Theorems · Theorem · potential theory

ContDiffWithinAt.laplacianWithin_CLM_comp_left

∀ {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E]
  {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace ℝ F] {G : Type u_4}
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace ℝ G] {f : E → F} {x : E} {s : Set E} {l : F →L[ℝ] G},
  ContDiffWithinAt ℝ 2 f s x →
    UniqueDiffOn ℝ s →
      x ∈ s → InnerProductSpace.laplacianWithin (⇑l ∘ f) s x = (⇑l ∘ InnerProductSpace.laplacianWithin f s) x

The Laplacian commutes with left composition by continuous linear maps.

Defined in

Mathlib.Analysis.InnerProductSpace.Laplacian

Cited by

1 results in Mathlib

Foundations

Depth 245 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceFiniteDimensionalNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace