Research proof index

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.

Weak Harnack inequality for positive supersolutions on B₁. For u > 0 with -∇·(A∇u) ≥ 0 on B₁, and 0 < q < 1: ‖u‖_{L^{q*}(B_{1/4})} ≤ (C(d)/(1-q)^{d/c'})^{Λ^{1/2}} · essInf_{B_{1/4}} u The estimate is stated on B_{1/4}, the ball naturally produced by the forward and inverse Moser steps together with the crossover estimate.

Weak Harnack on an arbitrary ball, obtained by rescaling the unit-ball Chapter 06 statement.

Second stage of weak Harnack: forward low-power iteration for positive supersolutions. For u > 0 a supersolution on B₁ and 0 < p < q < 1: ‖u‖_{L^{qd/(d-2)}(B_{1/2})}^p ≤ C_fwd(d) (Λp²/(1-q)² + 1)^{dχ/2} ‖u‖_{Lᵖ(B₁)}^p The theorem is stated for arbitrary 0 < p < q < 1, with the nonlinear factor raised to dχ/2. Proof: choose m large enou...

First stage of weak Harnack: inverse-power iteration for positive supersolutions. For u > 0 a supersolution on B₁ and every p₀ > 0: (Λ p₀² + 1)^{-d/2} · (∫_{B₁} |u⁻¹|^{p₀})⁻¹ ≤ C · (inf_{B_{1/2}} u)^{p₀} Proof: the Moser iteration (Steps 1-2 above) gives an a.e. L^∞ bound on u⁻¹ over B_{1/2}. Since u > 0 pointwise, this converts to a...

Component Cauchy bound for weak gradients of smooth extensions. Uses HasWeakPartialDeriv.ae_eq + lintegral_rpow_abs_component_le.