Research proof index

Weighted Cauchy-Schwarz inequality for the bilinear form. For any f : E → ℝ serving as a weight function: |∫ f · ⟪A∇u, ∇v⟫| ≤ (∫ f² · ⟪A∇u, ∇u⟫)^{1/2} · (Λ · ∫ ‖∇v‖²)^{1/2} This follows from the pointwise bound |⟪Aξ, η⟫|² ≤ ⟪Aξ,ξ⟫ · ⟪Aη,η⟫ ≤ ⟪Aξ,ξ⟫ · Λ‖η‖² (the first inequality is Cauchy-Schwarz for the A-inner product, the second is the upper...

Pointwise expansion of the bilinear-form integrand against a smooth-product witness η · w. The weak gradient of η · w is η · ∇w + (∇η) · w (from mul_smooth_bounded_p), so the bilinear-form integrand is: ⟪A∇u, ∇(η·w)⟫ = η · ⟪A∇u, ∇w⟫ + w · ⟪A∇u, ∇η⟫ where ∇η is encoded as the vector with components (fderiv ℝ η x)(single i 1). Note: this...

Integrability of a bounded scalar times the bilinear-form integrand. If |f(x)| ≤ C everywhere and the bilinear-form integrand ⟪A∇u,∇u⟫ is integrable (which it always is for u ∈ W^{1,2}), then f · ⟪A∇u,∇u⟫ is integrable. This is a key API lemma for the Caccioppoli/Moser absorption argument. Proved in a standalone context to keep elaboration managea...