Linfty subsolution De Giorgi ae zero of smallness
DeGiorgi.linfty_subsolution_DeGiorgi_ae_zero_of_smallness
Plain-language statement
Small-data De Giorgi endpoint: if the canonical De Giorgi energy sequence tends to zero, then the final truncation vanishes almost everywhere on the half-ball. This is the formally correct endpoint available from the current Chapter 05 infrastructure before any representative/continuity upgrade.
Exact Lean statement
theorem linfty_subsolution_DeGiorgi_ae_zero_of_smallness
{u : E → ℝ} {x₀ : E} {K lamStar : ℝ}
(hd : 2 < (d : ℝ))
(hK : 0 < K) (hLamStar : 0 < lamStar)
(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)))
(hsmall :
deGiorgiEnergySeq u x₀ lamStar 0 ≤
(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))),
positivePartSub u lamStar x = 0Formal artifact
Lean source
theorem linfty_subsolution_DeGiorgi_ae_zero_of_smallness {u : E → ℝ} {x₀ : E} {K lamStar : ℝ} (hd : 2 < (d : ℝ)) (hK : 0 < K) (hLamStar : 0 < lamStar) (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))) (hsmall : deGiorgiEnergySeq u x₀ lamStar 0 ≤ (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))), positivePartSub u lamStar x = 0 := by have hA_nonneg : ∀ n, 0 ≤ deGiorgiEnergySeq u x₀ lamStar n := by intro n simp [deGiorgiEnergySeq] refine integral_nonneg ?_ intro x positivity have hvanish : Tendsto (deGiorgiEnergySeq u x₀ lamStar) atTop (nhds 0) := deGiorgi_preiter_vanishing hd hK hLamStar hA_nonneg hpre hsmall exact ae_eq_zero_of_forall_setIntegral_sq_le_of_tendsto_zero hStarInt (integral_sq_halfBall_le_deGiorgiEnergySeq hLamStar hStarInt hAint) hvanish- Project
- DeGiorgi
- License
- Apache-2.0
- Commit
- 4c1b3077d378
- Source
- DeGiorgi/DeGiorgiIteration/Linfty.lean:1006-1039
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