laplacian_congr_nhds — Mathlib

English

If two functions agree in nhdsWithin, then their Laplacians within s agree in nhdsWithin.

Русский

Если две функции совпадают в nhdsWithin, то их лапласианы внутри s совпадают в nhdsWithin.

LaTeX

$$nhdsWithin_congr_nhdsWithin (h) (hs) : Δ[s] f1 = Δ[s] f2 in nhdsWithin$$

Lean4

/-- If two functions agree in a neighborhood of a point, then so do their Laplacians.
-/
theorem laplacian_congr_nhds (h : f₁ =ᶠ[𝓝 x] f₂) : Δ f₁ =ᶠ[𝓝 x] Δ f₂ :=
  by
  filter_upwards [EventuallyEq.iteratedFDeriv ℝ h 2] with x hx
  simp [laplacian_eq_iteratedFDeriv_stdOrthonormalBasis, hx]

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