Research proof index

In ℝ, the set of isolated points of any subset S is countable. Proof: cover by ⋃_{p,q ∈ ℚ} {unique element of S in (p,q)}. Each fiber has at most one element (uniqueness), and ℚ × ℚ is countable.

1D Stampacchia (a.e.): for f differentiable a.e., f' = 0 a.e. on {f = 0}. Proof: {f=0 ∧ f'≠0 ∧ f differentiable} ⊆ {isolated points of {f=0}}, which is countable (hence measure 0). The remaining set {f not differentiable} has measure 0 by hypothesis.

Core lemma: if f is differentiable at x with f(x) = 0 and f'(x) ≠ 0, then f(x+h) ≠ 0 for all sufficiently small h ≠ 0. Proof: the little-o condition gives |f(x+h) - c·h| ≤ (|c|/2)·|h| for small h. Since f(x) = 0, triangle inequality gives |f(x+h)| ≥ |c|·|h|/2 > 0.

Du Bois-Reymond lemma: if g ∈ L^1(a,b) and ∫ g · φ' = 0 for all smooth compactly supported φ in (a,b), then g is constant a.e. Proof (Evans, Appendix C): fix ψ₀ ∈ C_c^∞(a,b) with ∫ψ₀ = 1, set c = ∫g·ψ₀. For any test η, decompose η = [η - (∫η)·ψ₀] + (∫η)·ψ₀. The first part has zero integral, hence is φ' for some test φ (antideri...

1D Stampacchia for W^{1,1}: the weak derivative g vanishes a.e. on {u = 0}. Proof: replace u by its AC representative F (from w11_ae_eq_ac_representative). Then F' = g a.e. by the FTC. Apply deriv_eq_zero_ae_on_zeroSet to F.

For u ∈ W^{1,1}(a,b) with weak derivative g, u agrees a.e. with x ↦ C + ∫_a^x g(t) dt for some constant C. Proof: define F(x) = ∫_a^x g. By AC-IBP, F also has weak derivative g on (a,b). Then u - F has zero weak derivative. By du_bois_reymond, u - F = D a.e.