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

existence

• An absolutely continuous function has a weak derivative.

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 ${\displaystyle f(x)=|x|}$ ${\displaystyle x=0}$ ${\displaystyle ]a,b[}$ ${\displaystyle a<0<b}$ ${\displaystyle f^{\prime }\colon {]a,b[}\to \mathbb {R} }$

${\displaystyle f^{\prime }(x)={\begin{cases}-1&:x<0\\0&:x=0\\+1&:x>0\end{cases}}}$

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 . ${\displaystyle 0}$ ${\displaystyle I}$ ${\displaystyle \int _{I}f^{\prime }(t)\varphi (t)\,\mathrm {d} t}$ ${\displaystyle \varphi \in C_{c}^{\infty }(I)}$

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 .