# 13. Generalized derivatives#

In the following, let \(\Omega \subset {\mathbb R}^d\) be an open and connected set. If not stated otherwise, we also assume \(\Omega\) is bounded.

Let \(\alpha = (\alpha_1, \ldots , \alpha_d) \in {\mathbb N}_0^d\) be a multi-index, \(| \alpha | = \sum \alpha_i\), and define the classical differential operator for functions in \(C^\infty (\Omega)\)

For a function \(u \in C(\Omega)\) (with \(\Omega\) not necessarily bounded), the support is defined as

This is a compact set if and only if it is bounded. We say \(u\) has compact support in \(\Omega\), if \(\operatorname{supp} u \subset \Omega\) is compact. If \(\Omega\) is a bounded domain, then \(u\) has compact support in \(\Omega\) if and only if \(u\) vanishes in a neighbourhood of \(\partial \Omega\).

The space of smooth functions with compact support is denoted as

For a smooth function \(u \in C^{|\alpha|}(\Omega)\), there holds the formula of integration by parts

The \(L_2\) inner product with a function \(u\) in \(C(\Omega)\) defines the linear functional on \({\cal D}\)

We call these functionals in \({\cal D}^\prime\) distributions. When \(u\) is a function, we identify it with the generated distribution. The integration by parts formula is valid for functions \(u \in C^\alpha\). The strong regularity is needed only on the left hand side. Thus, we use the less demanding right hand side to extend the definition of differentiation for distributions:

Definition:For \(u \in {\cal D}^\prime\), we define \(g \in {\cal D}^\prime\) to be the generalized derivative of \(u\), denoted as \(g = D_g^\alpha u\) by\[ \left<g, \varphi \right>_{{\cal D}^\prime \times {\cal D}} = (-1)^{|\alpha|} \left<u, D^\alpha \varphi \right>_{{\cal D}^\prime \times {\cal D}} \qquad \forall \, \varphi \in {\cal D}. \]

If \(u \in C^\alpha\), then \(D_g^\alpha\) coincides with \(D^\alpha\).

The function space of *locally integrable* functions on \(\Omega\)
is called

It contains functions which can behave very badly near \(\partial \Omega\). E.g., \(e^{e^{1/x}}\) is in \(L_{loc}^1 (0,1)\). If \(\Omega\) is unbounded, then the constant function \(1\) is in \(L_1^{loc}\), but not in \(L_1\).

Definition:For \(u \in L_1^{loc}\), we call \(g\) the weak derivative \(D_w^\alpha u\), if \(g \in L_1^{loc}\) and it satisfies\[ \int_\Omega g(x) \varphi(x) \, dx = (-1)^{|\alpha|} \int_\Omega u(x) D^\alpha \varphi (x) \, dx \qquad \forall \, \varphi \in {\cal D}. \]

The weak derivative is more general than the classical derivative, but more restrictive than the generalized derivative.

**Example:**
Let \(\Omega = (-1,1)\) and

Then,

is the first generalized derivative \(D^1_g\) of \(u\), which is also a weak derivative.

*Proof:* We use the definition of the generalized derivative, and split the integral
into parts:

Next, we apply integration py parts on each sub-interval:

Finally, we compute the derivatives of \(u\) on each sub-interval, use that \(\varphi(-1) = \varphi(1) = 0\), and observe that the contributions at \(x=0\) cancel out:

Since \(g\) is a locally integrable function, the generalized derivative is also a weak derivative.

**Example:**
The second generalized derivative \(h = D_g^2 u\) is

*Proof:* The second derivative of \(u\) is the first derivative of \(g\):

The distribution \(h : C_0^\infty \rightarrow {\mathbb R}\) is (up to the factor \(-2\)) the point evaluation functional (aka \(\delta\)-distribution). It cannot be represented by an \(L_1^{loc}\) function, thus it is not a weak derivative.

**Example:** The third generalized derivative \(D_g^3 u\) is

*Proof:* Follows directly from the definition.

In the following, we will focus on weak derivatives. Unless it is essential we will skip the sub-scripts \(w\) and \(g\).