DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-sol)audited 2026-09-30

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Weak derivative of a locally integrable function

Definition

Let $Ω \subseteq R^{n}$ be open, $n \geq 1$ , let $u, v \in L_{l o c}^{1} (Ω)$ , and let $α \in N_{0}^{n}$ be the multi-index of $C^{k}$ maps and multi-index derivative notation in Euclidean space. We say that $v$ is the weak derivative $D^{α} u$ if $\int_{Ω} u D^{α} φ d x = (- 1)^{∣ α ∣} \int_{Ω} v φ d x for every φ \in C_{c}^{\infty} (Ω) .$ Both integrals are finite because the test and all its derivatives are bounded and compactly supported. The test identity is choice-free. By Locally integrable functions as regular distributions and the signed transpose convention in Distributional derivative, it is equivalent to $\partial^{α} T_{u} = T_{v}$ as distributions. For $α = 0$ , this identity reduces to $T_{u} = T_{v}$ , hence to $u = v$ almost everywhere by the published embedding theorem under The Axiom of Countable Choice ( ${A C}_{ω}$ ).

Well-definedness. Changing either representative on a null set leaves each test integral unchanged. Weak derivatives are therefore statements about almost-everywhere classes; existence and uniqueness of their locally integrable value classes are audited separately.

Depends on

Used by

The Sobolev space W^k,p is an algebra above the critical index Corollary
Weak differentiation has a closed graph on its natural domains Corollary
A gradient-only Poincare estimate needs normalisation Counterexample
A hypersurface jump is not W^1,p Counterexample
A mild solution need not be classical Counterexample
A nonzero boundary value creates a zero-extension jump Counterexample
A step has no locally integrable weak derivative Counterexample
Bounded discontinuous elliptic coefficients need not give H² solutions Counterexample
Cantor function has singular distributional derivative Counterexample
Degenerate ellipticity allows nonconstant solutions with interior zero sets Counterexample
Higher elliptic regularity cannot gain more than two derivatives Counterexample
Matching patch geometry does not repair a field trace jump Counterexample
Poincare-Wirtinger fails on disconnected bounded domains Counterexample
The translation semigroup is not analytic Counterexample
W^1,n is not contained in L^∞ Counterexample
Covariant and contravariant Piola maps and H(div)/H(curl) spaces Definition
Integer-order Sobolev spaces and their norms Definition
The ACL and Sobolev analytic definition of quasiconformality Definition
The Beltrami coefficient and the maximal dilatation Definition
The real H(curl) graph space on a polyhedron Definition
Uniformly elliptic divergence-form operators and their sesquilinear forms Definition
Weak solutions of the Beltrami equation Definition
Weighted L2 spaces and maximal dbar operators Definition
A piecewise-smooth coefficient gives an H² solution that is not twice classically differentiable Example
Absolute value has a Dirac second derivative Example
Matching C¹ pieces across a hyperplane have no jump derivative Example
Measurable coefficients with a Holder-regular weak solution Example
Mollifying the absolute-value corner Example
Radial powers approach the Morrey borderline exponent Example
Sharp Sobolev threshold for a radial power Example
The absolute value has a weak first derivative Example
The reentrant sector singularity has an explicit Sobolev threshold Example
The right-translation semigroup on Lp has the weak derivative as generator Example
A normalized parabolic Bellman equation has a bounded weak solution Lemma
ACL representatives recover their weak gradients by Fubini Lemma
Affine pullback of weak derivatives and Sobolev seminorms Lemma
Bi-Lipschitz NURBS maps preserve H1 and energy norms Lemma
Classical derivatives agree with weak derivatives Lemma
Compactly supported Sobolev functions extend by zero in every integer order Lemma
Continuous piecewise-affine functions with zero endpoints lie in H1-zero Lemma

…and 40 more results.

Dependency tree · two levels

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Sources

Juha Kinnunen, Sobolev Spaces (2026), Chapter 1 §1.1 (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (2014), §3.1 (standard reference, not scraped)