Linfty subsolution De Giorgi ae of unit Ball initial energy smallness
DeGiorgi.linfty_subsolution_DeGiorgi_ae_of_unitBall_initial_energy_smallness
Project documentation
Unit-ball smallness packaging for the De Giorgi endpoint. This is the direct successor to the raw hsmall-based theorem: instead of assuming a bound on the abstract initial De Giorgi energy A₀, it is enough to assume the same recurrence threshold for the concrete unit-ball quantity ∫_{B₁} (u₊)^2. The radius/level bookkeeping then supplies the A₀ bo...
Exact Lean statement
theorem linfty_subsolution_DeGiorgi_ae_of_unitBall_initial_energy_smallness
{u : E → ℝ} {x₀ : E} {K lamStar : ℝ}
(hd : 2 < (d : ℝ))
(hK : 0 < K) (hLamStar : 0 < lamStar)
(hposInt :
IntegrableOn (fun x => |max (u x) 0| ^ 2)
(Metric.ball x₀ 1) volume)
(hStarInt :
IntegrableOn (fun x => |positivePartSub u lamStar x| ^ 2)
(Metric.ball x₀ (1 / 2 : ℝ)) volume)
(hAint :
∀ n,
IntegrableOn (fun x => |positivePartSub u (deGiorgiLevel lamStar n) x| ^ 2)
(Metric.ball x₀ (deGiorgiRadius n)) volume)
(hpre :
∀ n,
deGiorgiEnergySeq u x₀ lamStar (n + 1) ≤
K / ((deGiorgiRadius n - deGiorgiRadius (n + 1)) ^ 2 *
(deGiorgiLevel lamStar (n + 1) - deGiorgiLevel lamStar n) ^ (4 / (d : ℝ))) *
deGiorgiEnergySeq u x₀ lamStar n ^ (1 + 2 / (d : ℝ)))
(hsmall0 :
∫ x in Metric.ball x₀ 1, |max (u x) 0| ^ 2 ∂volume ≤
(deGiorgiRecurrenceCoeff d K lamStar) ^ (-(1 : ℝ) / (2 / (d : ℝ))) *
(deGiorgiRecurrenceBase d) ^ (-(1 : ℝ) / (2 / (d : ℝ)) ^ 2)) :
∀ᵐ x ∂(volume.restrict (Metric.ball x₀ (1 / 2 : ℝ))),
max (u x) 0 ≤ lamStarFormal artifact
Lean source
theorem linfty_subsolution_DeGiorgi_ae_of_unitBall_initial_energy_smallness {u : E → ℝ} {x₀ : E} {K lamStar : ℝ} (hd : 2 < (d : ℝ)) (hK : 0 < K) (hLamStar : 0 < lamStar) (hposInt : IntegrableOn (fun x => |max (u x) 0| ^ 2) (Metric.ball x₀ 1) volume) (hStarInt : IntegrableOn (fun x => |positivePartSub u lamStar x| ^ 2) (Metric.ball x₀ (1 / 2 : ℝ)) volume) (hAint : ∀ n, IntegrableOn (fun x => |positivePartSub u (deGiorgiLevel lamStar n) x| ^ 2) (Metric.ball x₀ (deGiorgiRadius n)) volume) (hpre : ∀ n, deGiorgiEnergySeq u x₀ lamStar (n + 1) ≤ K / ((deGiorgiRadius n - deGiorgiRadius (n + 1)) ^ 2 * (deGiorgiLevel lamStar (n + 1) - deGiorgiLevel lamStar n) ^ (4 / (d : ℝ))) * deGiorgiEnergySeq u x₀ lamStar n ^ (1 + 2 / (d : ℝ))) (hsmall0 : ∫ x in Metric.ball x₀ 1, |max (u x) 0| ^ 2 ∂volume ≤ (deGiorgiRecurrenceCoeff d K lamStar) ^ (-(1 : ℝ) / (2 / (d : ℝ))) * (deGiorgiRecurrenceBase d) ^ (-(1 : ℝ) / (2 / (d : ℝ)) ^ 2)) : ∀ᵐ x ∂(volume.restrict (Metric.ball x₀ (1 / 2 : ℝ))), max (u x) 0 ≤ lamStar := by have hsmall : deGiorgiEnergySeq u x₀ lamStar 0 ≤ (deGiorgiRecurrenceCoeff d K lamStar) ^ (-(1 : ℝ) / (2 / (d : ℝ))) * (deGiorgiRecurrenceBase d) ^ (-(1 : ℝ) / (2 / (d : ℝ)) ^ 2) := by exact (deGiorgiEnergySeq_zero_le_integral_sq_posPartZero_on_unitBall hLamStar (hAint 0) hposInt).trans hsmall0 exact linfty_subsolution_DeGiorgi_ae_of_smallness hd hK hLamStar hStarInt hAint hpre hsmall- Project
- DeGiorgi
- License
- Apache-2.0
- Commit
- 4c1b3077d378
- Source
- DeGiorgi/DeGiorgiIteration/Linfty.lean:1080-1114
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Person-level attribution pending.