The localized Calderón–Zygmund estimate on a domain

Let 0 < n, 1 < p < ∞, and let Ω ⊆ ℝⁿ be open. This file localizes the whole-space W^{2,p} estimate for coefficients of small oscillation (TauCeti.PDE.exists_norm_iteratedGradient_le_mul_norm_of_sum_mul_inner_eq_of_abs_sub_le) to a strong solution on Ω, by a smooth cutoff (TauCeti.PDE.exists_lpNorm_smul_iteratedGradient_le_of_sum_mul_inner_eq_of_abs_sub_le).

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, let ζ be smooth with compact support in Ω, with ‖∇ζ‖ ≤ M₁ and ‖D²ζ‖ ≤ M₂, and let a be a measurable matrix-valued coefficient with |aᵢⱼ - Aᵢⱼ| ≤ ε ≤ δ λ and |aᵢⱼ| ≤ Λ almost everywhere on the support of ζ. Then every u ∈ W^{2,p}(Ω) with ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu = f ∈ Lᵖ(Ω) almost everywhere on Ω satisfies

‖ζ D²u‖_{Lᵖ(Ω)} ≤ (C / λ) ‖ζ f‖_{Lᵖ(Ω)} + (C n² Λ / λ + 1) (2 M₁ ‖∇u‖_{Lᵖ(Ω)} + M₂ ‖u‖_{Lᵖ(Ω)}).

No boundary condition is imposed on u and no regularity of Ω is used: the estimate is interior. The lower-order terms on the right are the price of localization. In the interior W^{2,p} estimate for continuous coefficients, where ζ runs over cutoffs of balls small enough that the oscillation of a is below δ λ, the gradient term is absorbed by an interpolation inequality for ‖∇u‖.

The argument

The cutoff ζ u lies in W^{2,p}_0(Ω) (TauCeti.Wkp.exists_mem_wkp0Submodule_value_ae_eq_mul), and its extension by zero lies in W^{2,p}(ℝⁿ) and vanishes off the support of ζ. By the Leibniz rule for the weak Hessian (TauCeti.Wkp.iteratedGradient_one_ae_eq_of_value_ae_eq_mul), D²(ζ u) = ζ D²u + T with T = ∇ζ ⊗ ∇u + ∇u ⊗ ∇ζ + u D²ζ, so

∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼ(ζ u) = ζ f + ∑ᵢⱼ aᵢⱼ Tᵢⱼ.

The whole-space estimate bounds ‖D²(ζ u)‖ by (C / λ) times the Lᵖ norm of the right side, the coefficient bound gives |∑ᵢⱼ aᵢⱼ Tᵢⱼ| ≤ n² Λ ‖T‖, and ‖T‖ ≤ 2 M₁ ‖∇u‖ + M₂ |u| pointwise. Finally ζ D²u = D²(ζ u) - T.

References

• D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, proof of Theorem 9.11.

theorem TauCeti.PDE.exists_lpNorm_smul_iteratedGradient_le_of_sum_mul_inner_eq_of_abs_sub_le {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))} {zeta : EuclideanSpace ℝ (Fin n) → ℝ} {M₁ M₂ : ℝ}, ContDiff ℝ (↑⊤) zeta → HasCompactSupport zeta → tsupport zeta ⊆ ↑Omega → (∀ (x : EuclideanSpace ℝ (Fin n)), ‖gradient zeta x‖ ≤ M₁) → (∀ (x : EuclideanSpace ℝ (Fin n)), ‖fderiv ℝ (gradient zeta) x‖ ≤ M₂) → ∀ {a : EuclideanSpace ℝ (Fin n) → Matrix (Fin n) (Fin n) ℝ} {ε Λ : ℝ}, MeasureTheory.AEStronglyMeasurable a (MeasureTheory.volume.restrict ↑Omega) → ε ≤ δ * lam → (∀ᵐ (x : EuclideanSpace ℝ (Fin n)), x ∈ tsupport zeta → ∀ (i j : Fin n), |a x i j - A i j| ≤ ε) → 0 ≤ Λ → (∀ᵐ (x : EuclideanSpace ℝ (Fin n)), x ∈ tsupport zeta → ∀ (i j : Fin n), |a x i j| ≤ Λ) → ∀ (u : Wkp MeasureTheory.volume Omega p 2) (f : ↥(MeasureTheory.Lp ℝ p (MeasureTheory.volume.restrict ↑Omega))), (∀ᵐ (x : EuclideanSpace ℝ (Fin n)) ∂MeasureTheory.volume.restrict ↑Omega, ∑ 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) = ↑↑f x) → MeasureTheory.lpNorm (fun (x : EuclideanSpace ℝ (Fin n)) => zeta x • ↑↑(Wkp.iteratedGradient 1 u) x) p (MeasureTheory.volume.restrict ↑Omega) ≤ ↑C / lam * MeasureTheory.lpNorm (fun (x : EuclideanSpace ℝ (Fin n)) => zeta x * ↑↑f x) p (MeasureTheory.volume.restrict ↑Omega) + (↑C / lam * ↑n ^ 2 * Λ + 1) * (2 * M₁ * ‖Wkp.iteratedGradient 0 (Wkp.lowerOrder 1 u)‖ + M₂ * ‖Wkp.value 2 u‖)

The localized Calderón–Zygmund estimate 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 ζ be smooth with compact support in Ω, ‖∇ζ‖ ≤ M₁ and ‖D²ζ‖ ≤ M₂. Let a be a measurable matrix-valued coefficient on Ω with |aᵢⱼ(x) - Aᵢⱼ| ≤ ε and |aᵢⱼ(x)| ≤ Λ for almost every x in the support of ζ, where ε ≤ δ λ and 0 ≤ Λ. Then every u ∈ W^{2,p}(Ω) with ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu = f almost everywhere on Ω, with f ∈ Lᵖ(Ω), satisfies

‖ζ D²u‖_{Lᵖ(Ω)} ≤ (C / λ) ‖ζ f‖_{Lᵖ(Ω)} + (C n² Λ / λ + 1) (2 M₁ ‖∇u‖_{Lᵖ(Ω)} + M₂ ‖u‖_{Lᵖ(Ω)}),

where D²u is the weak Hessian TauCeti.Wkp.iteratedGradient 1 u and ∇u is the weak gradient TauCeti.Wkp.iteratedGradient 0 (TauCeti.Wkp.lowerOrder 1 u). No boundary condition on u and no regularity of Ω is assumed.