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$W^{2, p}$ regularity implies classical or H"older regularity when $p$ is large

Statement

Assume the Axiom of Choice. Let $n \geq 2$ , $1 < p < \infty$ , and let $Ω$ be a bounded $W^{2, p}$ -extension domain (the extension property is for this displayed $k = 2$ and exponent $p$ ). (i) If $p > n / 2$ , every $u \in W^{2, p} (Ω)$ has a representative in $C^{0, γ} (\overset{ˉ}{Ω})$ for every $0 < γ < \min {1, 2 - n / p}$ , with the norm controlled by $∥ u ∥_{W^{2, p}}$ . (ii) If $p > n$ , every such $u$ has a representative in $C^{1, γ} (\overset{ˉ}{Ω})$ for every $0 < γ < 1 - n / p$ , with the corresponding norm bound; in particular $D^{2} u \in L^{p}$ . For any nondivergence expression $L u = \sum_{∣ β ∣ \leq 2} a_{β} D^{β} u$ with $a_{β} \in L^{\infty} (Ω)$ , this gives $L u \in L^{p} (Ω)$ ; whenever $L u = f$ also holds distributionally for some $f \in L^{p} (Ω)$ , the equality then holds almost everywhere. (iii) If $p > n / (1 - α)$ for $0 < α < 1$ , the representative is $C^{1, α} (\overset{ˉ}{Ω})$ . No finite $p$ gives $C^{2, α}$ regularity in general: on a ball, the function $u (x) = (x_{1})_{+}^{2}$ belongs to $W^{2, p}$ for every finite $p$ , while $D_{11} u = 21_{{x_{1} > 0}}$ has no continuous representative.

Facts & Assumptions

Given: the Axiom of Choice, $n \geq 2$ , $1 < p < \infty$ , the bounded $W^{2, p}$ -extension domain $Ω$ , and $u \in W^{2, p} (Ω)$ .

[A1]

The Axiom of Choice is the standing hypothesis, used through the extension operator and the embedding theorem below. (The Axiom of Choice)

[F1]

By the extension property there is a bounded linear $E : W^{2, p} (Ω) \to W^{2, p} (R^{n})$ with $(E u) ∣_{Ω} = u$ almost everywhere and $∥ E u ∥_{W^{2, p} (R^{n})} \leq C_{E} ∥ u ∥_{W^{2, p} (Ω)}$ . (Sobolev extension domains and extension operators)

[F2]

Higher-order Sobolev embedding (Higher-order Sobolev embedding): for a bounded extension domain and $k \geq 1$ , $1 \leq q < \infty$ , (a) if $k q < n$ then $W^{k, q} ↪ L^{r}$ for every $1 \leq r \leq n q / (n - k q)$ ; (b) if $k q = n$ then $W^{k, q} ↪ L^{r}$ for every finite $r$ ; (c) if $k q > n$ then every $u \in W^{k, q}$ has a representative in $C^{m, β} (\overset{ˉ}{Ω})$ for every integer $m \geq 0$ and $0 < β < 1$ with $m + β < k - n / q$ , with the norm bounded by a constant times $∥ u ∥_{W^{k, q}}$ . Applied with $k = 2$ and $q = p$ , the three cases are $2 p < n$ , $2 p = n$ , $2 p > n$ ; the case $2 p > n$ is exactly $p > n / 2$ . (Higher-order Sobolev embedding)

[F3]

The witness on the unit ball $Ω = B (0, 1)$ : for $u (x) = (x_{1})_{+}^{2} : = \max {x_{1}, 0}^{2}$ , the weak derivatives are $\partial_{1} u = 2 (x_{1})_{+}$ , $\partial_{11} u = 21_{{x_{1} > 0}}$ and $\partial_{i j} u = 0$ otherwise. Thus $u \in W^{2, p} (Ω)$ for every finite $p$ . On the disk section $Ω \cap {x_{1} = 0}$ the one-sided values of $\partial_{11} u$ differ, so this weak derivative has no continuous representative; consequently $u \notin C^{2, α} (\overset{ˉ}{Ω})$ for every $0 < α < 1$ . (Integer-order Sobolev spaces and their norms)

[F4]

The unit ball is a bounded $C^{2}$ domain and hence a $W^{2, p}$ -extension domain for every $1 \leq p \leq \infty$ (Bounded C^k domains and boundary charts, Bounded C^k domains admit integer-order Sobolev extension).

[F5]

If two locally integrable functions represent the same distribution on $Ω$ , they agree almost everywhere; this is the uniqueness of the zeroth weak derivative (Uniqueness of a weak derivative as an almost-everywhere class).

Proof

technique · direct application of the bounded-domain embedding

1.1F1F2givenA1

Domain hypothesis. By the definition of a $W^{2, p}$ -extension domain [F1], $Ω$ satisfies the bounded-domain premise of [F2] for $k = 2$ and exponent $p$ . The embedding conclusion of [F2] is already on $\overset{ˉ}{Ω}$ ; no embedding on the unbounded space $R^{n}$ is used.

1.2F3F4givenalgebra

The ball witness: no finite $p$ gives $C^{2, α}$ . On $Ω = B (0, 1)$ let $u (x) = \max {x_{1}, 0}^{2}$ . The function is $C^{1}$ with $\partial_{1} u = 2 \max {x_{1}, 0}$ , and integration by parts on the two sides of $x_{1} = 0$ gives the weak derivative $\partial_{11} u = 21_{{x_{1} > 0}}$ ; the interface term vanishes because $\max {x_{1}, 0}$ is continuous there. All second derivatives are bounded, so $u \in W^{2, p} (Ω)$ for every finite $p$ . If $\partial_{11} u$ had a continuous representative, it would equal $0$ on the negative open half-ball and $2$ on the positive open half-ball: the almost-everywhere equalities force these values on each open side by continuity. They cannot extend continuously across the interior disk $Ω \cap {x_{1} = 0}$ . Thus $u$ has no $C^{2, α}$ representative for any $0 < α < 1$ . The ball is in the stated extension-domain class by [F4].

2.1step 1.1F2algebra

Part (i): $p > n / 2$ . Then $2 p > n$ . For every $0 < γ < \min {1, 2 - n / p}$ the embedding [F2] with $k = 2$ , $q = p$ , $m = 0$ gives a representative $u^{*} \in C^{0, γ} (\overset{ˉ}{Ω})$ and $∥ u^{*} ∥_{C^{0, γ} (\overset{ˉ}{Ω})} \leq C ∥ u ∥_{W^{2, p} (Ω)}$ . If $2 - n / p \geq 1$ , the same strict inequality allows every $0 < γ < 1$ .

2.2step 1.1F2F5algebra

Part (ii): $p > n$ . Then $2 - n / p > 1$ . For each $0 < γ < 1 - n / p$ , one has $1 + γ < 2 - n / p$ , so [F2] with $k = 2$ , $q = p$ , $m = 1$ gives a representative $u^{*} \in C^{1, γ} (\overset{ˉ}{Ω})$ and the stated norm bound. The weak derivatives $D^{β} u$ with $∣ β ∣ \leq 2$ are $L^{p}$ classes by the definition of $W^{2, p}$ . Thus for $L = \sum_{∣ β ∣ \leq 2} a_{β} D^{β}$ with bounded coefficients, $L u$ is an $L^{p}$ class. If also $L u = f$ distributionally with $f \in L^{p}$ , then [F5] gives equality of the represented classes almost everywhere.

2.3step 1.1F2algebra

Part (iii): $p > n / (1 - α)$ . Then $1 + α < 2 - n / p$ , so [F2] with $k = 2$ , $q = p$ , $m = 1$ and exponent $α$ gives a representative in $C^{1, α} (\overset{ˉ}{Ω})$ with norm bounded by $C ∥ u ∥_{W^{2, p} (Ω)}$ . Parts (i), (ii), (iii) are direct applications of the bounded-domain higher-order embedding.

3.1step 2.1step 2.2step 2.3step 1.2F2∎

Conclusion. The higher-order embedding gives the asserted $C^{0, γ}$ representatives when $p > n / 2$ , $C^{1, γ}$ representatives when $p > n$ , and $C^{1, α}$ representatives when $p > n / (1 - α)$ . The ball witness of step 1.2 shows that no finite $p$ forces $C^{2, α}$ regularity in general, so the Sobolev and Schauder scales differ at the top order.

Remarks

The hypothesis is indexed by $(k, p) = (2, p)$ : a domain that is an extension domain for one pair need not be for another, and the statement uses only the displayed pair. The counterexample on the ball shows that the extension property alone, or any finite $p$ , cannot produce two H"older derivatives.
The strict exponent ranges in the statement are sufficient, rather than an assertion of optimality. For $p > n$ , the endpoint clause of Higher-order Sobolev embedding also gives $C^{1, 1 - n / p} (\overset{ˉ}{Ω})$ , since $2 - n / p \in (1, 2)$ is nonintegral; for example $p = 2 n$ gives $C^{1, 1 / 2}$ . The ball witness shows that no finite $p$ forces two Hölder derivatives.

Depends on

Used by

The Schauder and W^2,p scales are different, not interchangeable Remark

Dependency tree · two levels

42 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Villavert, Elementary Theory and Methods for Elliptic Partial Differential Equations (2017; complete 220-page lecture notes) (standard reference, not scraped)
Armin Schikorra, Partial Differential Equations I & II (version October 1, 2025; complete 281-page graduate lecture notes) (standard reference, not scraped)

W2,pW2,p regularity implies classical or H"older regularity when pp is large