| import Mathlib |
| import OAI.Geometry.PrescribedPotential.BoundedPoisson |
| import OAI.Geometry.PrescribedPotential.ComplexOperator |
| import OAI.Geometry.PrescribedPotential.PatchCutoffs |
|
|
| namespace OAI |
|
|
| /-! Bounded Smooth Poisson. -/ |
|
|
| section |
|
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| |
|
|
| noncomputable section |
| open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory |
| open scoped SchwartzMap ContDiff Classical |
| namespace GlobalElliptic |
| open Anticanonical SourceSmooth EllipticKernel SobolevChart |
| variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X] |
| [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι] |
| namespace GluingData |
| variable {g : KaehlerMetric A} (D : GluingData g ι) |
|
|
| theorem exists_complex_poisson_bounded : ∃ C : ℝ, 0 ≤ C ∧ ∀ F : Smooth A, |
| ∃ (u : Smooth A) (c : ℂ), complexL g u = F + Smooth.const c ∧ |
| ‖D.localizers.embed 0 u‖ ≤ C * ‖D.localizers.embed 0 F‖ := by |
| obtain ⟨m, hm, he⟩ := D.exists_completedError_small |
| obtain ⟨P⟩ := D.localizers.constantProjection_exists |
| let T := D.resolventZero m hm he ∘L D.poissonInput m hm he P |
| refine ⟨‖T‖, norm_nonneg T, fun F => ?_⟩ |
| let w := D.poissonInput m hm he P (D.localizers.embed 0 F) |
| have hT := T.le_opNorm (D.localizers.embed 0 F) |
| let c := D.poissonResidue m hm he P (D.localizers.embed 0 F) |
| have hc : P.proj w = D.localizers.embed 0 (Smooth.const c) := |
| (P.coefficient_spec w).symm |
| have hw : w = D.localizers.embed 0 (-F - Smooth.const c) + |
| (m^2 : ℝ) • D.resolventZero m hm he w := by |
| have hh := D.poissonInput_equation m hm he P (D.localizers.embed 0 F) |
| change w = -D.localizers.embed 0 F - P.proj w + _ at hh |
| rw [hc] at hh |
| simpa only [map_sub, map_neg] using hh |
| obtain ⟨u, hu⟩ := D.allRegular_smooth (D.resolvent_feedback_regular m hm he w _ hw) |
| refine ⟨u, c, ?_, ?_⟩ |
| · have hs := (D.smooth_shifted_equation m hm he w u hu).trans hw |
| rw [← hu, ← map_smul, ← map_add] at hs |
| have hh := D.localizers.embed_injective 0 hs |
| apply sub_right_inj.mp |
| calc |
| (m^2 : ℝ) • u - complexL g u = -F - Smooth.const c + (m^2 : ℝ) • u := hh |
| _ = (m^2 : ℝ) • u - (F + Smooth.const c) := by abel |
| · rw [hu] |
| simpa only [T, ContinuousLinearMap.comp_apply, w] using hT |
|
|
| end GluingData |
| end GlobalElliptic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
