Integrable bounded mul bilin Form Integrand
DeGiorgi.integrable_bounded_mul_bilinFormIntegrand
Project documentation
Integrability of a bounded scalar times the bilinear-form integrand. If |f(x)| ≤ C everywhere and the bilinear-form integrand ⟪A∇u,∇u⟫ is integrable (which it always is for u ∈ W^{1,2}), then f · ⟪A∇u,∇u⟫ is integrable. This is a key API lemma for the Caccioppoli/Moser absorption argument. Proved in a standalone context to keep elaboration managea...
Exact Lean statement
theorem integrable_bounded_mul_bilinFormIntegrand
{Ω : Set E}
{u : E → ℝ} {f : E → ℝ} {C : ℝ}
(A : EllipticCoeff d Ω)
(hu : MemW1pWitness 2 u Ω)
(hf_meas : AEStronglyMeasurable f (volume.restrict Ω))
(hf_bound : ∀ x, |f x| ≤ C) :
Integrable (fun x => f x * bilinFormIntegrandOfCoeff A hu hu x) (volume.restrict Ω)Formal artifact
Lean source
theorem integrable_bounded_mul_bilinFormIntegrand {Ω : Set E} {u : E → ℝ} {f : E → ℝ} {C : ℝ} (A : EllipticCoeff d Ω) (hu : MemW1pWitness 2 u Ω) (hf_meas : AEStronglyMeasurable f (volume.restrict Ω)) (hf_bound : ∀ x, |f x| ≤ C) : Integrable (fun x => f x * bilinFormIntegrandOfCoeff A hu hu x) (volume.restrict Ω) := by have hbilin_int := integrable_bilinFormIntegrandOfCoeff A hu hu have hC_nonneg : 0 ≤ C := le_trans (abs_nonneg (f 0)) (hf_bound 0) -- Use: |f · g| ≤ C · |g|, and C · |g| is integrable since g is. apply Integrable.mono' ((hbilin_int.norm).const_mul C) (hf_meas.mul (aestronglyMeasurable_bilinFormIntegrandOfCoeff A hu hu)) filter_upwards with x simp only [Pi.mul_apply, Real.norm_eq_abs] calc ‖f x * bilinFormIntegrandOfCoeff A hu hu x‖ = |f x| * |bilinFormIntegrandOfCoeff A hu hu x| := by rw [Real.norm_eq_abs, abs_mul] _ ≤ C * |bilinFormIntegrandOfCoeff A hu hu x| := mul_le_mul_of_nonneg_right (hf_bound x) (abs_nonneg _) _ = C * ‖bilinFormIntegrandOfCoeff A hu hu x‖ := by rw [Real.norm_eq_abs]- Project
- DeGiorgi
- License
- Apache-2.0
- Commit
- 4c1b3077d378
- Source
- DeGiorgi/WeakFormulation/WeightedEstimates.lean:372-393
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
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.