The interior Schauder estimate on small balls #
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 α-Hölder with constant L on the closed ball closedBall x₀ R, and such
that a(x₀) has ellipticity constants 0 < λ ≤ Λ. If R is small compared with the Hölder
modulus of the coefficients, in the sense of the scale condition of
TauCeti.PDE.exists_holderWith_iteratedFDeriv_two_of_support_subset_closedBall, then for every
ρ < R there is a constant C, independent of the solution, such that every u which is C² on
the open ball B_R with α-Hölder Hessian there satisfies
[D²u]_{α; closedBall x₀ ρ} ≤ C ([L u]_{α; B_R} + N),
where N bounds |u|, ‖Du‖ and ‖D²u‖ on B_R. The Hölder constant of the Hessian on B_R
enters only qualitatively: the estimate does not depend on it. The constant C depends on the
coefficients only through λ, Λ, L and a bound M for their entries on the ball, and not on
the centre x₀. This is the localized form of the Schauder estimate used to pass from
coefficients of small oscillation on a ball to Hölder continuous coefficients, by covering a
compact set with finitely many small balls, or the whole space with balls of one radius.
The argument #
Choose a smooth cutoff ζ equal to one near closedBall x₀ ρ with support in B_R, the
translate of one fixed cutoff about the origin so that its bounds do not depend on x₀, and put
w = ζ u, which is C² on the whole space (ContDiffOn.contDiff_smul_of_tsupport_subset). By the
second-order product rule (TauCeti.fderiv_fderiv_mul_apply_of_tsupport_subset),
∂ᵢ∂ⱼw = ζ ∂ᵢ∂ⱼu + Tᵢⱼ, where Tᵢⱼ = u ∂ᵢ∂ⱼζ + ∂ᵢζ ∂ⱼu + ∂ⱼζ ∂ᵢu,
so L w = ζ L u + ∑ᵢⱼ aᵢⱼ Tᵢⱼ. Every term is a product of a globally Hölder factor built from
ζ, vanishing off B_R, and a factor that is bounded and Hölder on B_R; such products are
globally Hölder (HolderWith.smul_of_eq_zero). On the ball the lower-order factors u and ∂ⱼu
are Lipschitz, with constant N, hence α-Hölder. This bounds the Hölder constant of L w by
C (K + N), where K is the Hölder constant of L u, and ‖D²w‖_{L²} ≤ C N; the Hessian of
w is supported in B_R and globally Hölder a priori. The compact-support estimate then bounds the
Hölder constant of D²w, which equals D²u near closedBall x₀ ρ.
Main declaration #
TauCeti.PDE.exists_holderOnWith_iteratedFDeriv_two_closedBall_of_sum_smul_fderiv_fderiv: the interior Schauder estimate on small balls.
References #
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Chapter 6, proof of Theorem 6.2 (localization of the Schauder estimate by cutoffs).
The interior Schauder estimate on small balls. Let 0 < α < 1. There is C₀ > 0,
depending only on dimension and α, with the following property. Let 0 < λ, Λ, L, M and
R be given, put κ = C₀ (Λ / λ)^(1 + (n+1)/4 + α/2), and assume the scale condition
2 κ (n+1)² (1 + 2ᵅ) L Rᵅ ≤ λ.
Then for every ρ < R there is C, depending only on n, α, λ, Λ, L, M, R and ρ,
with the following property. Let a be α-Hölder entrywise with constant L and bounded entrywise
by M on the closed ball closedBall x₀ R, and suppose a x₀ has ellipticity bounds λ, Λ. Put
L u = ∑ᵢⱼ aᵢⱼ ∂ᵢⱼu. Let u be C² on the open ball B_R, with |u|, ‖Du‖ and ‖D²u‖ bounded
by N there and with D²u α-Hölder on B_R (with any constant). If L u is α-Hölder on B_R
with constant K, then D²u is α-Hölder on closedBall x₀ ρ with constant C (K + N).
The constant C does not depend on the centre x₀ or on the coefficient field a beyond the
bounds λ, Λ, L and M; this uniformity is what makes the estimate usable on infinitely many
balls at once.