The interior Schauder estimate for Hölder coefficients

Let 0 < α < 1, let Ω ⊆ ℝ^{n+1} be open and K ⊆ Ω compact, and let L u = ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu be a non-divergence-form operator whose coefficients are α-Hölder near each point of K, with a(x) elliptic at each x ∈ K: λₓ ‖ξ‖² ≤ ξᵀ a(x) ξ for some λₓ > 0. This file proves the interior Schauder estimate (TauCeti.PDE.exists_holderOnWith_iteratedFDeriv_two_of_sum_smul_fderiv_fderiv_of_holderOnWith): there is a constant C such that every u which is C² on Ω, with D²u α-Hölder on each compact subset of Ω, satisfies

[D²u]_{α; K} ≤ C ([L u]_{α; Ω} + N),

where [·]_{α; S} is the α-Hölder constant on S and N bounds |u|, ‖Du‖ and ‖D²u‖ on Ω. The constant depends on n, α, K, Ω and the coefficient field a, but not on u. The Hölder continuity of D²u on compact subsets of Ω enters only qualitatively: the estimate does not depend on its Hölder constants. No boundary condition on u and no regularity of Ω are used, and uniform ellipticity on K with explicit constants is not needed as a hypothesis.

The argument

Around each x ∈ K choose a radius Rₓ with closedBall x Rₓ ⊆ Ω, so small that the Hölder constant of the coefficients near x satisfies the scale condition of the interior estimate on small balls (TauCeti.PDE.exists_holderOnWith_iteratedFDeriv_two_closedBall_of_sum_smul_fderiv_fderiv), with the coefficients frozen at x. That estimate bounds the Hölder constant of D²u on closedBall x (Rₓ / 2). Finitely many of the balls B(x, Rₓ / 2) cover K. If δ is a Lebesgue number of this cover (lebesgue_number_lemma_of_metric), two points of K at distance less than δ lie in one of these balls, while at larger distances ‖D²u(x) - D²u(y)‖ ≤ 2N ≤ 2N δ^(-α) ‖x - y‖^α (HolderOnWith.of_dist_lt_of_dist_le).

References

• D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Chapter 6, Theorem 6.2 and Corollary 6.3 (interior Schauder estimates on compact subsets).

theorem TauCeti.PDE.exists_holderOnWith_iteratedFDeriv_two_of_sum_smul_fderiv_fderiv_of_holderOnWith {n : ℕ} {α : NNReal} (hα : 0 < α) (hα1 : α < 1) {Ω K : Set (EuclideanSpace ℝ (Fin (n + 1)))} (hΩ : IsOpen Ω) (hK : IsCompact K) (hKΩ : K ⊆ Ω) {a : EuclideanSpace ℝ (Fin (n + 1)) → Matrix (Fin (n + 1)) (Fin (n + 1)) ℝ} (ha : ∀ x ∈ K, ∃ r > 0, ∃ (L : NNReal), ∀ (i j : Fin (n + 1)), HolderOnWith L α (fun (y : EuclideanSpace ℝ (Fin (n + 1))) => a y i j) (Metric.ball x r)) (hell : ∀ x ∈ K, ∃ (lam : ℝ), 0 < lam ∧ ∀ (ξ : EuclideanSpace ℝ (Fin (n + 1))), lam * ‖ξ‖ ^ 2 ≤ (a x).toQuadraticForm' ξ.ofLp) : ∃ (C : NNReal), ∀ {u : EuclideanSpace ℝ (Fin (n + 1)) → ℝ} {N M : NNReal}, ContDiffOn ℝ 2 u Ω → (∀ x ∈ Ω, |u x| ≤ ↑N) → (∀ x ∈ Ω, ‖fderiv ℝ u x‖ ≤ ↑N) → (∀ x ∈ Ω, ‖iteratedFDeriv ℝ 2 u x‖ ≤ ↑N) → (∀ S ⊆ Ω, IsCompact S → ∃ (V : NNReal), HolderOnWith V α (iteratedFDeriv ℝ 2 u) S) → HolderOnWith M α (fun (x : EuclideanSpace ℝ (Fin (n + 1))) => ∑ i : Fin (n + 1), ∑ j : Fin (n + 1), a x i j • ((fderiv ℝ (fderiv ℝ u) x) (EuclideanSpace.single i 1)) (EuclideanSpace.single j 1)) Ω → HolderOnWith (C * (M + N)) α (iteratedFDeriv ℝ 2 u) K

The interior Schauder estimate for Hölder coefficients. Let 0 < α < 1, let Ω ⊆ ℝ^{n+1} be open and K ⊆ Ω compact, and let a be a matrix-valued coefficient which is α-Hölder on a ball around each x ∈ K, with a x elliptic at each x ∈ K, that is, λₓ ‖ξ‖² ≤ ξᵀ a(x) ξ for some λₓ > 0. Put L u = ∑ᵢⱼ aᵢⱼ ∂ᵢ∂ⱼu. Then there is a constant C, depending on a, K and Ω but not on u, with the following property. Let u be C² on Ω, with |u|, ‖Du‖ and ‖D²u‖ bounded by N on Ω and with D²u α-Hölder on every compact subset of Ω (with any constants). If L u is α-Hölder on Ω with constant M, then D²u is α-Hölder on K with constant C (M + N).