Theorems · Theorem · potential theory

InnerProductSpace.HarmonicOnNhd.contDiffOn

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E]
  {F : Type u_2} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace ℝ F] {f : E → F} {s : Set E},
  InnerProductSpace.HarmonicOnNhd f s → ContDiffOn ℝ 2 f s

Harmonic functions are two times continuously differentiable.

Defined in: Mathlib.Analysis.InnerProductSpace.Harmonic.Basic
Cited by: 1 results in Mathlib
Foundations: Depth 245 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup InnerProductSpace FiniteDimensional NormedAddCommGroup NormedSpace

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Setstatement and proof · cited by 53,352
Realstatement and proof · cited by 25,697
NormedAddCommGroupstatement and proof · cited by 15,752
NormedSpacestatement and proof · cited by 12,499
ENatstatement · cited by 4,985
WithTopstatement · cited by 3,754
InnerProductSpacestatement and proof · cited by 3,523
FiniteDimensionalstatement and proof · cited by 1,854
ContDiffOnstatement · cited by 294
ContDiffAt.contDiffWithinAtproof · cited by 31
InnerProductSpace.HarmonicOnNhdstatement and proof · cited by 24

Cited by1

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InnerProductSpace.HarmonicOnNhd.exists_analyticOnNhd_univ_re_eqproof · cited by 1