All proofs
Project-declaredLean 4.29.0-rc6 · mathlib@5c8398df5281

Exists smooth W12 approx on unit Ball

DeGiorgi.exists_smooth_W12_approx_on_unitBall

Project documentation

specialization of the unit-ball smooth approximation theorem.

Exact Lean statement

theorem exists_smooth_W12_approx_on_unitBall
    {u : E → ℝ}
    (hw : MemW1pWitness 2 u (Metric.ball (0 : E) 1)) :
    ∃ ψ : ℕ → E → ℝ,
      (∀ n, ContDiff ℝ (⊤ : ℕ∞) (ψ n)) ∧
      (∀ n, HasCompactSupport (ψ n)) ∧
      Tendsto
        (fun n => eLpNorm (fun x => ψ n x - u x)
          2 (volume.restrict (Metric.ball (0 : E) 1)))
        atTop (nhds 0) ∧
      (∀ i : Fin d,
        Tendsto
          (fun n => eLpNorm
            (fun x => (fderiv ℝ (ψ n) x) (EuclideanSpace.single i 1) - hw.weakGrad x i)
            2 (volume.restrict (Metric.ball (0 : E) 1)))
          atTop (nhds 0))

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
theorem exists_smooth_W12_approx_on_unitBall    {u : E  }    (hw : MemW1pWitness 2 u (Metric.ball (0 : E) 1)) :     ψ :   E  ,      ( n, ContDiff  (⊤ : ∞) (ψ n))       ( n, HasCompactSupport (ψ n))       Tendsto        (fun n => eLpNorm (fun x => ψ n x - u x)          2 (volume.restrict (Metric.ball (0 : E) 1)))        atTop (nhds 0)       ( i : Fin d,        Tendsto          (fun n => eLpNorm            (fun x => (fderiv  (ψ n) x) (EuclideanSpace.single i 1) - hw.weakGrad x i)            2 (volume.restrict (Metric.ball (0 : E) 1)))          atTop (nhds 0)) := by  let hw' : MemW1pWitness (ENNReal.ofReal (2 : )) u (Metric.ball (0 : E) 1) :=    { memLp := by        simpa using hw.memLp      weakGrad := hw.weakGrad      weakGrad_component_memLp := by        intro i        simpa using hw.weakGrad_component_memLp i      isWeakGrad := hw.isWeakGrad }  rcases exists_smooth_W1p_approx_on_unitBall (d := d) (p := (2 : )) (by norm_num) hw' with    ψ, hψ_smooth, hψ_compact, hψ_fun, hψ_grad  refine ψ, hψ_smooth, hψ_compact, ?_, ?_  · simpa [hw'] using hψ_fun  · intro i    simpa [hw'] using hψ_grad i
Project
DeGiorgi
License
Apache-2.0
Commit
4c1b3077d378
Source
DeGiorgi/UnitBallApproximationCore/Approximation.lean:1376-1405

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

Project-declaredLean 4.29.0-rc6

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.

partial differential equationsregularity theoryanalysis

Source project: DeGiorgi

Person-level attribution pending.

View proof record
Project-declaredLean 4.29.0-rc6

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.

partial differential equationsregularity theoryanalysis

Source project: DeGiorgi

Person-level attribution pending.

View proof record
Project-declaredLean 4.29.0-rc6

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) < ∞.

partial differential equationsregularity theoryanalysis

Source project: DeGiorgi

Person-level attribution pending.

View proof record