| import OAI.Geometry.Immersion.ClosedSurface.OscillationBounds |
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| namespace OAI |
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| noncomputable section |
| open Set Complex Bundle Manifold |
| open scoped ContDiff Matrix Topology Manifold BigOperators |
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| namespace ClosedSurfaceR4.RealModes |
| open ClosedSurfaceR4.SmallModes ClosedSurfaceR4.WeightedEstimates |
| open ClosedSurfaceR4.QuadraticMean (realPartCLM) |
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| lemma norm_realPartCLM_le (n : ℕ) : ‖realPartCLM n‖ ≤ 1 := by |
| apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one |
| intro Z |
| rw [one_mul] |
| exact (pi_norm_le_iff_of_nonneg (norm_nonneg Z)).2 fun i => |
| ((Complex.abs_re_le_norm (Z i)).trans (norm_le_pi_norm Z i)) |
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| def realOsc (τ : ℝ) {n : ℕ} (Z : Field n) : RField n := |
| fun p => QuadraticMean.realPart (oscillate τ Z p) |
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| lemma realOsc_eq_displacement (τ : ℝ) {n : ℕ} (Z : Field n) : |
| realOsc τ Z = ClosedSurfaceR4.QuadraticMean.displacement τ Prod.fst Z := by |
| funext p |
| unfold realOsc ClosedSurfaceR4.QuadraticMean.displacement |
| ClosedSurfaceR4.QuadraticMean.realMode oscillate |
| congr 2 |
| simp only [unitMode, ClosedSurfaceR4.QuadraticMean.phase, Complex.ofReal_div] |
| congr 1 |
| ring |
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| lemma contDiffOn_realOsc {n : ℕ} {U : Set Base} {Z : Field n} |
| (hZ : ContDiffOn ℝ ∞ Z U) (τ : ℝ) : ContDiffOn ℝ ∞ (realOsc τ Z) U := |
| (realPartCLM n).contDiff.comp_contDiffOn (contDiffOn_oscillate hZ τ) |
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| lemma weighted_realPart {n : ℕ} {U : Set Base} (hU : UniqueDiffOn ℝ U) |
| {s C : ℝ} {m : ℕ} {Z : Field n} (hs : 0 ≤ s) (hC : 0 ≤ C) |
| (hZ : ContDiffOn ℝ ∞ Z U) (hb : WeightedBound U s m C Z) : |
| WeightedBound U s m C (fun p => QuadraticMean.realPart (Z p)) := by |
| have hh := hb.linear hU hs hZ (realPartCLM n) |
| exact hh.mono_const (mul_le_of_le_one_left hC (norm_realPartCLM_le n)) |
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| lemma weighted_realOsc {n : ℕ} {U : Set Base} (hU : IsOpen U) |
| {τ s C : ℝ} {m : ℕ} {Z : Field n} (hτ : 0 < τ) (hτs : τ ≤ s) |
| (hC : 0 ≤ C) (hZ : ContDiffOn ℝ ∞ Z U) (hb : WeightedBound U s m C Z) : |
| WeightedBound U τ m (2 ^ m * C) (realOsc τ Z) := |
| weighted_realPart hU.uniqueDiffOn hτ.le (mul_nonneg (by positivity) hC) |
| (contDiffOn_oscillate hZ τ) (weighted_oscillate hU hτ hτs hC hZ hb) |
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| lemma coordDeriv_realPart {n : ℕ} {Z : Field n} {p : Base} |
| (hZ : DifferentiableAt ℝ Z p) (v : Base) : |
| coordDeriv v (fun q => QuadraticMean.realPart (Z q)) p = QuadraticMean.realPart (coordDeriv v Z p) := by |
| exact congrArg (fun A : Base →L[ℝ] RVec n => A v) |
| (((realPartCLM n).hasFDerivAt.comp p hZ.hasFDerivAt).fderiv) |
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| lemma complexify_dot_realPart {n : ℕ} (X : RVec n) (Z : Ambient n) : |
| (complexify X ⬝ᵥ Z).re = X ⬝ᵥ QuadraticMean.realPart Z := by |
| simp [dotProduct, QuadraticMean.realPart, Complex.mul_re] |
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| def realMetric {n : ℕ} (F : RField n) (v w p : Base) : ℝ := |
| coordDeriv v F p ⬝ᵥ coordDeriv w F p |
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| def realLinearized {n : ℕ} (F U : RField n) (v w p : Base) : ℝ := |
| coordDeriv v F p ⬝ᵥ coordDeriv w U p + coordDeriv w F p ⬝ᵥ coordDeriv v U p |
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| lemma realMetric_add {n : ℕ} {F U : RField n} {p : Base} |
| (hF : DifferentiableAt ℝ F p) (hU : DifferentiableAt ℝ U p) (v w : Base) : |
| realMetric (fun q => F q + U q) v w p = |
| realMetric F v w p + realLinearized F U v w p + realMetric U v w p := by |
| unfold realMetric realLinearized |
| rw [partial_add hF hU v, partial_add hF hU w] |
| simp only [add_dotProduct, dotProduct_add] |
| rw [dotProduct_comm (coordDeriv v U p)] |
| ring |
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| lemma realLinearized_realPart {n : ℕ} {F : RField n} {Z : Field n} {p : Base} |
| (hF : DifferentiableAt ℝ F p) (hZ : DifferentiableAt ℝ Z p) (v w : Base) : |
| realLinearized F (fun q => QuadraticMean.realPart (Z q)) v w p = |
| (linearizedMetric (fun q => complexify (F q)) Z v w p).re := by |
| simp only [realLinearized, linearizedMetric, coordDeriv_realPart hZ, |
| coordDeriv_complexify hF, Complex.add_re, complexify_dot_realPart] |
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| lemma realLinearized_realOsc {n : ℕ} {F : RField n} {Z : Field n} {p : Base} |
| (hF : DifferentiableAt ℝ F p) (hZ : DifferentiableAt ℝ Z p) (τ : ℝ) (v w : Base) : |
| realLinearized F (realOsc τ Z) v w p = |
| (unitMode τ p * conjugatedMetric τ (fun q => complexify (F q)) Z v w p).re := by |
| have hO : DifferentiableAt ℝ (oscillate τ Z) p := by |
| unfold oscillate |
| exact (unitMode_hasFDerivAt τ p).differentiableAt.smul hZ |
| unfold realOsc |
| rw [realLinearized_realPart hF hO, linearizedMetric_oscillate τ hZ] |
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| def realMetricTensor {n : ℕ} (F : RField n) : Base → Fin 3 → ℝ := |
| fun p => ![realMetric F dx dx p, realMetric F dx dy p, realMetric F dy dy p] |
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| def realLinearizedTensor {n : ℕ} (F U : RField n) : Base → Fin 3 → ℝ := |
| fun p => ![realLinearized F U dx dx p, realLinearized F U dx dy p, |
| realLinearized F U dy dy p] |
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| lemma realLinearized_residual {n : ℕ} {F : RField n} {Z : Field n} {p : Base} |
| (hF : DifferentiableAt ℝ F p) (hZ : DifferentiableAt ℝ Z p) (τ : ℝ) (f : Tensor) : |
| realLinearizedTensor F (realOsc τ Z) p - realOsc τ f p = |
| realOsc τ (SmallModes.residual τ (fun q => complexify (F q)) f Z) p := by |
| ext i |
| fin_cases i <;> |
| simp [realLinearizedTensor, realLinearized_realOsc hF hZ, realOsc, QuadraticMean.realPart, |
| oscillate, SmallModes.residual, conjugatedD, tensorOf, mul_sub, Matrix.vecHead, Matrix.vecTail] |
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| theorem real_finite_forced_mode {n : ℕ} {F : RField n} (hF : ContDiff ℝ ∞ F) |
| {U : Set Base} (h : ModeDomain (fun p => complexify (F p)) U) |
| {τ s K C : ℝ} {f : SmallModes.Tensor} (hτ : 0 < τ) (hs : 0 < s) |
| (hτs : τ ≤ s) (hs1 : s ≤ 1) (hK : 0 ≤ K) (hC : 0 ≤ C) |
| (hf : ContDiffOn ℝ ∞ f U) (q m : ℕ) |
| (hc : ReconstructionCoefficientBound (fun p => complexify (F p)) U s (m + q + 1) K) |
| (hb : WeightedBound U s (m + q + 1) C f) : |
| let Z := modeApprox τ (fun p => complexify (F p)) (fun _ => 0) f q |
| ContDiffOn ℝ ∞ (realOsc τ Z) U ∧ |
| WeightedBound U τ m (2 ^ m * (forcedModeConstant n m K q * C)) (realOsc τ Z) ∧ |
| WeightedBound U τ m |
| (2 ^ m * (fullErrorConstant n (m + q) K ^ (q + 1) * (τ / s) ^ (q + 1) * C)) |
| (fun p => realLinearizedTensor F (realOsc τ Z) p - realOsc τ f p) := by |
| dsimp only |
| have hm := contDiffOn_modeApprox τ h (V := fun _ => 0) contDiffOn_const hf q |
| have hz : WeightedBound U s (m + q + 1) C (fun _ : Base => (0 : Ambient n)) := |
| (weightedBound_zero U s _).mono_const hC |
| have hX : ∀ p ∈ U, coordDeriv dx (fun p => complexify (F p)) p ⬝ᵥ (0 : Ambient n) = 0 := by |
| intro p hp; simp |
| have hY : ∀ p ∈ U, coordDeriv dy (fun p => complexify (F p)) p ⬝ᵥ (0 : Ambient n) = 0 := by |
| intro p hp; simp |
| have hV : ∀ p ∈ U, goodSecond (fun p => complexify (F p)) p ⬝ᵥ (0 : Ambient n) = 0 := by |
| intro p hp; simp |
| have ha := weighted_modeApprox hτ (contDiff_complexify hF) h hs hτs hs1 hK hC |
| hf (V := fun _ => 0) contDiffOn_const hX hY hV q m hc hb hz |
| have hr := weighted_residual_modeApprox hτ (contDiff_complexify hF) h hs hs1 hK hC |
| hf (V := fun _ => 0) contDiffOn_const hX hY hV q m hc.toFullModeCoefficientBound hb hz |
| refine ⟨contDiffOn_realOsc hm τ, weighted_realOsc h.isOpen hτ hτs |
| (mul_nonneg (forcedModeConstant_nonneg n m q hK) hC) hm ha, ?_⟩ |
| have hsmooth := contDiffOn_residual τ h hm hf |
| have hn : 0 ≤ fullErrorConstant n (m + q) K ^ (q + 1) * (τ / s) ^ (q + 1) * C := |
| mul_nonneg (mul_nonneg (pow_nonneg (fullErrorConstant_nonneg _ _ hK) _) |
| (pow_nonneg (div_nonneg hτ.le hs.le) _)) hC |
| have hr' := weighted_realOsc h.isOpen hτ hτs hn hsmooth hr |
| apply hr'.congr |
| intro p hp |
| exact realLinearized_residual (hF.differentiable (by simp) p) |
| (((hm p hp).contDiffAt (h.isOpen.mem_nhds hp)).differentiableAt (by simp)) τ f |
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| end ClosedSurfaceR4.RealModes |
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| namespace ClosedSurfaceR4.WeightedEstimates |
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| lemma WeightedBound.enlarge_scale {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] {U : Set E} {s C : ℝ} {m : ℕ} {f : E → F} |
| (h : WeightedBound U s m C f) (hC : 0 ≤ C) {c : ℝ} (hc : 1 ≤ c) : |
| WeightedBound U (c * s) m (c ^ m * C) f := by |
| intro j hj p hp |
| calc |
| _ = c ^ j * (s ^ j * ‖iteratedFDerivWithin ℝ j f U p‖) := by rw [mul_pow]; ring |
| _ ≤ c ^ j * C := mul_le_mul_of_nonneg_left (h j hj p hp) (pow_nonneg (by linarith) _) |
| _ ≤ c ^ m * C := mul_le_mul_of_nonneg_right (pow_le_pow_right₀ hc hj) hC |
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| end ClosedSurfaceR4.WeightedEstimates |
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| end |
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| end OAI |
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