The Calderón–Zygmund estimate in W^{2,p}(ℝⁿ) for coefficients of small oscillation

Let 0 < n and 1 < p < ∞. For u ∈ W^{2,p}(ℝⁿ) write D²u for its weak Hessian TauCeti.Wkp.iteratedGradient 1 u, an Lᵖ field of endomorphisms of ℝⁿ, so that ⟪D²u(x) eᵢ, eⱼ⟫ is the weak second derivative ∂ᵢ∂ⱼu. This file proves the strong-solution estimate for the non-divergence-form operator ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu with coefficients of small oscillation (TauCeti.PDE.exists_norm_iteratedGradient_le_mul_norm_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 s be closed, and let a be a matrix-valued coefficient with |aᵢⱼ(x) - Aᵢⱼ| ≤ ε ≤ δ λ for almost every x ∈ s. Then every u ∈ W^{2,p}(ℝⁿ) vanishing outside s with ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu = f almost everywhere satisfies ‖D²u‖_{Lᵖ} ≤ (C / λ) ‖f‖_{Lᵖ}. No regularity of a is assumed.

The argument

Freeze the coefficients at A: ∑ᵢⱼ Aᵢⱼ ∂ᵢ∂ⱼu = f + ∑ᵢⱼ (Aᵢⱼ - aᵢⱼ) ∂ᵢ∂ⱼu. Outside s the weak Hessian vanishes (TauCeti.Wkp.iteratedGradient_ae_eq_zero_of_value_ae_eq_zero), and on s the error term is at most n² ε ‖D²u‖ pointwise (Matrix.abs_contraction_le). The constant-coefficient estimate (TauCeti.exists_norm_iteratedGradient_le_mul_norm_of_sum_mul_inner_eq) gives ‖D²u‖ ≤ (C₀ / λ) (‖f‖ + n² ε ‖D²u‖), and the choice of δ makes the coefficient of ‖D²u‖ on the right at most 1 / 2, so it is absorbed on the left (TauCeti.PDE.le_two_mul_div_mul_of_le_div_mul_of_le_add). This is the step of the W^{2,p} theory for continuous coefficients that is applied, after a partition of unity, on each ball small enough that the oscillation of the coefficients is below δ λ.

References

• D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorem 9.11 and its proof (freezing the coefficients and absorbing the error).

theorem TauCeti.PDE.le_two_mul_div_mul_of_le_div_mul_of_le_add {C N ε μ d g f : ℝ} (hμ : 0 < μ) (hC : 0 ≤ C) (hN : 0 < N) (hε : ε ≤ 1 / (2 * (C + 1) * N) * μ) (hd : 0 ≤ d) (hdg : d ≤ C / μ * g) (hg : g ≤ f + N * ε * d) : d ≤ 2 * C / μ * f

The absorption step of the perturbation argument for coefficients of small oscillation: if d ≤ (C / μ) g and g ≤ f + N ε d, where 0 ≤ d and ε ≤ μ / (2 (C + 1) N), then d ≤ (2 C / μ) f. The choice of ε makes the error coefficient C N ε / μ at most 1 / 2.

theorem TauCeti.PDE.exists_norm_iteratedGradient_le_mul_norm_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) → ∀ {a : EuclideanSpace ℝ (Fin n) → Matrix (Fin n) (Fin n) ℝ} {s : Set (EuclideanSpace ℝ (Fin n))} {ε : ℝ}, IsClosed s → ε ≤ δ * lam → (∀ᵐ (x : EuclideanSpace ℝ (Fin n)), x ∈ s → ∀ (i j : Fin n), |a x i j - A i j| ≤ ε) → ∀ (u : Wkp MeasureTheory.volume ⊤ p 2) (f : ↥(MeasureTheory.Lp ℝ p (MeasureTheory.volume.restrict ↑⊤))), (∀ᵐ (x : EuclideanSpace ℝ (Fin n)), x ∉ s → ↑↑(Wkp.value 2 u) x = 0) → (∀ᵐ (x : EuclideanSpace ℝ (Fin n)), ∑ 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) → ‖Wkp.iteratedGradient 1 u‖ ≤ ↑C / lam * ‖f‖

The Calderón–Zygmund estimate in W^{2,p}(ℝⁿ) 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 s be a closed set, and let a be a matrix-valued coefficient with |aᵢⱼ(x) - Aᵢⱼ| ≤ ε for almost every x ∈ s, where ε ≤ δ λ. Then every u ∈ W^{2,p}(ℝⁿ) vanishing almost everywhere outside s and solving ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu = f almost everywhere, with f ∈ Lᵖ(ℝⁿ), satisfies

‖D²u‖_{Lᵖ} ≤ (C / λ) ‖f‖_{Lᵖ},

where D²u is the weak Hessian TauCeti.Wkp.iteratedGradient 1 u. No measurability or continuity of a is assumed.