The interior Calderón–Zygmund estimate on balls

Let 0 < n and 1 < p < ∞. This file proves the interior W^{2,p} estimate on a ball for strong solutions of a non-divergence-form equation ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu + ⟪b, ∇u⟫ + c u = f whose principal coefficients have small oscillation and whose lower-order coefficients are bounded (TauCeti.PDE.exists_mul_lpNorm_iteratedGradient_ball_le_of_sum_mul_inner_add_inner_add_mul_eq). There are C and δ > 0, depending only on n and p, with the following property. Let A be a matrix with λ ‖ξ‖² ≤ ξᵀ A ξ and 0 < λ, write B_ρ for the ball B(x₀, ρ), and let B_R ⊆ Ω. If |aᵢⱼ - Aᵢⱼ| ≤ δ λ, |aᵢⱼ| ≤ Λ, ‖b‖ ≤ β and |c| ≤ γ almost everywhere on B_R, then every u ∈ W^{2,p}(Ω) solving the equation almost everywhere on B_R satisfies, for r < R,

(R - r)² ‖D²u‖_{Lᵖ(B_r)} ≤ C ((R² / λ) ‖f‖_{Lᵖ(B_R)} + (1 + (Λ + β R + γ R²) / λ)² ‖u‖_{Lᵖ(B_R)}).

In contrast to the localized estimate TauCeti.PDE.exists_lpNorm_smul_iteratedGradient_le_of_sum_mul_inner_eq_of_abs_sub_le, the right side involves only f and u, not the gradient of u, even when b ≠ 0. The lower-order coefficients enter only through the scale-invariant ratios β R / λ and γ R² / λ. When the principal coefficients are continuous, every point of Ω is the centre of a ball on which their oscillation is below δ λ, so this is the local estimate behind the interior W^{2,p} estimate for continuous coefficients.

The argument

For 0 < s < t ≤ R, a cutoff between B_s and B_t whose first and second derivatives are bounded by c / (t - s) and c / (t - s)² (TauCeti.exists_forall_contDiff_cutoff_closedBall) gives two inequalities:

• the localized estimate on B_t, applied to the equation ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu = g with g = f - ⟪b, ∇u⟫ - c u, bounds ‖D²u‖_{B_s} by ‖g‖_{B_t}, (t - s)⁻¹ ‖∇u‖_{B_t} and (t - s)⁻² ‖u‖_{B_R}. Since ‖g‖_{B_t} ≤ ‖f‖_{B_R} + β ‖∇u‖_{B_t} + γ ‖u‖_{B_R} and t - s ≤ R, the lower-order terms are absorbed into the last two terms: their coefficient grows by a constant multiple of (β R + γ R²) / λ, and the constant from the cutoff bounds grows by 1 (TauCeti.le_add_mul_of_le_add_of_le_add_mul_add_mul);
• the interpolation inequality between concentric balls (TauCeti.Wkp.exists_lpNorm_gradient_ball_le) bounds ‖∇u‖_{B_s} by ε ‖D²u‖_{B_t} plus lower-order terms, for every ε > 0.

In terms of the weighted quantities Φ₁ = sup (R - σ) ‖∇u‖_{B_σ} and Φ₂ = sup (R - σ)² ‖D²u‖_{B_σ} over 0 < σ < R, which are finite because u ∈ W^{2,p}(Ω), applying both inequalities at σ and at (σ + R) / 2, with ε a small multiple of R - σ, gives Φ₂ ≤ C (R² ‖f‖ / λ + K (Φ₁ + ‖u‖)) and Φ₁ ≤ θ (Φ₂ + Φ₁) + C_θ ‖u‖ with θ as small as needed. Absorbing Φ₁ bounds Φ₂, and (R - r)² ‖D²u‖_{B_r} ≤ Φ₂; this real-variable step is TauCeti.sq_mul_le_of_forall_le.

References

• D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorem 9.11 and its proof (the weighted interior seminorms), and Theorem 7.28 (interpolation).

theorem TauCeti.PDE.exists_mul_lpNorm_iteratedGradient_ball_le_of_sum_mul_inner_add_inner_add_mul_eq {n : ℕ} {p : ENNReal} [Fact (1 ≤ p)] (hn : 0 < n) (hp : 1 < p) (hp_top : p ≠ ⊤) : ∃ (C : NNReal) (δ : ℝ), 0 < δ ∧ ∀ {A : Matrix (Fin n) (Fin n) ℝ} {lam : ℝ}, 0 < lam → (∀ (ξ : EuclideanSpace ℝ (Fin n)), lam * ‖ξ‖ ^ 2 ≤ A.toQuadraticForm' ξ.ofLp) → ∀ {Omega : TopologicalSpace.Opens (EuclideanSpace ℝ (Fin n))} {x₀ : EuclideanSpace ℝ (Fin n)} {r R : ℝ}, r < R → Metric.ball x₀ R ⊆ ↑Omega → ∀ {a : EuclideanSpace ℝ (Fin n) → Matrix (Fin n) (Fin n) ℝ} {b : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin n)} {c : EuclideanSpace ℝ (Fin n) → ℝ} {ε Λ β γ : ℝ}, MeasureTheory.AEStronglyMeasurable a (MeasureTheory.volume.restrict (Metric.ball x₀ R)) → ε ≤ δ * lam → (∀ᵐ (x : EuclideanSpace ℝ (Fin n)), x ∈ Metric.ball x₀ R → ∀ (i j : Fin n), |a x i j - A i j| ≤ ε) → (∀ᵐ (x : EuclideanSpace ℝ (Fin n)), x ∈ Metric.ball x₀ R → ∀ (i j : Fin n), |a x i j| ≤ Λ) → MeasureTheory.AEStronglyMeasurable b (MeasureTheory.volume.restrict (Metric.ball x₀ R)) → (∀ᵐ (x : EuclideanSpace ℝ (Fin n)), x ∈ Metric.ball x₀ R → ‖b x‖ ≤ β) → MeasureTheory.AEStronglyMeasurable c (MeasureTheory.volume.restrict (Metric.ball x₀ R)) → (∀ᵐ (x : EuclideanSpace ℝ (Fin n)), x ∈ Metric.ball x₀ R → |c x| ≤ γ) → ∀ (u : Wkp MeasureTheory.volume Omega p 2) (f : EuclideanSpace ℝ (Fin n) → ℝ), (∀ᵐ (x : EuclideanSpace ℝ (Fin n)) ∂MeasureTheory.volume.restrict (Metric.ball x₀ R), ∑ i : Fin n, ∑ j : Fin n, a x i j * inner ℝ ((↑↑(Wkp.iteratedGradient 1 u) x) (EuclideanSpace.single i 1)) (EuclideanSpace.single j 1) + inner ℝ (b x) (↑↑(Wkp.iteratedGradient 0 (Wkp.lowerOrder 1 u)) x) + c x * ↑↑(Wkp.value 2 u) x = f x) → (R - r) ^ 2 * MeasureTheory.lpNorm (↑↑(Wkp.iteratedGradient 1 u)) p (MeasureTheory.volume.restrict (Metric.ball x₀ r)) ≤ ↑C * (R ^ 2 / lam * MeasureTheory.lpNorm f p (MeasureTheory.volume.restrict (Metric.ball x₀ R)) + (1 + (Λ + β * R + γ * R ^ 2) / lam) ^ 2 * MeasureTheory.lpNorm (↑↑(Wkp.value 2 u)) p (MeasureTheory.volume.restrict (Metric.ball x₀ R)))

The interior Calderón–Zygmund estimate on balls for coefficients of small oscillation. Let 0 < n and 1 < p < ∞. There are constants C and δ > 0, depending only on n and p, with the following property. Let A be a matrix with λ ‖ξ‖² ≤ ξᵀ A ξ and 0 < λ, let Ω be open, and let B_ρ denote the ball B(x₀, ρ). Let r < R with B_R ⊆ Ω, let a be a measurable matrix-valued coefficient with |aᵢⱼ(x) - Aᵢⱼ| ≤ ε and |aᵢⱼ(x)| ≤ Λ for almost every x ∈ B_R, where ε ≤ δ λ, and let b and c be measurable lower-order coefficients with ‖b(x)‖ ≤ β and |c(x)| ≤ γ for almost every x ∈ B_R. Then every u ∈ W^{2,p}(Ω) with ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu + ⟪b, ∇u⟫ + c u = f almost everywhere on B_R satisfies

(R - r)² ‖D²u‖_{Lᵖ(B_r)} ≤ C ((R² / λ) ‖f‖_{Lᵖ(B_R)} + (1 + (Λ + β R + γ R²) / λ)² ‖u‖_{Lᵖ(B_R)}),

where D²u is the weak Hessian TauCeti.Wkp.iteratedGradient 1 u and ∇u the weak gradient TauCeti.Wkp.iteratedGradient 0 (TauCeti.Wkp.lowerOrder 1 u). No boundary condition on u and no regularity of the coefficients beyond the oscillation bound and the bounds on b and c are assumed, and f is not assumed to be in Lᵖ: the equation and the coefficient bounds force it. Unlike TauCeti.PDE.exists_lpNorm_smul_iteratedGradient_le_of_sum_mul_inner_eq_of_abs_sub_le, the right side does not involve the gradient of u, even when b ≠ 0.