The interior Calderón–Zygmund estimate for continuous coefficients #
Let 0 < n, 1 < p < ∞, and let Ω ⊆ ℝⁿ be open. This file proves the interior W^{2,p}
estimate for strong solutions of a non-divergence-form equation
∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu + ⟪b, ∇u⟫ + c u = f whose principal coefficients are continuous on Ω and whose
lower-order coefficients are bounded, in
PDE.exists_lpNorm_iteratedGradient_le_of_sum_mul_inner_add_inner_add_mul_eq_of_continuousAt.
Let K ⊆ Ω be compact, and let a be measurable on Ω, continuous at every point x ∈ K, with
a x elliptic there: λₓ ‖ξ‖² ≤ ξᵀ a(x) ξ for some λₓ > 0. Coefficients continuous on Ω
are the main case. Let b and c be measurable with ‖b‖ ≤ β and |c| ≤ γ almost everywhere
on Ω. Then there is a constant C such that every u ∈ W^{2,p}(Ω) solving the equation with
f ∈ Lᵖ(Ω) almost everywhere on Ω satisfies
‖D²u‖_{Lᵖ(K)} ≤ C (‖f‖_{Lᵖ(Ω)} + ‖u‖_{Lᵖ(Ω)}).
The constant depends on n, p, K, Ω, the coefficient field a and the bounds β and γ,
but not on u or f. No boundary condition is imposed on u and no regularity of Ω is used.
Uniform ellipticity on K with explicit constants is not needed as a hypothesis: ellipticity at
each point of K suffices, and the continuity of a at the points of K supplies the rest. Away
from K the principal coefficients need not be bounded, which is why f is assumed to lie in
Lᵖ(Ω). The case b = 0, c = 0 is the estimate for the principal part alone.
The argument #
The interior estimate on balls
(TauCeti.PDE.exists_mul_lpNorm_iteratedGradient_ball_le_of_sum_mul_inner_add_inner_add_mul_eq)
gives constants C₀ and δ > 0, depending only on n and p. By continuity of a, every
x ∈ K is the centre of a ball B(x, 2rₓ) ⊆ Ω on which |aᵢⱼ - aᵢⱼ(x)| ≤ δ λₓ
(ContinuousAt.exists_ball_subset_forall_abs_sub_le), so the ball
estimate with the coefficients frozen at A = a(x) bounds ‖D²u‖_{Lᵖ(B(x, rₓ))} by f and u,
the bounds β and γ on b and c entering its constant.
By compactness finitely many of the balls B(x, rₓ) cover K, and the Lᵖ norm on K is at
most the sum of the Lᵖ norms on these balls
(MeasureTheory.eLpNorm_restrict_biUnion_finset_le).
References #
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorem 9.11 and its proof (covering by balls on which the coefficients have small oscillation).
The interior Calderón–Zygmund estimate for continuous coefficients. Let 0 < n and
1 < p < ∞, let Ω ⊆ ℝⁿ be open and K ⊆ Ω compact, and let a be a matrix-valued coefficient,
measurable on Ω and continuous at each x ∈ K, such that a x is elliptic at each x ∈ K,
that is, λₓ ‖ξ‖² ≤ ξᵀ a(x) ξ for some λₓ > 0. Let b and c be measurable lower-order
coefficients with ‖b(x)‖ ≤ β and |c(x)| ≤ γ for almost every x ∈ Ω. Then there is a
constant C, depending on a, β, γ, K and Ω but not on u or f, such that every
u ∈ W^{2,p}(Ω) with ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu + ⟪b, ∇u⟫ + c u = f ∈ Lᵖ(Ω) almost everywhere on Ω
satisfies
‖D²u‖_{Lᵖ(K)} ≤ C (‖f‖_{Lᵖ(Ω)} + ‖u‖_{Lᵖ(Ω)}),
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 Ω are assumed, and the gradient of u does not appear on the right.