potential (v x) := by |
| have H := modica_linear hv hb hAC x |
| have hs := pow_le_pow_left₀ (mul_nonneg (Real.sqrt_nonneg _) (norm_nonneg _)) H 2 |
| rw [mul_pow, Real.sq_sqrt (by norm_num)] at hs |
| dsimp [potential] |
| linarith |
|
|
| theorem diffuseDefect_nonneg {n : ℕ} {v : Space n → ℝ} |
| (hv : ContDiff ℝ 2 v) (hi : ∀ x, v x ∈ Set.Ioo (-1 : ℝ) 1) |
| (hAC : AllenCahn v) (x : Space n) : 0 ≤ diffuseDefect v x := by |
| have hb (y : Space n) : v y ∈ Set.Icc (-1 : ℝ) 1 := ⟨(hi y).1.le, (hi y).2.le⟩ |
| have hq : 0 < defectWeight v x := weight_pos (hi x) |
| have hnorm : ‖gradient (inverseProfile v) x‖ ≤ 1 := by |
| rw [gradient_inverseProfile (hv.differentiable (by norm_num) x) (hi x), norm_smul, |
| Real.norm_eq_abs, abs_of_nonneg (div_nonneg (Real.sqrt_nonneg _) hq.le), |
| div_mul_eq_mul_div, div_le_iff₀ hq] |
| simpa only [one_mul, defectWeight] using modica_linear hv hb hAC x |
| dsimp [diffuseDefect] |
| nlinarith [norm_nonneg (gradient (inverseProfile v) x)] |
|
|
| end DeGiorgi |
| end |
|
|
| noncomputable section |
| namespace DeGiorgi |
| open MeasureTheory |
| open scoped ENNReal |
| open Filter Topology |
| open scoped ContDiff |
|
|
| lemma smooth_C2 {n : ℕ} {f : Space n → ℝ} (hf : ContDiff ℝ ∞ f) : |
| ContDiff ℝ 2 f := hf.of_le (WithTop.coe_le_coe.mpr le_top) |
|
|
| lemma coordDeriv_smooth {n : ℕ} {f : Space n → ℝ} |
| (hf : ContDiff ℝ ∞ f) (i : Fin n) : ContDiff ℝ ∞ (coordDeriv f i) := by |
| exact (hf.fderiv_right (m := ∞) (by simp)).clm_apply contDiff_const |
|
|
| lemma coordDeriv_sum {n : ℕ} {ι : Type*} (s : Finset ι) {f : ι → Space n → ℝ} |
| (hf : ∀ i ∈ s, Differentiable ℝ (f i)) (j : Fin n) (x : Space n) : |
| coordDeriv (fun y => ∑ i ∈ s, f i y) j x = ∑ i ∈ s, coordDeriv (f i) j x := by |
| simp only [coordDeriv, fderiv_fun_sum (fun i hi => hf i hi x), sum_apply] |
|
|
| lemma coordDeriv_comm {n : ℕ} {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) |
| (i j : Fin n) : coordDeriv (coordDeriv f i) j = coordDeriv (coordDeriv f j) i := by |
| funext x |
| have hs := (hf.contDiffAt (x := x)).isSymmSndFDerivAt (by simp : minSmoothness ℝ 2 ≤ (2 : ℕ∞ω)) |
| have hd := (hf.fderiv_right (m := 1) (by norm_num)).differentiable (by norm_num) x |
| change fderiv ℝ (fun y => fderiv ℝ f y (basis n i)) x (basis n j) = |
| fderiv ℝ (fun y => fderiv ℝ f y (basis n j)) x (basis n i) |
| simp [fderiv_clm_apply hd (differentiableAt_const _)] |
| exact hs _ _ |
|
|
| lemma laplacian_coordDeriv {n : ℕ} {f : Space n → ℝ} |
| (hf : ContDiff ℝ ∞ f) (i : Fin n) (x : Space n) : |
| Laplacian.laplacian (coordDeriv f i) x = coordDeriv (Laplacian.laplacian f) i x := by |
| have hf2 := smooth_C2 hf |
| have he : Laplacian.laplacian f = fun y => ∑ j : Fin n, coordDeriv (coordDeriv f j) j y := |
| funext (laplacian_eq_sum_coordDeriv hf2) |
| rw [laplacian_eq_sum_coordDeriv (smooth_C2 (coordDeriv_smooth hf i)), he, |
| coordDeriv_sum _ (fun j _ => (coordDeriv_smooth (coordDeriv_smooth hf j) j).differentiable (by simp))] |
| apply Finset.sum_congr rfl |
| intro j _ |
| rw [coordDeriv_comm hf2 i j, |
| coordDeriv_comm (smooth_C2 (coordDeriv_smooth hf j)) i j] |
|
|
| lemma laplacian_sum {n : ℕ} {ι : Type*} (s : Finset ι) {f : ι → Space n → ℝ} |
| (hf : ∀ i ∈ s, ContDiff ℝ 2 (f i)) (x : Space n) : |
| Laplacian.laplacian (fun y => ∑ i ∈ s, f i y) x = ∑ i ∈ s, Laplacian.laplacian (f i) x := by |
| rw [laplacian_eq_sum_coordDeriv (ContDiff.sum (fun i hi => hf i hi))] |
| have he (j : Fin n) : coordDeriv (fun y => ∑ i ∈ s, f i y) j = |
| fun y => ∑ i ∈ s, coordDeriv (f i) j y := |
| funext (coordDeriv_sum s (fun i hi => (hf i hi).differentiable (by norm_num)) j) |
| simp_rw [he] |
| simp_rw [coordDeriv_sum _ (fun i hi => (coordDeriv_contDiff (hf i hi) _).differentiable (by norm_num))] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro i hi |
| exact (laplacian_eq_sum_coordDeriv (hf i hi) x).symm |
|
|
| lemma laplacian_sq {n : ℕ} {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) (x : Space n) : |
| Laplacian.laplacian (fun y => f y ^ 2) x = |
| 2 * ‖gradient f x‖ ^ 2 + 2 * f x * Laplacian.laplacian f x := by |
| have hd := hf.differentiable (by norm_num) |
| have hfirst (i : Fin n) : coordDeriv (fun y => f y ^ 2) i = |
| fun y => 2 * f y * coordDeriv f i y := by |
| funext y |
| simp [coordDeriv, fderiv_fun_pow 2 (hd y), mul_assoc] |
| have hsecond (i : Fin n) : coordDeriv (fun y => 2 * f y * coordDeriv f i y) i x = |
| 2 * (coordDeriv f i x) ^ 2 + 2 * f x * coordDeriv (coordDeriv f i) i x := by |
| rw [coordDeriv_mul (hd.const_mul 2) ((coordDeriv_contDiff hf i).differentiable (by norm_num))] |
| simp only [coordDeriv, fderiv_const_mul (hd x), smul_apply, smul_eq_mul] |
| ring |
| rw [laplacian_eq_sum_coordDeriv (hf.pow 2)] |
| simp_rw [hfirst, hsecond] |
| rw [Finset.sum_add_distrib, ← Finset.mul_sum, ← Finset.mul_sum, |
| ← gradient_norm_sq, ← laplacian_eq_sum_coordDeriv hf] |
|
|
| lemma laplacian_gradient_norm_sq {n : ℕ} {f : Space n → ℝ} |
| (hf : ContDiff ℝ ∞ f) (x : Space n) : |
| Laplacian.laplacian (fun y => ‖gradient f y‖ ^ 2) x = |
| 2 * (∑ i : Fin n, ‖gradient (coordDeriv f i) x‖ ^ 2) + |
| 2 * (∑ i : Fin n, coordDeriv f i x * coordDeriv (Laplacian.laplacian f) i x) := by |
| have he : (fun y => ‖gradient f y‖ ^ 2) = fun y => ∑ i : Fin n, coordDeriv f i y ^ 2 := |
| funext (gradient_norm_sq f) |
| rw [he, laplacian_sum _ (fun i _ => (smooth_C2 (coordDeriv_smooth hf i)).pow 2)] |
| simp_rw [laplacian_sq (smooth_C2 (coordDeriv_smooth hf _)), laplacian_coordDeriv hf] |
| rw [Finset.sum_add_distrib, ← Finset.mul_sum, Finset.mul_sum] |
| congr 1 |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
|
|
| lemma inverseProfile_smooth {n : ℕ} {v : Space n → ℝ} (hv : ContDiff ℝ ∞ v) |
| (hi : ∀ x, v x ∈ Set.Ioo (-1 : ℝ) 1) : ContDiff ℝ ∞ (inverseProfile v) := by |
| have hm (x : Space n) : 1 - v x ≠ 0 := ne_of_gt (sub_pos.mpr (hi x).2) |
| have hp (x : Space n) : 1 + v x ≠ 0 := ne_of_gt (by linarith [(hi x).1]) |
| have hq : ContDiff ℝ ∞ (fun x => (1 + v x) / (1 - v x)) := |
| (contDiff_const.add hv).div (contDiff_const.sub hv) hm |
| have hl : ContDiff ℝ ∞ (fun x => (Real.sqrt 2 / 2) * Real.log ((1 + v x) / (1 - v x))) := |
| contDiff_const.mul (hq.log (fun x => div_ne_zero (hp x) (hm x))) |
| convert hl using 1 |
| funext x |
| dsimp [inverseProfile] |
| rw [Real.artanh_eq_half_log ⟨(hi x).1.le, (hi x).2.le⟩] |
| ring |
|
|
| lemma gradient_norm_sq_smooth {n : ℕ} {f : Space n → ℝ} (hf : ContDiff ℝ ∞ f) : |
| ContDiff ℝ ∞ (fun x => ‖gradient f x‖ ^ 2) := by |
| have he : (fun x => ‖gradient f x‖ ^ 2) = fun x => ∑ i : Fin n, coordDeriv f i x ^ 2 := |
| funext (gradient_norm_sq f) |
| rw [he] |
| exact ContDiff.sum (fun i _ => (coordDeriv_smooth hf i).pow 2) |
|
|
| lemma diffuseDefect_smooth {n : ℕ} {v : Space n → ℝ} (hv : ContDiff ℝ ∞ v) |
| (hi : ∀ x, v x ∈ Set.Ioo (-1 : ℝ) 1) : ContDiff ℝ ∞ (diffuseDefect v) := |
| contDiff_const.sub (gradient_norm_sq_smooth (inverseProfile_smooth hv hi)) |
|
|
| lemma coordDeriv_profile {n : ℕ} {v : Space n → ℝ} (hv : Differentiable ℝ v) |
| (hi : ∀ x, v x ∈ Set.Ioo (-1 : ℝ) 1) (i : Fin n) (x : Space n) : |
| coordDeriv v i x = defectWeight v x / Real.sqrt 2 * coordDeriv (inverseProfile v) i x := by |
| have hq : defectWeight v x ≠ 0 := ne_of_gt (weight_pos (hi x)) |
| have hs : Real.sqrt 2 ≠ 0 := (Real.sqrt_pos.mpr (by norm_num)).ne' |
| dsimp only [coordDeriv] |
| rw [fderiv_inverseProfile (hv x) (hi x)] |
| simp only [smul_apply, smul_eq_mul] |
| field_simp |
|
|
| lemma inverseProfile_laplacian_eq {n : ℕ} {v : Space n → ℝ} |
| (hv : ContDiff ℝ 2 v) (hi : ∀ x, v x ∈ Set.Ioo (-1 : ℝ) 1) |
| (hpde : AllenCahn v) (x : Space n) : |
| Laplacian.laplacian (inverseProfile v) x = -Real.sqrt 2 * v x * diffuseDefect v x := by |
| have hq : 0 < defectWeight v x := weight_pos (hi x) |
| have hs : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num) |
| rw [laplacian_inverseProfile hv hi, hpde] |
| dsimp only [diffuseDefect] |
| rw [gradient_inverseProfile (hv.differentiable (by norm_num) x) (hi x), norm_smul, |
| Real.norm_eq_abs, abs_of_pos (div_pos hs hq)] |
| dsimp only [defectWeight] at * |
| field_simp |
| rw [Real.sq_sqrt (by norm_num)] |
| ring |
|
|
| lemma inner_gradient_eq_sum {n : ℕ} (f g : Space n → ℝ) (x : Space n) : |
| inner ℝ (gradient f x) (gradient g x) = |
| ∑ i : Fin n, coordDeriv f i x * coordDeriv g i x := by |
| have h := (basis n).sum_inner_mul_inner (gradient f x) (gradient g x) |
| simpa only [coordDeriv, real_inner_comm, inner_gradient_left] using h.symm |
|
|
| lemma diffuseDefect_laplacian {n : ℕ} {v : Space n → ℝ} |
| (hv : ContDiff ℝ ∞ v) (hi : ∀ x, v x ∈ Set.Ioo (-1 : ℝ) 1) |
| (hpde : AllenCahn v) (x : Space n) : |
| Laplacian.laplacian (diffuseDefect v) x = |
| -2 * (∑ i : Fin n, ‖gradient (coordDeriv (inverseProfile v) i) x‖ ^ 2) + |
| 2 * Real.sqrt 2 * v x * inner ℝ (gradient (inverseProfile v) x) (gradient (diffuseDefect v) x) + |
| 2 * defectWeight v x * diffuseDefect v x * ‖gradient (inverseProfile v) x‖ ^ 2 := by |
| have hU := inverseProfile_smooth hv hi |
| have hD := diffuseDefect_smooth hv hi |
| have hdv := hv.differentiable (by simp) |
| have hdD := hD.differentiable (by simp) |
| have he : Laplacian.laplacian (inverseProfile v) = |
| fun y => -Real.sqrt 2 * v y * diffuseDefect v y := |
| funext (inverseProfile_laplacian_eq (smooth_C2 hv) hi hpde) |
| have hd (i : Fin n) : coordDeriv (Laplacian.laplacian (inverseProfile v)) i x = |
| -Real.sqrt 2 * v x * coordDeriv (diffuseDefect v) i x - |
| defectWeight v x * diffuseDefect v x * coordDeriv (inverseProfile v) i x := by |
| rw [he, coordDeriv_mul (hdv.const_mul _) hdD] |
| have hh : coordDeriv (fun y => -Real.sqrt 2 * v y) i x = |
| -Real.sqrt 2 * coordDeriv v i x := by |
| simp [coordDeriv, fderiv_const_mul (hdv x)] |
| rw [hh, coordDeriv_profile hdv hi] |
| have hs : Real.sqrt 2 ≠ 0 := (Real.sqrt_pos.mpr (by norm_num)).ne' |
| field_simp |
| ring |
| have hsub : Laplacian.laplacian (diffuseDefect v) x = |
| -Laplacian.laplacian (fun y => ‖gradient (inverseProfile v) y‖ ^ 2) x := by |
| change Laplacian.laplacian ((fun _ : Space n => (1 : ℝ)) - |
| (fun y => ‖gradient (inverseProfile v) y‖ ^ 2)) x = _ |
| rw [contDiff_const.contDiffAt.laplacian_sub (smooth_C2 (gradient_norm_sq_smooth hU)).contDiffAt, |
| InnerProductSpace.laplacian_const] |
| simp |
| rw [hsub, laplacian_gradient_norm_sq hU] |
| simp_rw [hd] |
| rw [inner_gradient_eq_sum, gradient_norm_sq] |
| have hsum : ∑ i : Fin n, coordDeriv (inverseProfile v) i x * |
| (-Real.sqrt 2 * v x * coordDeriv (diffuseDefect v) i x - |
| defectWeight v x * diffuseDefect v x * coordDeriv (inverseProfile v) i x) = |
| -Real.sqrt 2 * v x * (∑ i : Fin n, coordDeriv (inverseProfile v) i x * coordDeriv (diffuseDefect v) i x) - |
| defectWeight v x * diffuseDefect v x * (∑ i : Fin n, coordDeriv (inverseProfile v) i x ^ 2) := by |
| rw [Finset.mul_sum, Finset.mul_sum, ← Finset.sum_sub_distrib] |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
| rw [hsum] |
| ring |
|
|
| lemma inverseProfile_gradient_bound {n : ℕ} {v : Space n → ℝ} |
| (hv : ContDiff ℝ 2 v) (hi : ∀ x, v x ∈ Set.Ioo (-1 : ℝ) 1) |
| (hAC : AllenCahn v) (x : Space n) : ‖gradient (inverseProfile v) x‖ ≤ 1 := by |
| have h := diffuseDefect_nonneg hv hi hAC x |
| dsimp only [diffuseDefect] at h |
| nlinarith [norm_nonneg (gradient (inverseProfile v) x)] |
|
|
| lemma diffuseDrift_bound {n : ℕ} {v : Space n → ℝ} |
| (hv : ContDiff ℝ 2 v) (hi : ∀ x, v x ∈ Set.Ioo (-1 : ℝ) 1) |
| (hAC : AllenCahn v) (x : Space n) : |
| ‖(2 * Real.sqrt 2 * v x) • gradient (inverseProfile v) x‖ ≤ 2 * Real.sqrt 2 := by |
| have hvabs : |v x| ≤ 1 := abs_le.mpr ⟨(hi x).1.le, (hi x).2.le⟩ |
| rw [norm_smul, norm_mul, Real.norm_of_nonneg (mul_nonneg (by norm_num) (Real.sqrt_nonneg _)), Real.norm_eq_abs] |
| calc 2 * Real.sqrt 2 * |v x| * ‖gradient (inverseProfile v) x‖ ≤ |
| 2 * Real.sqrt 2 * 1 * 1 := mul_le_mul |
| (mul_le_mul_of_nonneg_left hvabs (by positivity)) |
| (inverseProfile_gradient_bound hv hi hAC x) (norm_nonneg _) (by positivity) |
| _ = _ := by ring |
|
|
| end DeGiorgi |
| end |
|
|
| end OAI |
|
|
