Weak derivative

A weak derivative is a function or distribution defined by transferring differentiation onto smooth, compactly supported test functions inside an integral. It extends ordinary differentiation to functions that may not be classically differentiable.

Test function: A smooth function with compact support, used to probe functions or distributions through integration. Weak derivatives are defined by requiring an integration identity for every such function.
Lebesgue integral: An integral defined using measure, accommodating broad classes of measurable functions. Weak derivative identities use integrals that remain meaningful beyond continuous functions.
Weak solution: A function satisfying a differential equation after the equation is expressed through integral identities against test functions. Weak derivatives let equations be imposed without requiring pointwise differentiability.
Cantor function: A continuous, nondecreasing function that is constant on the intervals removed in constructing the Cantor set. Its singular variation is not captured by an integrable weak derivative.
Distribution: A continuous linear functional on a space of smooth test functions, generalizing functions and their derivatives. The test-function definition first produces a distribution, even when no ordinary derivative exists.
Almost-everywhere equality: Equality of functions except on a set of measure zero. Weak derivatives represented by functions are determined only up to changes on null sets.
Finite element method: A numerical method that approximates solutions of differential equations using finite-dimensional spaces of piecewise-defined functions. Its trial functions often have weak derivatives despite having derivative jumps.
Absolute continuity: A function is absolutely continuous when small total interval length forces small total variation across those intervals. On an interval, absolute continuity characterizes functions recoverable by integrating their weak derivative.
Distributional derivative: The derivative of a distribution, defined by transferring differentiation to test functions with a sign change. A weak derivative is a distributional derivative represented by a function.
Sobolev space: A function space whose elements and specified weak derivatives are integrable to chosen powers. Sobolev spaces collect functions according to the integrability of their weak derivatives.

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