| import OAI.Analysis.AllenCahn.DefectCalculus |
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| namespace OAI |
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| noncomputable section |
| namespace DeGiorgi |
| open MeasureTheory |
| open scoped ENNReal |
| open Filter Topology |
| open scoped ContDiff |
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| lemma diffuseDefect_drift_inequality {n : ℕ} {v : Space n → ℝ} |
| (hv : ContDiff ℝ ∞ v) (hi : ∀ x, v x ∈ Set.Ioo (-1 : ℝ) 1) |
| (hAC : AllenCahn v) (x : Space n) : |
| Laplacian.laplacian (diffuseDefect v) x ≤ 2 * diffuseDefect v x + |
| fderiv ℝ (diffuseDefect v) x ((2 * Real.sqrt 2 * v x) • gradient (inverseProfile v) x) := by |
| have hD := diffuseDefect_nonneg (smooth_C2 hv) hi hAC x |
| have hp := inverseProfile_gradient_bound (smooth_C2 hv) hi hAC x |
| have hpn := norm_nonneg (gradient (inverseProfile v) x) |
| have hp2 : ‖gradient (inverseProfile v) x‖ ^ 2 ≤ 1 := by nlinarith |
| have hq : defectWeight v x ≤ 1 := by dsimp [defectWeight]; nlinarith [sq_nonneg (v x)] |
| have hq0 : 0 ≤ defectWeight v x := (weight_pos (hi x)).le |
| have hh : 0 ≤ ∑ i : Fin n, ‖gradient (coordDeriv (inverseProfile v) i) x‖ ^ 2 := |
| Finset.sum_nonneg (fun i _ => sq_nonneg _) |
| have hprod : defectWeight v x * ‖gradient (inverseProfile v) x‖ ^ 2 ≤ 1 := by |
| calc _ ≤ 1 * 1 := mul_le_mul hq hp2 (sq_nonneg _) (by norm_num) |
| _ = 1 := one_mul _ |
| have H := mul_le_mul_of_nonneg_right hprod hD |
| rw [diffuseDefect_laplacian hv hi hAC, map_smul, smul_eq_mul, ← inner_gradient_left] |
| rw [real_inner_comm (gradient (diffuseDefect v) x)] |
| nlinarith |
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| end DeGiorgi |
| end |
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| end OAI |
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