# 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 ${\ displaystyle I = (a, b)}$${\ displaystyle f}$${\ displaystyle f ^ {\ prime}}$${\ displaystyle L _ {\ mathrm {loc}} ^ {1}}$${\ displaystyle I}$ ${\ displaystyle \ varphi \ in C_ {c} ^ {\ infty} (I)}$${\ displaystyle \ varphi}$

${\ displaystyle \ int _ {I} f ^ {\ prime} (t) \ varphi (t) \, \ mathrm {d} t = - \ int _ {I} f (t) \ varphi ^ {\ prime} (t) \, \ mathrm {d} t}$.

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. ${\ displaystyle \ left (\ varphi (a) = 0, \ varphi (b) = 0 \ right)}$

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 ${\ displaystyle f}$${\ displaystyle L _ {\ mathrm {loc}} ^ {1}}$${\ displaystyle f}$${\ displaystyle g \ in L _ {\ mathrm {loc}} ^ {1} (I)}$

${\ displaystyle \ int _ {I} g (t) \ varphi (t) \, \ mathrm {d} t = - \ int _ {I} f (t) \ varphi ^ {\ prime} (t) \, \ mathrm {d} t}$

fulfilled for each test function. Such a function is called the weak derivative of . You write like the classical derivative . ${\ displaystyle \ varphi}$${\ displaystyle g}$${\ displaystyle f}$${\ displaystyle f ^ {\ prime}: = g}$

### 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 . ${\ displaystyle \ Omega \ subseteq \ mathbb {R} ^ {n}}$${\ displaystyle f \ colon \ Omega \ rightarrow \ mathbb {R}}$${\ displaystyle f \ in L _ {\ mathrm {loc}} ^ {1} (\ Omega)}$${\ displaystyle \ alpha = (\ alpha _ {1}, \ dotsc, \ alpha _ {n}) \ in \ mathbb {N} _ {0} ^ {n}}$

A function is called the -th weak derivative of if the following applies to all test functions : ${\ displaystyle g \ in L _ {\ mathrm {loc}} ^ {1} (\ Omega)}$${\ displaystyle \ alpha}$${\ displaystyle f}$${\ displaystyle \ varphi}$

${\ displaystyle \ int _ {\ Omega} g (x) \ varphi (x) \, \ mathrm {d} x = (- 1) ^ {| \ alpha |} \ int _ {\ Omega} f (x) D ^ {\ alpha} \ varphi (x) \, \ mathrm {d} x}$.

Here is and . Often one writes . ${\ displaystyle \ textstyle | \ alpha | = \ sum _ {i = 1} ^ {n} \ alpha _ {i}}$${\ displaystyle D ^ {\ alpha} = {\ frac {\ partial ^ {| \ alpha |}} {\ partial ^ {\ alpha _ {1}} x_ {1} \ dotso \ partial ^ {\ alpha _ { n}} x_ {n}}}}$${\ displaystyle g = D ^ {\ alpha} f}$

You can place apparently only for calling. The subset of functions in which weak derivatives exist is a so-called Sobolev space . ${\ displaystyle f, g \ in L _ {\ mathrm {loc}} ^ {1} (\ Omega)}$${\ displaystyle f, g \ in L ^ {p} (\ Omega)}$${\ displaystyle 1 \ leq p \ leq \ infty}$${\ displaystyle L ^ {p}}$${\ displaystyle n \ geq 1}$

If there is a function , then one requires the weak differentiability in each of the image components. ${\ displaystyle f \ colon \ Omega \ rightarrow \ mathbb {R} ^ {m}}$${\ displaystyle m}$

### 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. ${\ displaystyle \ left (\ right.}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R} ^ {n} \ left. \ right)}$

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 ${\ displaystyle g_ {1}}$${\ displaystyle g_ {2}}$

${\ displaystyle \ int _ {I} (g_ {1} (t) -g_ {2} (t)) \ varphi (t) \, \ mathrm {d} t = 0}$

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 ). ${\ displaystyle \ varphi}$ ${\ displaystyle g_ {1} = g_ {2}}$${\ displaystyle L _ {\ mathrm {loc}} ^ {1}}$${\ displaystyle L ^ {p}}$${\ displaystyle 1 \ leq p <\ infty}$

### 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. ${\ displaystyle f}$${\ displaystyle f ^ {\ prime}}$${\ displaystyle L _ {\ mathrm {loc}} ^ {1}}$${\ displaystyle L _ {\ mathrm {loc}} ^ {1}}$

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 . ${\ displaystyle n}$${\ displaystyle k}$${\ displaystyle C ^ {k}}$${\ displaystyle n> k \ geq 0}$

## Examples

Weak derivation absolute amount
• 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${\ displaystyle f (x) = | x |}$${\ displaystyle x = 0}$${\ displaystyle] a, b [}$${\ displaystyle a <0 ${\ displaystyle f ^ {\ prime} \ colon {] a, b [} \ to \ mathbb {R}}$
${\ displaystyle f ^ {\ prime} (x) = {\ begin {cases} -1 &: x <0 \\ 0 &: x = 0 \\ + 1 &: x> 0 \ end {cases}}}$
and any test function just${\ displaystyle \ varphi \ colon] a, b [\ to \ mathbb {R}}$
{\ displaystyle {\ begin {aligned} \ int _ {a} ^ {b} \ varphi '(x) \ cdot f (x) \, \ mathrm {d} x = & \ int _ {a} ^ {0 } - \ varphi '(x) x \, \ mathrm {d} x + \ int _ {0} ^ {b} \ varphi' (x) x \, \ mathrm {d} x \\ = & - \ left ( \ int _ {a} ^ {0} \ varphi (x) \ cdot (-1) \, \ mathrm {d} x + \ int _ {0} ^ {b} \ varphi (x) \ cdot 1 \, \ mathrm {d} x \ right) \\ = & - \ int _ {a} ^ {b} \ varphi (x) \ cdot f ^ {\ prime} (x) \, \ mathrm {d} x. \ end {aligned}}}
Thus is a weak derivative of .${\ displaystyle f ^ {\ prime}}$${\ displaystyle f}$
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 .${\ displaystyle \ {0 \}}$
• The function
${\ displaystyle f (x) = {\ begin {cases} x ^ {2} \ sin ({\ frac {1} {x ^ {2}}}) &: x \ neq 0 \\ 0 &: x = 0 \ end {cases}}}$
is classically differentiable on the interval , but not weakly differentiable. The problem is that the derivative${\ displaystyle I = (- 1,1)}$
${\ displaystyle f ^ {\ prime} (x) = {\ begin {cases} 2x \ sin ({\ frac {1} {x ^ {2}}}) - {\ frac {2} {x}} \ cos ({\ frac {1} {x ^ {2}}}) &: x \ neq 0 \\ 0 &: x = 0 \ end {cases}}}$
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 .${\ displaystyle 0}$${\ displaystyle I}$${\ displaystyle \ int _ {I} f ^ {\ prime} (t) \ varphi (t) \, \ mathrm {d} t}$${\ displaystyle \ varphi \ in C_ {c} ^ {\ infty} (I)}$