Weak derivative
A weak derivative is in the functional analysis, a branch of mathematics, an extension of the concept of ordinary (classical) derivation. It enables functions to assign a derivation that are not (strongly or in the classical sense ) is differentiable.
Weak derivatives play an important role in the theory of partial differential equations. Rooms weakly differentiable functions are the Sobolev spaces. An even more general term of the derivative is the distribution derivative.
- 2.1 uniqueness
- 2.2 Relation to the classical (strong) derivative
- 2.3 existence
Definition
Weak derivative of real functions
Looking at one on an open interval ( classically ) differentiable function and a test function ( that is, infinitely differentiable and has compact support ), then
Here, the integration by parts was used, where the boundary terms disappear due to the properties of the test functions.
If a function, then, even if not differentiable (more precisely, no differentiable representative in the equivalence class has ), there exist a function, the equation
Met for each test function. Such a function is called a weak derivative of. We write as in the classical derivation.
Higher weak derivatives
Analogously to the case described above, weak derivatives can also be defined for functions on higher dimensional spaces. Accordingly, one can also define the higher weak derivatives.
There are a square integrable function, that is, and a multi- section.
A function is called -th weak derivative of, if for all test functions:
Here, and. Often one writes.
You can ask for instead more generally. The subset of functions in the weak derivatives exist is a so-called Sobolev space.
If there is a function, so calls to the weak differentiability in each of the image components.
Extensions
The definition of the weak derivative can be applied to unlimited quantities so wholly, or expand rooms or spaces of periodic functions on the sphere, or higher-dimensional spheres.
In a further generalization can also be derivatives of fractional order to win.
Properties
Unambiguity
The weak derivative, if it exists, is unique: If there were two weak derivatives and would then have by definition
Apply to all test functions, but what about after the Lemma of Du Bois- Reymond means ( in - sense, ie almost everywhere ), since the test functions are dense in ( for ).
Relationship to the classical (strong) derivative
In each classically differentiable function, the weak derivative exists and coincides with the classical derivation, so that one can speak of a generalization of the derivation of the term. In contrast to the classical derivation of the weak derivative is not pointwise, but only defined for the entire function. Pointwise does not even exist a weak derivative. Equality is therefore to be understood in the sense -, that is, Two functions are equal if and only if the following holds.
It can be shown that sufficiently often existing weak differentiability again draws differentiable in the classical sense by themselves. This is just the statement of the embedding theorem of Sobolev: Under certain conditions exist embeddings of Sobolev space with weak derivatives in space - times differentiable functions with.
Existence
- An absolutely continuous function has a weak derivative.
Examples
The absolute value function ( see Example not differentiable ) at every point except classical differentiable and therefore has to completely no classical derivation. On the function
The classical derivation of. This function can also be continued to a weak derivative of the value function ( at all ), since a null set, and therefore in the integration is insignificant (it is 0, set the value at the point desired, as well as on any other subset with dimension 0). This, to continue operation, ie signum function. The signum function itself is no longer weakly differentiable, but you can in the sense of distributions derived.
Applications
Weak derivatives are systematically studied in Sobolev spaces and used for example in solving partial differential equations.