Research proof index

Ball version of cutoff admissibility driven by the concrete Chapter 02 positive-part witness constructor.

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.

Existence of weak solutions to the inhomogeneous Dirichlet problem with boundary datum u₀ and divergence-form right-hand side div F.

Energy estimate for the shifted Dirichlet solution u - u₀.

Inhomogeneous Dirichlet solutions are unique up to a.e. equality.