Weak derivation
In functional analysis , a branch of mathematics , a weak derivation is an extension of the term ordinary (classical) derivation . It makes it possible to assign a derivative to functions that are not (strongly or in the classical sense) differentiable.
Weak derivatives play a big role in the theory of partial differential equations . Spaces with weakly differentiable functions are the Sobolev spaces . An even more general term of derivation is the distribution derivation .
definition
Weak derivative for real functions
If one considers a function differentiable on an open interval (classically) , the derivative of which is a function (locally integrable), and a test function (that is, it can be differentiated any number of times and has a compact carrier ), then applies
- .
Here was partial integration used, the boundary terms due to the characteristics of the test functions cease . If the requirement for the integrability of the derivative is omitted, the integral on the left-hand side of the above equation is generally not well-defined.
If it is itself a -function, then, even if it is not differentiable (more precisely: has no differentiable representative in the equivalence class), a function can exist that corresponds to the equation
fulfilled for each test function. Such a function is called the weak derivative of . You write like the classical derivative .
Higher weak derivatives
Similar to the case described above, weak derivatives can also be defined for functions on higher-dimensional spaces. The higher weak derivatives can also be defined accordingly.
Let , be a locally integrable function, that is , and a multi-index .
A function is called the -th weak derivative of if the following applies to all test functions :
- .
Here is and . Often one writes .
You can place apparently only for calling. The subset of functions in which weak derivatives exist is a so-called Sobolev space .
If there is a function , then one requires the weak differentiability in each of the image components.
Extensions
The definition of the weak derivative can be extended to unlimited sets, i.e. whole or , spaces of periodic functions or spaces on the sphere or higher-dimensional spheres.
In a further generalization, derivatives of fractional orders can also be obtained.
properties
Uniqueness
The weak derivative, if it exists, is unambiguous: If there were two weak derivatives and , then according to the definition
apply to all test functions , which, according to the lemma of Du Bois-Reymond, means (in the -sense, i.e. almost everywhere ), since the test functions are close to (for ).
Relationship to the classical (strong) derivative
For every classically differentiable function whose derivative is a -function, the weak derivative exists and agrees with the classical derivative, so that one can speak of a generalization of the derivative concept. In contrast to the classical derivative, the weak derivative is not defined pointwise, but only for the entire function. Pointwise, a weak derivative does not even have to exist. Equality is therefore to be understood in the sense, i.e. H. almost everywhere.
It can be shown that weak differentiability that is sufficiently often present also entails differentiability in the classical sense. This is precisely the statement of Sobolev's embedding theorem : Under certain conditions, there are embeddings of a Sobolew space with weak derivatives in spaces with functions that can be differentiated by multiple .
existence
- An absolutely continuous function has a weak derivative.
Examples
- The absolute value function (see example of non-differentiable function ) is differentiable in every point except classically and therefore has no classical derivative in the interval for . However, applies to with
- and any test function just
- Thus is a weak derivative of .
- Since there is a zero set and is therefore insignificant in the integration, the value at 0 can be set as desired. The derivation chosen above is the signum function . The signum function itself is no longer weakly differentiable, but it can be derived in terms of distributions .
- The function
- is classically differentiable on the interval , but not weakly differentiable. The problem is that the derivative
- is not Lebesgue integrable on any containing, compact subset of . This means that the integral in particular is not well-defined for all test functions .
literature
- Dirk Werner : functional analysis , Springer-Verlag, Berlin, ISBN 978-3-540-72533-6 .
- Lawrence Evans: Partial Differential Equations , American Mathematical Society, ISBN 0-8218-0772-2 .