Research proof index

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...

The bilinear-form integrand is additive in the left slot.

The bilinear-form integrand is homogeneous in the left slot.

Localization step for weighted Caccioppoli on nested sets.

Ball-specialized weighted Caccioppoli inequality, using the generalized cutoff admissibility theorem from the previous section.

Ball-specialized weighted Caccioppoli inequality with the truncation witness constructed from the concrete Chapter 02 positive-part API.