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159357f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | import Mathlib
import OAI.Analysis.Conductivity.Sobolev.CentralPhysicalVariational
import OAI.Analysis.Conductivity.Sources.CentralWeightedVariational
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory
def angularArea : ℝ := (2*Real.pi)^2
lemma angularArea_ge_one : 1≤angularArea := by
have hp := Real.pi_gt_three
dsimp only [angularArea]
nlinarith [sq_nonneg (Real.pi-3)]
theorem central_unnormalized_variational_exists (s slopes : Fin 3 → ℝ)
(hs : ∑ i,slopes i=0) :
∃ p : centralEnergySpace s,centralM s p=0 ∧
(∀ v : centralEnergySpace s,
inner ℝ (centralD s p) (centralD s v)+
angularArea*(∑ i : Fin 3,inner ℝ (spectralGraphWeight (torusRate s) (centralT s i p))
(spectralGraphWeight (torusRate s) (centralT s i v)))=
angularArea*(∑ i : Fin 3,slopes i*spectralGraphMean (torusRate s) (centralT s i v))) ∧
(∀ q : centralEnergySpace s,(∀ v : centralEnergySpace s,
inner ℝ (centralD s q) (centralD s v)+
angularArea*(∑ i : Fin 3,inner ℝ (spectralGraphWeight (torusRate s) (centralT s i q))
(spectralGraphWeight (torusRate s) (centralT s i v)))=
angularArea*(∑ i : Fin 3,slopes i*spectralGraphMean (torusRate s) (centralT s i v))) →
q-p=centralM s q • centralOne s) := by
exact @central_variational_exists_graph_weighted (centralEnergySpace s) CentralGradients
inferInstance (centralEnergy_inner s) inferInstance inferInstance inferInstance
TorusModes inferInstance angularArea angularArea_ge_one (torusRate s) (centralD s) (centralT s)
(centralM s) (centralOne s)
(centralAmbientM_one s) (centralAmbientD_one s) (centralAmbientWeight_one s)
(centralOne_mean s) (centralCoercivityConstant s) (centralCoercivityConstant_pos s)
(central_graph_coercivity s) slopes hs
end ScalarConductivity
end
end OAI
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