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Bounded measurable coefficients do not give Schauder estimates

Statement refuted

The assertion that uniform ellipticity with merely bounded (even bounded continuous) principal coefficients suffices for a $C^{2, α}$ conclusion for classical $C^{2}$ solutions of $L u = f$ is false; the counterexample below exhibits a bounded continuous uniformly elliptic $L$ , a classical $C^{2}$ solution of $L u = 1$ , and a coefficient that fails to make $D^{2} u$ $α$ -Hölder for any exponent larger than the coefficient modulus. This does not contradict the interior Schauder theorem of this page, which assumes $C^{0, α}$ principal coefficients.

Facts & Assumptions

Given: $n \geq 2$ , $0 < β < α < 1$ , the unit ball $B_{1} (0) \subseteq R^{n}$ , the diagonal matrix field $A (x) = diag (1 + ∣ x_{1} ∣^{β}, 1, \dots, 1)$ , and the function $u (x) = \int_{0}^{x_{1}} \frac{x_{1} - s}{1 + ∣ s ∣^{β}} d s$ .

[A1]

The only choice assumption is Countable Choice ${A C}_{ω}$ ; no full Axiom of Choice is used. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

Uniform ellipticity and the nondivergence operator were fixed in Uniformly elliptic nondivergence-form operators and their frozen coefficients: $L = A^{i j} \partial_{i} \partial_{j}$ is uniformly elliptic with constants $λ, Λ$ when the symmetric matrix field satisfies $λ ∣ ξ ∣^{2} \leq A^{i j} (x) ξ_{i} ξ_{j} \leq Λ ∣ ξ ∣^{2}$ . The Hölder classes and their seminorms are those of Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains. Coordinates $x_{i}$ and partials $\partial_{i}$ use the operator's one-based relabelling of coordinate $i - 1$ and $\partial_{i - 1}$ in $C^{k}$ maps and multi-index derivative notation in Euclidean space; multi-index derivatives retain that dependency's canonical order.

Counterexample

technique · direct

1.1F1givenalgebra

The coefficient field. For $x \in B_{1} (0)$ the matrix $A (x) = diag (1 + ∣ x_{1} ∣^{β}, 1, \dots, 1)$ is symmetric with eigenvalues $1 + ∣ x_{1} ∣^{β} \in [1, 2]$ , so it is bounded, continuous and uniformly elliptic on $B_{1} (0)$ with $λ = 1$ , $Λ = 2$ ; its off-diagonal entries vanish, and its $C^{0, α}$ seminorm on $B_{1} (0)$ is infinite because $∣ x_{1} ∣^{β}$ is not $α$ -Hölder at $0$ when $α > β$ .

1.2F1givenalgebra

The solution. Since the integrand $\frac{x_{1} - s}{1 + ∣ s ∣^{β}}$ vanishes at $s = x_{1}$ , differentiating under the integral sign gives $u^{'} (x_{1}) = \int_{0}^{x_{1}} \frac{d s}{1 + ∣ s ∣^{β}}$ and $u^{''} (x_{1}) = \frac{1}{1 + ∣ x_{1} ∣^{β}}$ ; both are continuous on $R$ , so $u \in C^{2}$ , and $u$ depends on $x_{1}$ only. Hence $A^{i j} (x) \partial_{i} \partial_{j} u (x) = A^{11} (x) u^{''} (x_{1}) = (1 + ∣ x_{1} ∣^{β}) \cdot \frac{1}{1 + ∣ x_{1} ∣^{β}} = 1$ for every $x \in B_{1} (0)$ , that is $L u = 1$ with $f \equiv 1$ , which is as smooth as possible.

2.1step 1.2F1algebra

Failure of the Hölder estimate. For $h \to 0$ one has $u^{''} (h) - u^{''} (0) = \frac{1}{1 + ∣ h ∣^{β}} - 1 = - \frac{∣ h ∣^{β}}{1 + ∣ h ∣^{β}}$ , so $\frac{∣ u^{''} (h) - u^{''} (0) ∣}{∣ h ∣^{α}} = \frac{∣ h ∣^{β - α}}{1 + ∣ h ∣^{β}} \to + \infty$ because $β < α$ ; hence $[u^{''}]_{0, α; B_{1} (0)} = + \infty$ and $u \notin C^{2, α} (B_{1} (0))$ for the given $α \in (0, 1)$ .

3.1step 1.1step 1.2step 2.1A1given∎

Conclusion. Steps 1.1, 1.2 and 2.1 give a uniformly elliptic nondivergence operator with bounded continuous (in particular bounded measurable) principal coefficients and a classical $C^{2}$ solution of $L u = 1$ on $B_{1} (0)$ whose second derivative fails to be $α$ -Hölder. Therefore bounded measurability of the coefficients does not force $C^{2, α}$ regularity; such a statement is false as stated, and the Hölder hypothesis on $A$ in the interior Schauder estimate of this page is not a technical convenience. The example uses no divergence-form interpretation and no choice beyond [A1].

Remarks

The failure is driven by the coefficient's own modulus: $A^{11} - 1 = ∣ x_{1} ∣^{β}$ is $β$ -Hölder, and the solution inherits exactly that modulus in $u^{''}$ ; a $C^{0, α}$ coefficient with $α > β$ would require $u^{''}$ to gain that Hölder regularity, which the example shows cannot be expected without the hypothesis.
The coefficient field here is continuous, so the example also refutes the stronger claim with "continuous" in place of "measurable"; its coefficient modulus is Dini and has vanishing mean oscillation as well. Those weaker hypotheses cannot force the particular $C^{2, α}$ conclusion refuted here; they may support different regularity conclusions.

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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)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, complete 392 pages) (standard reference, not scraped)