The Calderón–Zygmund estimate in W^{2,p}(ℝⁿ) for uniformly continuous coefficients #
Let 0 < n and 1 < p < ∞. This file proves the global W^{2,p} estimate on the whole space for
strong solutions of a non-divergence-form equation ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu = f whose coefficients are
uniformly elliptic, bounded and uniformly continuous: there is a constant C such that every
u ∈ W^{2,p}(ℝⁿ) with ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu = f ∈ Lᵖ(ℝⁿ) almost everywhere satisfies
‖D²u‖_{Lᵖ} ≤ C (‖f‖_{Lᵖ} + ‖u‖_{Lᵖ}),
where D²u is the weak Hessian TauCeti.Wkp.iteratedGradient 1 u.
The dependence of C on the coefficients is explicit
(TauCeti.PDE.exists_norm_iteratedGradient_le_of_sum_mul_inner_eq_of_dist_lt_abs_sub_le). There
are C₀ and δ > 0, depending only on n and p, such that if λ ‖ξ‖² ≤ ξᵀ a(x) ξ with
0 < λ, |aᵢⱼ(x)| ≤ Λ, and the entries of a vary by at most δ λ between points at distance
less than ρ, then
‖D²u‖_{Lᵖ} ≤ C₀ (‖f‖_{Lᵖ} / λ + (1 + Λ / λ)² / ρ² ‖u‖_{Lᵖ}).
Uniformly continuous coefficients admit such a ρ for every δ, which gives the qualitative
form (TauCeti.PDE.exists_norm_iteratedGradient_le_of_sum_mul_inner_eq_of_uniformContinuous).
No condition at infinity is imposed on a beyond these uniform bounds. This is the a priori
estimate behind strong solvability in W^{2,p}(ℝⁿ) by the method of continuity.
The argument #
The interior estimate on balls
(TauCeti.PDE.exists_mul_lpNorm_iteratedGradient_ball_le_of_sum_mul_inner_add_inner_add_mul_eq),
with the coefficients frozen at a(y), bounds ‖D²u‖_{Lᵖ(B(y, ρ/2))} by the Lᵖ(B(y, ρ)) norms of
f and u, with constants independent of the centre y. Averaging the p-th powers of these
local estimates over all centres y ∈ ℝⁿ turns them into the global estimate, at the cost of
the volume ratio |B(0, ρ)| / |B(0, ρ/2)| = 2ⁿ
(TauCeti.lpNorm_le_of_ae_lpNorm_restrict_ball_le_add). This replaces a covering of ℝⁿ by balls
with bounded overlap.
References #
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorem 9.11 and its proof (local estimates on balls of small oscillation).
The Calderón–Zygmund estimate in W^{2,p}(ℝⁿ) for coefficients of small local
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 measurable matrix-valued coefficient with
λ ‖ξ‖² ≤ ξᵀ a(x) ξ, 0 < λ, and |aᵢⱼ(x)| ≤ Λ for every x, whose entries vary by at most
δ λ between points at distance less than ρ, where 0 < ρ. Then every u ∈ W^{2,p}(ℝⁿ) with
∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu = f ∈ Lᵖ(ℝⁿ) almost everywhere satisfies
‖D²u‖_{Lᵖ} ≤ C (‖f‖_{Lᵖ} / λ + (1 + Λ / λ)² / ρ² ‖u‖_{Lᵖ}),
where D²u is the weak Hessian TauCeti.Wkp.iteratedGradient 1 u.
The Calderón–Zygmund estimate in W^{2,p}(ℝⁿ) for uniformly continuous coefficients.
Let 0 < n and 1 < p < ∞, and let a be a uniformly continuous matrix-valued coefficient with
λ ‖ξ‖² ≤ ξᵀ a(x) ξ, 0 < λ, and |aᵢⱼ(x)| ≤ Λ for every x. Then there is a constant C,
depending on n, p, λ, Λ and the modulus of continuity of a, such that every
u ∈ W^{2,p}(ℝⁿ) with ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu = f ∈ Lᵖ(ℝⁿ) almost everywhere satisfies
‖D²u‖_{Lᵖ} ≤ C (‖f‖_{Lᵖ} + ‖u‖_{Lᵖ}),
where D²u is the weak Hessian TauCeti.Wkp.iteratedGradient 1 u.