CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Sobolev maxima and minima form a lattice

Sources

Juha Kinnunen, Sobolev Spaces, Chapter 2 §2.2, Remark 2.4(3) (printed p. 31), where the lattice property of $W^{1, p}$ is recorded: $\max {u, v}$ and $\min {u, v}$ lie in $W^{1, p}$ , the gradients are $D u$ and $D v$ on the respective comparison regions, and $D u = D v$ almost everywhere on ${u = v}$ . The source derives the rule from $\max {u, v} = \frac{1}{2} (u + v + ∣ u - v ∣)$ and the absolute-value rule; the proof below instead uses the positive-part calculus of the preceding corollary together with the pointwise identities $u \lor v = v + (u - v)^{+}$ and $u \land v = u - (u - v)^{+}$ , as the design directs.
John K. Hunter, Notes on Partial Differential Equations, Chapter 3, §§3.1–3.5, for the weak-derivative convention, the Sobolev spaces and the lattice operations on Sobolev functions.
Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Chapter 8 §8.2, where the corresponding truncation and absolute-value rules appear as exercises; the argument below is reconstructed from the library interfaces cited in the Facts block and does not import those statements.

Statement

Assume the Axiom of Choice. Let $Ω \subseteq R^{n}$ be open, $n \geq 1$ , let $1 \leq p \leq \infty$ , and let $u, v \in W^{1, p} (Ω; R)$ be real Sobolev classes. On measurable representatives define $u \lor v : = \max {u, v}$ and $u \land v : = \min {u, v}$ pointwise; the resulting almost-everywhere classes are well defined. Then $u \lor v \in W^{1, p} (Ω)$ and $u \land v \in W^{1, p} (Ω)$ , and for every $i \in {1, \dots, n}$ , almost everywhere on $Ω$ , $D_{i} (u \lor v) = 1_{{u > v}} D_{i} u + 1_{{u \leq v}} D_{i} v, D_{i} (u \land v) = 1_{{u < v}} D_{i} u + 1_{{u \geq v}} D_{i} v .$ Moreover the two gradients agree on the coincidence set: on ${u = v}$ one has $D_{i} (u \lor v) = D_{i} (u \land v)$ almost everywhere, equivalently $D_{i} u = D_{i} v$ almost everywhere there. If $Ω = \emptyset$ every assertion holds vacuously, the endpoints $p = 1$ and $p = \infty$ are included, and no assertion is made about the pointwise derivative of an arbitrary representative.

Facts & Assumptions

Given: The Axiom of Choice; an open $Ω \subseteq R^{n}$ with $n \geq 1$ ; an exponent $1 \leq p \leq \infty$ ; real Sobolev classes $u, v \in W^{1, p} (Ω; R)$ ; and the pointwise lattice operations of the Statement, applied to measurable representatives.

[F1]

$W^{1, p} (Ω; K)$ is the set of classes $u \in L^{p} (Ω; K)$ such that for every first-order multi-index there is an $L^{p}$ class with a locally integrable representative satisfying the signed test identity for every test function, and each such derivative determines one class $D_{i} u$ (Integer-order Sobolev spaces and their norms).

[F2]

For a function $f$ one has $f^{+} = \max {f, 0}$ , $f^{-} = \max {- f, 0}$ , $f = f^{+} - f^{-}$ and $∣ f ∣ = f^{+} + f^{-}$ pointwise (The positive and negative parts of a function).

[F3]

If $S \subseteq R$ and $m \in R$ , then $m$ is a maximum of $S$ when $m \in S$ and $s \leq m$ for every $s \in S$ , and a minimum of $S$ when $m \in S$ and $m \leq s$ for every $s \in S$ ; a set has at most one maximum and at most one minimum, written $\max S$ and $\min S$ (Maximum and minimum of a set).

[F4]

An ordered field is a field with a positive cone $P$ satisfying trichotomy and closure, with $a < b$ defined by $b - a \in P$ and $a \leq b$ by $a < b$ or $a = b$ (Ordered field).

[F5]

Assume Countable Choice. If two locally integrable classes agree almost everywhere, then one is a weak $α$ -derivative of a class if and only if the other is; in particular this applies to representatives of $L^{p} (Ω)$ classes for every $1 \leq p \leq \infty$ , which are locally integrable on each compact test support (Weak differentiation ignores null-set changes).

[F6]

Assume Countable Choice. Weak differentiation is complex-linear and passes to open subsets: if $v_{j} = D^{α} u_{j}$ weakly for $j = 1, 2$ and $a, b \in C$ , then $a v_{1} + b v_{2} = D^{α} (a u_{1} + b u_{2})$ weakly (Linearity, locality, and commutation of weak derivatives).

[F7]

Assume the Axiom of Choice. For $w \in W^{1, p} (Ω; R)$ one has $w^{+} \in W^{1, p} (Ω)$ with $D_{i} w^{+} = 1_{{w > 0}} D_{i} w$ almost everywhere, and $D_{i} w = 0$ almost everywhere on ${w = 0}$ (Positive, negative, and truncated Sobolev functions).

[F8]

If $f, g : X \to \overline{R}$ are measurable on a measurable space, then $\max (f, g)$ , $\min (f, g)$ , $f + g$ (where defined) and the pointwise product $f g$ are measurable (Arithmetic and lattice operations preserve measurability whenever they are defined).

[F9]

If $f$ is measurable and $g$ is Borel measurable on its codomain, then $g \circ f$ is measurable (Composition with a Borel measurable outer map preserves measurability).

[F10]

For nonnegative measurable $f \leq g$ one has $\int f \leq \int g$ , and $\int c f = c \int f$ for $c \geq 0$ (Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F11]

For $1 \leq p < \infty$ the class $L^{p} (μ)$ is a real vector space under pointwise addition and scalar multiplication, and so is $L^{\infty} (μ)$ ( $L^{p}$ and $L^{\infty}$ are vector spaces for $p \geq 1$ ).

[F12]

For $1 \leq p < \infty$ , $L^{p} (μ)$ consists of the measurable $f$ with $\int ∣ f ∣^{p} d μ < \infty$ , and $L^{p} (μ)$ denotes the quotient of $L^{p} (μ)$ by the almost-everywhere-zero functions (The function space $L^{p} (μ)$ for $0 < p < \infty$ ).

[F13]

On a measure space, $L^{p} (μ)$ for $0 < p < \infty$ and for $p = \infty$ is the set of almost-everywhere classes of $L^{p} (μ)$ respectively $L^{\infty} (μ)$ , and for $1 \leq p \leq \infty$ the displayed quotient agrees with the usual quotient-vector-space construction (The space $L^{p} (μ)$ as the quotient by null functions).

[F14]

$L^{\infty} (μ) = {f : X \to R : f$ measurable and $∥ f ∥_{\infty} < \infty}$ for a measure space $(X, A, μ)$ (The space $L^{\infty} (μ)$ of essentially bounded measurable functions).

[F15]

If $f$ is measurable with $∥ f ∥_{\infty} < \infty$ , then $∣ f ∣ \leq ∥ f ∥_{\infty}$ almost everywhere; and if $∣ f ∣ \leq M$ almost everywhere, then $∥ f ∥_{\infty} \leq M$ (The essential supremum is attained as the least essential bound).

[F16]

In ZF the Axiom of Choice implies Countable Choice and the prescribed-start form of Dependent Choice (AC supplies the countable and dependent choices used in Banach integration).

[F17]

The Axiom of Choice asserts a choice function for every family of nonempty sets (The Axiom of Choice).

Proof

technique · direct: write $u\vee v=v+(u-v)^+$ and $u\wedge v=u-(u-v)^+$, apply the positive-part calculus of the preceding corollary to $u-v$, and use the level-set identity on $\{u=v\}$ to make the two indicator conventions agree almost everywhere

1.1F5F12F13F16F17given

By [F16] the Axiom of Choice [F17] yields Countable Choice and Dependent Choice in ZF, so the choice hypotheses of [F5] and [F6] and the Axiom-of-Choice hypothesis of [F7] are in force. Fix one measurable representative $\hat{u}$ of $u$ , one measurable representative $\hat{v}$ of $v$ , and for each $i$ one measurable representative $u_{i}$ of $D_{i} u$ and one $v_{i}$ of $D_{i} v$ ; these are finitely many selections and need no choice principle. The classes here are the almost-everywhere classes of [F12] and [F13], and by [F5] every one of these representatives is locally integrable on each compact subset of $Ω$ .

1.2F2F3F4given

For all real $a, b$ one has $\max {a, b} = b + (a - b)^{+}$ and $\min {a, b} = a - (a - b)^{+}$ : if $a \geq b$ then $a - b \geq 0$ by [F4], so $(a - b)^{+} = \max {a - b, 0} = a - b$ by [F2] and [F3], giving $b + (a - b) = a = \max {a, b}$ and $a - (a - b) = b = \min {a, b}$ ; and if $a < b$ then $a - b < 0$ , so $(a - b)^{+} = 0$ and the two right-hand sides are $b$ and $a$ .

2.1F1F6F11F12F13step 1.1given

Put $w : = u - v$ . By step 1.1 the classes $u, v, D_{i} u, D_{i} v$ are locally integrable, so [F6] with $a = 1$ , $b = - 1$ gives $D_{i} u - D_{i} v = D_{i} (u - v)$ weakly on $Ω$ , hence almost everywhere on $Ω$ ; and $w \in L^{p} (Ω)$ with each $D_{i} w \in L^{p} (Ω)$ by [F11], [F12] and [F13]. By [F1] this says $w \in W^{1, p} (Ω; R)$ with the weak derivatives $D_{i} w = D_{i} u - D_{i} v$ .

3.1F7step 2.1given

The corollary [F7] applied to the class $w = u - v$ of step 2.1 gives $w^{+} \in W^{1, p} (Ω)$ with $D_{i} w^{+} = 1_{{w > 0}} D_{i} w$ almost everywhere on $Ω$ , and $D_{i} w = 0$ almost everywhere on ${w = 0}$ , so $1_{{w = 0}} D_{i} w = 0$ and $D_{i} u = D_{i} v$ almost everywhere on ${u = v} = {w = 0}$ .

4.1F1F2F6F8F12F13step 1.2step 2.1step 3.1given

Define the classes $u \lor v : = v + w^{+}$ and $u \land v : = u - w^{+}$ . Both lie in $W^{1, p} (Ω)$ with weak derivatives $D_{i} (u \lor v) = D_{i} v + D_{i} w^{+}$ and $D_{i} (u \land v) = D_{i} u - D_{i} w^{+}$ almost everywhere, by [F6] and [F1]; and by step 1.2 the pointwise identities $\max {\hat{u}, \hat{v}} = \hat{v} + (\hat{u} - \hat{v})^{+}$ and $\min {\hat{u}, \hat{v}} = \hat{u} - (\hat{u} - \hat{v})^{+}$ hold everywhere on $Ω$ with $(\hat{u} - \hat{v})^{+} = \max {\hat{u} - \hat{v}, 0}$ by [F2]. Since $\hat{u} - \hat{v}$ represents $w$ , the function $(\hat{u} - \hat{v})^{+}$ represents $w^{+}$ , so $\max {\hat{u}, \hat{v}}$ and $\min {\hat{u}, \hat{v}}$ are representatives of $u \lor v$ and of $u \land v$ ; they are measurable by [F8], so [F12] and [F13] make the pointwise maximum and minimum legitimate almost-everywhere classes with the same members as $u \lor v$ and $u \land v$ .

5.1F8F9F10F11F12F13F14F15step 2.1step 3.1step 4.1given

First derivative formula. By steps 2.1 and 3.1 the class $D_{i} (u \lor v)$ has the representative $v_{i} + 1_{{\hat{w} > 0}} (u_{i} - v_{i})$ almost everywhere, where $\hat{w} : = \hat{u} - \hat{v}$ ; on ${\hat{w} > 0}$ this equals $u_{i}$ and on ${\hat{w} \leq 0}$ it equals $v_{i}$ , so it agrees pointwise everywhere with $1_{{\hat{w} > 0}} u_{i} + 1_{{\hat{w} \leq 0}} v_{i}$ , which is measurable by [F8] and [F9]. Each product lies in $L^{p} (Ω)$ : for $p < \infty$ one has $∣ 1_{{\hat{w} > 0}} u_{i} ∣ \leq ∣ u_{i} ∣$ pointwise and $\int_{Ω} ∣ u_{i} ∣^{p} < \infty$ , so [F10] gives $\int_{Ω} ∣ 1_{{\hat{w} > 0}} u_{i} ∣^{p} \leq \int_{Ω} ∣ u_{i} ∣^{p} < \infty$ , and similarly for $v_{i}$ ; for $p = \infty$ one has $∣ 1_{{\hat{w} > 0}} u_{i} ∣ \leq ∣ u_{i} ∣ \leq ∥ u_{i} ∥_{\infty}$ almost everywhere by [F15], so $∥ 1_{{\hat{w} > 0}} u_{i} ∥_{\infty} \leq ∥ u_{i} ∥_{\infty} < \infty$ and the product lies in $L^{\infty} (Ω)$ by [F14]. Hence both products and their sum lie in $L^{p} (Ω)$ by [F11], [F12] and [F13], and therefore $D_{i} (u \lor v) = 1_{{u > v}} D_{i} u + 1_{{u \leq v}} D_{i} v$ almost everywhere on $Ω$ .

6.1F8F9F10F11F12F13F14F15step 2.1step 3.1step 4.1given

Second derivative formula. Likewise $D_{i} (u \land v)$ has the representative $u_{i} - 1_{{\hat{w} > 0}} (u_{i} - v_{i})$ , which agrees pointwise everywhere with $1_{{\hat{w} \leq 0}} u_{i} + 1_{{\hat{w} > 0}} v_{i}$ ; the same measurability and $L^{p}$ membership arguments as in step 5.1 apply, and on ${\hat{w} = 0}$ step 3.1 gives $u_{i} = v_{i}$ almost everywhere, so this representative also agrees almost everywhere with $1_{{\hat{w} < 0}} u_{i} + 1_{{\hat{w} \geq 0}} v_{i}$ . Therefore $D_{i} (u \land v) = 1_{{u < v}} D_{i} u + 1_{{u \geq v}} D_{i} v$ almost everywhere on $Ω$ .

7.1step 3.1step 5.1step 6.1given

Coincidence set. On ${u = v} = {\hat{w} = 0}$ , step 3.1 gives $D_{i} u = D_{i} v$ almost everywhere, while the formulas of steps 5.1 and 6.1 give $D_{i} (u \lor v) = D_{i} v$ and $D_{i} (u \land v) = D_{i} u$ there; hence $D_{i} (u \lor v) = D_{i} (u \land v)$ almost everywhere on ${u = v}$ , and the two gradient descriptions agree there.

8.1F5F6F7F10F14F15step 1.1step 2.1step 3.1step 4.1step 5.1step 6.1step 7.1given

Degenerate cases and accounting. If $Ω = \emptyset$ the only classes are zero and every assertion holds vacuously. If $u = v$ , then $w = 0$ , $w^{+} = 0$ , $u \lor v = u \land v = u$ , the two formulas both reduce to $D_{i} u = D_{i} v$ almost everywhere, and step 7.1 is consistent with that. The endpoints $p = 1$ and $p = \infty$ are included: step 3.1 uses [F7] for every $1 \leq p \leq \infty$ , and step 5.1 separates the finite and infinite exponent cases only through [F10], [F14] and [F15]. The case $n = 1$ is included because no step uses more than one coordinate direction. Only Countable Choice (through [F5] and [F6]) and the Axiom of Choice (through [F7]) are used, the representative selections of step 1.1 are finite, and no further selection is made. ∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

58 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

Juha Kinnunen, Sobolev Spaces (2026), Chapter 2 (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (2014), Chapter 3 (standard reference, not scraped)
Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011) (standard reference, not scraped)