The global Schauder estimate on the whole space #
Let 0 < α < 1, and let L u = ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu be a non-divergence-form operator on ℝ^{n+1}
whose coefficients are uniformly elliptic and bounded with constants 0 < λ ≤ Λ on the whole
space (TauCeti.PDE.UniformlyEllipticOn univ a λ Λ) and globally α-Hölder with constant L.
This file proves the global Schauder estimate on the whole space: there is a constant C,
depending only on n, α, λ, Λ and L, such that every C² function u with bounded,
globally α-Hölder Hessian satisfies
‖Du‖_∞ + ‖D²u‖_∞ + [D²u]_α ≤ C ([L u]_α + ‖u‖_∞),
where [·]_α is the global α-Hölder constant
(TauCeti.PDE.exists_holderWith_iteratedFDeriv_two_and_norm_le_of_sum_smul_fderiv_fderiv), or in
terms of the norm of the bounded space C_b^{2,α}(ℝ^{n+1}),
‖u‖_{C^{2,α}} ≤ C ([L u]_α + ‖u‖_∞) (TauCeti.PDE.exists_norm_le_of_sum_smul_fderiv_fderiv).
The Hölder constant of the Hessian enters only qualitatively: the estimate does not depend on it.
Unlike the interior estimate on compact sets
(PDE.exists_holderOnWith_iteratedFDeriv_two_of_sum_smul_fderiv_fderiv_of_holderOnWith),
the constant is uniform over the whole space, and only ‖u‖_∞ remains among the lower-order
terms.
The argument #
The interior estimate on small balls
(TauCeti.PDE.exists_holderOnWith_iteratedFDeriv_two_closedBall_of_sum_smul_fderiv_fderiv) has a
constant that depends on the coefficients only through λ, Λ, L and a bound for their
entries, which uniform ellipticity bounds by Λ
(TauCeti.PDE.UniformlyEllipticOn.abs_apply_le). The global bounds therefore give one radius R,
satisfying the scale condition, and one constant C₁ for balls of radius R about every point.
Two points at distance less than R / 2 lie in the half ball about one of them, while at larger
distances
‖D²u(x) - D²u(y)‖ ≤ 2N ≤ 2N (R / 2)^(-α) ‖x - y‖^α
(HolderOnWith.of_dist_lt_of_dist_le), where N bounds |u|, ‖Du‖ and ‖D²u‖. This is the
first estimate,
[D²u]_α ≤ C (K + N)
(TauCeti.PDE.exists_holderWith_iteratedFDeriv_two_of_sum_smul_fderiv_fderiv_of_holderWith), where
K is the Hölder constant of L u.
The first and second derivatives are then removed from the right side by the interpolation
inequalities ‖D²u‖_∞ ≤ 32 ε⁻² ‖u‖_∞ + 2 ε^α [D²u]_α and
‖Du‖_∞ ≤ 34 ε⁻¹ ‖u‖_∞ + 2 ε^(1+α) [D²u]_α
(HolderWith.norm_iteratedFDeriv_add_two_le_of_norm_le,
HolderWith.norm_iteratedFDeriv_succ_le_of_norm_le_of_bddAbove), which for ε ≤ 1 give the
common bound (34 ε⁻¹ + 32 ε⁻²) ‖u‖_∞ + 2 ε^α [D²u]_α
(HolderWith.norm_iteratedFDeriv_succ_le_and_norm_iteratedFDeriv_add_two_le). Taking for
[D²u]_α the Hölder norm nnHolderNorm, which is finite, and ε so small that
4 C ε^α ≤ 1 / 2, the resulting multiple of [D²u]_α is absorbed into the left side.
Main declarations #
TauCeti.PDE.exists_holderWith_iteratedFDeriv_two_of_sum_smul_fderiv_fderiv_of_holderWith: the global Hölder bound for the Hessian in terms of[L u]_αand bounds onu,DuandD²u.TauCeti.PDE.exists_holderWith_iteratedFDeriv_two_and_norm_le_of_sum_smul_fderiv_fderiv: the global Schauder estimate in terms of[L u]_αand‖u‖_∞alone.TauCeti.PDE.exists_norm_le_of_sum_smul_fderiv_fderiv: the same estimate for the norm ofCkHolderSpace 2 α.
References #
- N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics 12, AMS, 1996 (a priori estimates in the whole space for equations with variable Hölder coefficients).
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Chapter 6, Theorem 6.2 and Lemma 6.32 (interior estimates and interpolation).
The global Schauder estimate with lower-order bounds. Let 0 < α < 1, and let λ, Λ
and L be given. Then there is a constant C, depending only on n, α, λ, Λ and L, with
the following property. Let a be a matrix-valued coefficient on ℝ^{n+1} which is uniformly
elliptic and bounded with constants 0 < λ ≤ Λ (UniformlyEllipticOn univ a λ Λ) and globally
α-Hölder entrywise with constant L, and put L u = ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu. Let u be C², with |u|,
‖Du‖ and ‖D²u‖ bounded by N and with D²u α-Hölder on every compact set (with any
constants). If L u is globally α-Hölder with constant K, then D²u is globally α-Hölder
with constant C (K + N).
The global Schauder estimate on the whole space. Let 0 < α < 1, and let λ, Λ and
L be given. Then there is a constant C, depending only on n, α, λ, Λ and L, with the
following property. Let a be a matrix-valued coefficient on ℝ^{n+1} which is uniformly
elliptic and bounded with constants 0 < λ ≤ Λ (UniformlyEllipticOn univ a λ Λ) and globally
α-Hölder entrywise with constant L, and put L u = ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu. Let u be C² with
|u| ≤ N, and with D²u bounded and globally α-Hölder (with any constant). If L u is globally
α-Hölder with constant K, then D²u is globally α-Hölder with constant C (K + N), and
‖Du‖ and ‖D²u‖ are bounded by C (K + N).
The global Schauder estimate in the C_b^{2,α} norm. Let 0 < α < 1, and let λ, Λ
and L be given. Then there is a constant C, depending only on n, α, λ, Λ and L, with
the following property. Let a be a matrix-valued coefficient on ℝ^{n+1} which is uniformly
elliptic and bounded with constants 0 < λ ≤ Λ (UniformlyEllipticOn univ a λ Λ) and globally
α-Hölder entrywise with constant L, and put L u = ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu. Then every
u ∈ C_b^{2,α}(ℝ^{n+1}) with |u| ≤ N, for which L u is globally α-Hölder with constant K,
satisfies ‖u‖_{C^{2,α}} ≤ C (K + N).