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Freezing coefficients makes the Schauder error absorbable on a small ball

Statement

Let $n \geq 2$ , $0 < α < 1$ , $R > 0$ and $x_{0} \in R^{n}$ . Let $L = a^{i j} \partial_{i} \partial_{j} + b^{i} \partial_{i} + c$ be uniformly elliptic on $B_{R} (x_{0})$ with constants $0 < λ \leq Λ < \infty$ , $[A]_{0, α; B_{R} (x_{0})} \leq K$ , and $b, c \in C^{0, α} (B_{R} (x_{0}))$ . Put $L_{0} : = a^{i j} (x_{0}) \partial_{i} \partial_{j}$ and $M_{R} : = R^{α} K + R ∥ b ∥_{\infty} + R^{1 + α} [b]_{0, α} + R^{2} ∥ c ∥_{\infty} + R^{2 + α} [c]_{0, α} .$ Assume $M_{R} < \infty$ . For every $ε > 0$ there are $η_{ε} \in (0, 1]$ and $C_{ε} < \infty$ , depending only on $n, α, λ, Λ, M_{R}, ε$ , such that for every radius $0 < ρ \leq R η_{ε}$ and every $u$ with $∥ u ∥_{2, α; B_{ρ} (x_{0})}^{*} < \infty$ , $ρ^{2} ∥ (L - L_{0}) u ∥_{0, α; B_{ρ} (x_{0})}^{*} \leq ε ∥ u ∥_{2, α; B_{ρ} (x_{0})}^{*} + C_{ε} \sup_{B_{ρ} (x_{0})} ∣ u ∣,$ where the cutoff $R η_{ε}$ is at most $R$ and the constant is uniform over all smaller radii, and $∥ g ∥_{0, α; B_{ρ}}^{*} : = \sup_{B_{ρ}} ∣ g ∣ + ρ^{α} [g]_{0, α; B_{ρ}}$ .

Facts & Assumptions

Given: $n \geq 2$ , $0 < α < 1$ , $R > 0$ , $x_{0}$ , an operator $L$ with the coefficient bounds of the Statement, a fixed $ε > 0$ , and any $0 < ρ \leq R η_{ε}$ with $∥ u ∥_{2, α; B_{ρ} (x_{0})}^{*} < \infty$ .

[F1]

$L_{0}$ is the constant-coefficient operator with matrix $A (x_{0})$ , so $(L - L_{0}) u = (a^{i j} - a^{i j} (x_{0})) \partial_{i} \partial_{j} u + b^{i} \partial_{i} u + c u$ ; the coefficient bounds are recorded by $M_{R}$ in the Statement, and $x_{0}$ is the base point for every frozen coefficient. (Uniformly elliptic nondivergence-form operators and their frozen coefficients, Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains)

[F2]

In the normalized variables of step 1.1, put $K : = R^{α} K$ and let $B, C$ be the scaled $C^{0, α}$ bounds of $\tilde{b}, \tilde{c}$ ; then $K + B + C \leq M_{R}$ . On $B_{t}$ , $∣ {\tilde{a}}^{i j} (z) - {\tilde{a}}^{i j} (0) ∣ \leq K t^{α}$ and $[{\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0)]_{0, α; B_{t}} \leq K$ . Also $[D_{i} \tilde{u}]_{0, α; B_{t}} \leq (2 t)^{1 - α} \sup_{B_{t}} ∣ D^{2} \tilde{u} ∣$ and $[\tilde{u}]_{0, α; B_{t}} \leq (2 t)^{1 - α} \sup_{B_{t}} ∣ D \tilde{u} ∣$ . (Local Hölder and scaled C-two-alpha norms on balls, Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains)

[F3]

Interpolation with $ε$ -loss: for every $ε^{'} > 0$ there is $C_{ε^{'}}$ with (i) $ρ \sup_{B_{ρ}} ∣ D u ∣ \leq ε^{'} ∥ u ∥_{2, α; B_{ρ}}^{*} + C_{ε^{'}} \sup ∣ u ∣$ , (ii) $\sup ∣ D^{2} u ∣ \leq ε^{'} ρ^{α} [D^{2} u]_{0, α; B_{ρ}} + C_{ε^{'}} ρ^{- 2} \sup ∣ u ∣$ , and $ρ^{2 + α} [D^{2} u]_{0, α; B_{ρ}} \leq ∥ u ∥_{2, α; B_{ρ}}^{*}$ . (Ehrling-type Hölder and derivative interpolation with an epsilon loss)

[F4]

The product rule for the Hölder seminorm: $[f g]_{0, α} \leq [f]_{0, α} \sup ∣ g ∣ + \sup ∣ f ∣ [g]_{0, α}$ , and the elementary inequality $a b \leq ε a^{p} + C_{p} ε^{- 1 / (p - 1)} b^{p / (p - 1)}$ for $a, b \geq 0$ and $p > 1$ . (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Young's inequality for conjugate real exponents)

Proof

technique · direct

1.1F1F2F3given

Normalize the scale and record the coefficient oscillation. Put $z = (x - x_{0}) / R$ and $\tilde{u} (z) = u (x_{0} + R z)$ . In these variables the operator has coefficients $\tilde{a} (z) = a (x_{0} + R z)$ , $\tilde{b} (z) = R b (x_{0} + R z)$ and $\tilde{c} (z) = R^{2} c (x_{0} + R z)$ on $B_{1}$ , and their dimensionless Hölder bounds are controlled by $M_{R}$ in the Statement. Write $t : = ρ / R \leq 1$ . Since $x_{0}$ is the centre of the original ball, [F1] gives $\sup_{B_{t}} ∣ {\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0) ∣ \leq (R^{α} K) t^{α}$ and $[{\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0)]_{0, α; B_{t}} \leq R^{α} K$ . Fix $ε^{'} > 0$ to be chosen below and let $C_{ε^{'}}$ be the constant of [F3]; all estimates below are in the normalized variables on $B_{t}$ and the scaled norm is $∥ \tilde{u} ∥^{*} : = ∥ \tilde{u} ∥_{2, α; B_{t}}^{*}$ .

2.1step 1.1F2F3F4algebra

The second-order part. By [F4] and step 1.1, $[({\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0)) \partial_{i} \partial_{j} \tilde{u}]_{0, α} \leq K \sup ∣ D^{2} \tilde{u} ∣ + K t^{α} [D^{2} \tilde{u}]_{0, α}$ and $\sup ∣ ({\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0)) \partial_{i} \partial_{j} \tilde{u} ∣ \leq K t^{α} \sup ∣ D^{2} \tilde{u} ∣$ . Hence F3 and its last bound give $t^{2} (\sup ∣ ({\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0)) \partial_{i} \partial_{j} \tilde{u} ∣ + t^{α} [({\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0)) \partial_{i} \partial_{j} \tilde{u}]_{0, α}) \leq [ε^{'} (K + K t^{α}) + C K t^{α}] ∥ \tilde{u} ∥^{*} + C_{ε^{'}} K t^{α} \sup ∣ \tilde{u} ∣ .$ The $C K t^{α} ∥ \tilde{u} ∥^{*}$ contribution is the product-seminorm term $\sup ∣ {\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0) ∣ [D_{i} D_{j} \tilde{u}]_{0, α}$ after scaling; it has no interpolation factor $ε^{'}$ .

2.2step 1.1F2F3F4algebra

The lower-order part. Write $\tilde{M} : = ∥ \tilde{b} ∥_{C^{0, α}} + ∥ \tilde{c} ∥_{C^{0, α}} \leq M_{R}$ . By [F4] and [F2], the supremum of ${\tilde{b}}^{i} \partial_{i} \tilde{u} + \tilde{c} \tilde{u}$ is bounded by $\tilde{M} (\sup ∣ D \tilde{u} ∣ + \sup ∣ \tilde{u} ∣)$ , and its Hölder seminorm is bounded by $\tilde{M} (\sup ∣ D \tilde{u} ∣ + (2 t)^{1 - α} \sup ∣ D^{2} \tilde{u} ∣ + \sup ∣ \tilde{u} ∣ + (2 t)^{1 - α} \sup ∣ D \tilde{u} ∣)$ . Multiplying by $t^{2}$ and $t^{2 + α}$ , respectively, and inserting F3,(ii) shows that this contribution is at most $ε^{'} C_{1}^{l o w} (t) ∥ \tilde{u} ∥^{*} + C_{n, α, ε^{'}} (1 + \tilde{M}) \sup ∣ \tilde{u} ∣$ , where $C_{1}^{l o w} (t)$ is bounded for $0 < t \leq 1$ and has a finite limit as $t ↓ 0$ .

3.1step 1.1step 2.1step 2.2F3givenalgebra∎

Uniform choice of the normalized radius and conclusion. In normalized variables, collect the top-norm coefficients from steps 2.1 and 2.2 as $C_{e r r} (t) : = ε^{'} C_{i n t e r p} (t) + C_{o s c} K t^{α}$ , where $C_{i n t e r p} (t)$ is bounded on $0 < t \leq 1$ and has a finite limit at $0$ , and $C_{o s c}$ depends only on $n, α, λ, Λ$ . The second term explicitly includes the product-seminorm term of step 2.1, which has no $ε^{'}$ factor and tends to zero as $t ↓ 0$ . Given $ε > 0$ , choose first $ε^{'} > 0$ so that $ε^{'} C_{i n t e r p} (0) \leq ε / 4$ (if $C_{i n t e r p} (0) = 0$ , any positive $ε^{'}$ suffices); then choose $η_{ε} \in (0, 1]$ so small that $ε^{'} C_{i n t e r p} (t) \leq ε / 2$ and $C_{o s c} K t^{α} \leq ε / 2$ for every $0 < t \leq η_{ε}$ . Thus $C_{e r r} (t) \leq ε$ uniformly over every such radius. The lower-order remainder coefficients are also uniformly bounded there by $C_{ε}$ depending only on the displayed dimensionless parameters; the cap $t \leq 1$ is the small-scale condition used for those terms. Scaling back gives the same estimate for every physical radius $0 < ρ \leq R η_{ε}$ .

Remarks

The quantitative structure is the classical one: after normalization, the oscillation of the principal coefficients on a radius- $t$ ball is at most $(R^{α} K) t^{α}$ ; the frozen error carries the two extra derivatives scaled as $t^{2}$ , and interpolation converts the resulting powers into an arbitrarily small multiple of the full scaled norm plus a bounded multiple of $\sup ∣ u ∣$ .
The estimate is uniform over every smaller radius below $R η_{ε}$ . The cutoff fraction $η_{ε} \leq 1$ is chosen from the dimensionless coefficient bounds, including the small-scale cap needed for the lower-order terms.

Depends on

Used by

Freezing cannot absorb a fixed oscillation on arbitrarily small balls Counterexample
Interior Schauder estimate for uniformly elliptic equations Theorem

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Sources

Armin Schikorra, Partial Differential Equations I & II (version October 1, 2025; complete 281-page graduate lecture notes) (standard reference, not scraped)
John Villavert, Elementary Theory and Methods for Elliptic Partial Differential Equations (2017; complete 220-page lecture notes) (standard reference, not scraped)
Leon Simon, Lectures on Partial Differential Equations (Stanford University; complete 118-page author notes, Chapter 12 Schauder Theory) (standard reference, not scraped)