Research proof index

A-harmonic replacement: given u ∈ W^{1,2}(B_R), there exists h that is A-harmonic with h - u ∈ W₀^{1,2}(B_R).

A-harmonic replacements are unique up to a.e. equality.

The weak bilinear form is continuous with respect to the L² gradient seminorms.

The weak bilinear form is coercive with respect to the L² gradient seminorm.

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...