Research proof index

If ∫ ψ · f = ∫ ψ · g for all ψ ∈ Cc^∞(U), then f =ᵃᵉ g on U. This is the du Bois-Reymond lemma, the key uniqueness engine for weak derivatives.

A discrete Minkowski inequality: for p ≥ 1 and a finite sequence a : Fin n → ℝ≥0, the ℓᵖ norm of a is bounded above by the sum of its entries.

Swap the time derivative $\partial_t$ with the integral $\int$.

Swap the spatial derivative $\partial_x$ with the integral $\int$.

Swap the second spatial derivative $\partial_{xx}$ with the integral $\int$.

Uniqueness of weak multi-derivatives on U: any two candidates must agree almost everywhere on U. The proof reduces to the du Bois-Reymond lemma via the defining identity.