| import OAI.Geometry.Perimeter.Regularity.EllipticContraction |
| import OAI.Geometry.Perimeter.Regularity.EllipticCompactGradient |
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| namespace OAI |
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| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal ContDiff InnerProductSpace |
| namespace RPISingularHarnack |
| variable {n : ℕ} |
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| theorem C1_limit_of_uniform_gradients {f : ℕ → Euc n → ℝ} {g : Euc n → ℝ} |
| (hf : ∀ i, ContDiff ℝ 1 (f i)) {p : Euc n} {r : ℝ} |
| {W : Euc n → Euc n} (hW : ContinuousOn W (Metric.closedBall p r)) |
| (hgrad : TendstoUniformlyOn (fun i ↦ gradient (f i)) W atTop (Metric.closedBall p r)) |
| (hlim : ∀ x∈Metric.ball p r, Tendsto (fun i ↦ f i x) atTop (𝓝 (g x))) : |
| ContDiffOn ℝ 1 g (Metric.ball p r) ∧ ∀ x∈Metric.ball p r, gradient g x=W x := by |
| let D : Euc n → Euc n →L[ℝ] ℝ := fun x ↦ InnerProductSpace.toDual ℝ (Euc n) (W x) |
| have hDlim : TendstoUniformlyOn (fun i ↦ fderiv ℝ (f i)) D atTop (Metric.ball p r) := by |
| rw [Metric.tendstoUniformlyOn_iff] |
| intro ε hε |
| filter_upwards [(Metric.tendstoUniformlyOn_iff.mp hgrad) ε hε] with i hi x hx |
| have hh := hi x (Metric.ball_subset_closedBall hx) |
| change dist (InnerProductSpace.toDual ℝ (Euc n) (W x)) (fderiv ℝ (f i) x)<ε |
| rw [show fderiv ℝ (f i) x=InnerProductSpace.toDual ℝ (Euc n) (gradient (f i) x) from |
| (LinearIsometryEquiv.apply_symm_apply _ _).symm, |
| (InnerProductSpace.toDual ℝ (Euc n)).isometry.dist_eq] |
| exact hh |
| have hD (x : Euc n) (hx : x∈Metric.ball p r) : HasFDerivAt g (D x) x := |
| hasFDerivAt_of_tendstoUniformlyOn Metric.isOpen_ball hDlim |
| (fun i x _ ↦ (hf i).differentiable (by simp) x |>.hasFDerivAt) hlim hx |
| constructor |
| · change ContDiffOn ℝ (0+1) g (Metric.ball p r) |
| rw [contDiffOn_succ_iff_hasFDerivWithinAt_of_uniqueDiffOn Metric.isOpen_ball.uniqueDiffOn] |
| refine ⟨by simp,D,?_,fun x hx ↦ (hD x hx).hasFDerivWithinAt⟩ |
| apply contDiffOn_zero.mpr |
| exact (InnerProductSpace.toDual ℝ (Euc n)).continuous.comp_continuousOn |
| (hW.mono Metric.ball_subset_closedBall) |
| · intro x hx |
| rw [gradient,(hD x hx).fderiv] |
| exact LinearIsometryEquiv.symm_apply_apply _ _ |
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| theorem C1_limit_of_geometric_gradient_estimates {f : ℕ → Euc n → ℝ} {g : Euc n → ℝ} |
| (hf : ∀ i, ContDiff ℝ 1 (f i)) {p : Euc n} {r θ R C B : ℝ} |
| (hθ : 0<θ) (hR : 0<R) |
| (hb : ∀ i x, x∈Metric.closedBall p r → ‖gradient (f i) x‖≤B) |
| (hmod : ∀ i j x, x∈Metric.closedBall p r → ∀ y, y∈Metric.closedBall p r → |
| dist x y≤θ^j*R → ‖gradient (f i) x-gradient (f i) y‖≤C*(1/2)^j) |
| (hlim : ∀ x∈Metric.ball p r, Tendsto (fun i ↦ f i x) atTop (𝓝 (g x))) : |
| ContDiffOn ℝ 1 g (Metric.ball p r) ∧ |
| ∀ j x, x∈Metric.ball p r → ∀ y, y∈Metric.ball p r → dist x y≤θ^j*R → |
| ‖gradient g x-gradient g y‖≤C*(1/2)^j := by |
| obtain ⟨W,hW,σ,hσ,hconv⟩ := compact_field_subsequence (isCompact_closedBall p r) |
| (fun i ↦ (gradient_continuous_C1 (hf i)).continuousOn) hθ hR hb hmod |
| obtain ⟨hg,hgW⟩ := C1_limit_of_uniform_gradients (fun i ↦ hf (σ i)) hW hconv |
| (fun x hx ↦ (hlim x hx).comp hσ.tendsto_atTop) |
| refine ⟨hg,fun j x hx y hy hxy ↦ ?_⟩ |
| rw [hgW x hx,hgW y hy] |
| have ht := ((hconv.tendsto_at (Metric.ball_subset_closedBall hx)).sub |
| (hconv.tendsto_at (Metric.ball_subset_closedBall hy))).norm |
| exact le_of_tendsto ht (Eventually.of_forall (fun i ↦ |
| hmod (σ i) j x (Metric.ball_subset_closedBall hx) y (Metric.ball_subset_closedBall hy) hxy)) |
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| end RPISingularHarnack |
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| end |
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| end OAI |
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