Source-pinned research
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Clear filtersWkpU.limit_of_cauchySeq
Plain-language statement
A Cauchy sequence in W^{k,p}(U) has a limit f_lim in W^{k,p}(U), with the toLp and derivtoLp projections of the sequence converging to those of f_lim.
Source project: PDE
Person-level attribution pending.
WkpU.norm_derivtoLp_le
Plain-language statement
The Lp norm of the n-th weak derivative of f is bounded by the W^{k,p} norm.
Source project: PDE
Person-level attribution pending.
WkpU.tendsto_iff_all_derivELpNorm
Plain-language statement
Convergence in W^{k,p}(U) is equivalent to convergence of each weak-derivative eLpNorm component to zero: fₙ → f in W^{k,p} if and only if ‖D^s(fₙ - f)‖_{Lp} → 0 for all multi-indices s with |s| ≤ k.
Source project: PDE
Person-level attribution pending.
WkpUNorm_eq_zero_iff
Plain-language statement
The W^{k,p}(U) norm vanishes iff the function is zero a.e..
Source project: PDE
Person-level attribution pending.
zeroWeakmultiDerivU
Project documentation
Any weak multi-derivative Df on U agrees a.e. with the canonical choice. -/ theorem WeakmultiderivU_unique {d : ℕ+} {n : ℕ} {U : Set (Fin d → ℝ)} (hU : IsOpen U) (s : Fin n → Fin d) (f : Lp_locU d 1 U) (h : HasWeakMultiDerivU U f s) (Df : Lp_locU d 1 U) (hDf : IsWeakMultiDerivU U s f Df) : (WeakmultiderivU U f s h : (Fin d → ℝ) →ₘ[μU d U] ℝ) =ᵐ[volume...
Source project: PDE
Person-level attribution pending.