Linfty subsolution Moser
DeGiorgi.linfty_subsolution_Moser
Plain-language statement
Moser L^p → L∞ estimate for subsolutions on the unit ball, in the honest essential/a.e. form available before the continuity upgrade. This is the normalized-coefficient unit-ball form of the Moser local maximum estimate.
Exact Lean statement
theorem linfty_subsolution_Moser
(hd : 2 < (d : ℝ))
(A : NormalizedEllipticCoeff d (Metric.ball (0 : E) 1))
{u : E → ℝ} {p₀ : ℝ} (hp₀ : 1 < p₀)
(hsub : IsSubsolution A.1 u)
(hposInt :
IntegrableOn (fun x => |max (u x) 0| ^ p₀)
(Metric.ball (0 : E) 1) volume) :
∀ᵐ x ∂(volume.restrict (Metric.ball (0 : E) (1 / 2 : ℝ))),
|max (u x) 0| ^ p₀ ≤
C_Moser d * A.1.Λ ^ ((d : ℝ) / 2) *
(p₀ / (p₀ - 1)) ^ (d : ℝ) *
∫ x in Metric.ball (0 : E) 1, |max (u x) 0| ^ p₀ ∂volumeFormal artifact
Lean source
theorem linfty_subsolution_Moser (hd : 2 < (d : ℝ)) (A : NormalizedEllipticCoeff d (Metric.ball (0 : E) 1)) {u : E → ℝ} {p₀ : ℝ} (hp₀ : 1 < p₀) (hsub : IsSubsolution A.1 u) (hposInt : IntegrableOn (fun x => |max (u x) 0| ^ p₀) (Metric.ball (0 : E) 1) volume) : ∀ᵐ x ∂(volume.restrict (Metric.ball (0 : E) (1 / 2 : ℝ))), |max (u x) 0| ^ p₀ ≤ C_Moser d * A.1.Λ ^ ((d : ℝ) / 2) * (p₀ / (p₀ - 1)) ^ (d : ℝ) * ∫ x in Metric.ball (0 : E) 1, |max (u x) 0| ^ p₀ ∂volume := by have hiter_raw := moser_iteration_bound (d := d) hd A (u := u) hp₀ hsub hposInt have hiter : ∀ n, IntegrableOn (fun x => |max (u x) 0| ^ moserExponentSeq d p₀ n) (Metric.ball (0 : E) (moserRadius n)) volume ∧ moserIterNorm (d := d) (u := u) p₀ n ≤ moserLinftyMajorant (d := d) A (u := u) p₀ := by intro n rcases hiter_raw n with ⟨hInt_n, hbound_n⟩ exact ⟨hInt_n, hbound_n.trans (moser_geometric_majorant (d := d) hd A hp₀ n)⟩ simpa [moserLinftyBoundPow] using moser_ae_closeout (d := d) hd A (u := u) hp₀ hiter- Project
- DeGiorgi
- License
- Apache-2.0
- Commit
- 4c1b3077d378
- Source
- DeGiorgi/MoserIteration/Linfty.lean:364-389
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Related declarations
Ae eq of tendsto e Lp Norm sub
BareFunction.ae_eq_of_tendsto_eLpNorm_sub
Plain-language statement
Lp limit uniqueness: if f_n → g₁ and f_n → g₂ in eLpNorm, then g₁ =ᵐ g₂.
Source project: DeGiorgi
Person-level attribution pending.
E Lp Norm pi le sum component
BareFunction.eLpNorm_pi_le_sum_component
Plain-language statement
Vector eLpNorm ≤ sum of component eLpNorms for Pi-valued functions. Uses eLpNorm_mono_real for the pointwise bound together with eLpNorm_sum_le for ℝ-valued functions, avoiding Pi instance synthesis.
Source project: DeGiorgi
Person-level attribution pending.
Mem Lp of tendsto e Lp Norm
BareFunction.memLp_of_tendsto_eLpNorm
Plain-language statement
If f n → g in eLpNorm and each f n ∈ Lp, then g ∈ Lp, provided g is AEStronglyMeasurable. Avoids the Lp type entirely. The key observation: eLpNorm (f n - g) → 0 means eLpNorm (f N - g) < 1 for some N. Then eLpNorm g ≤ eLpNorm (f N - g) + eLpNorm (f N) < ∞.
Source project: DeGiorgi
Person-level attribution pending.