Generalized derivatives

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 $C^\infty (\Omega)$

\[ D^\alpha = \left( \frac{\partial}{\partial x_1} \right)^{\alpha_1} \cdots \left( \frac{\partial}{\partial x_n} \right)^{\alpha_d}. \]

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

\[ \operatorname{supp} \{ u \} := \overline{ \{ x \in \Omega : u(x) \neq 0 \} }. \]

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

\[ {\cal D} (\Omega) := C_0^\infty (\Omega) := \{ u \in C^\infty(\Omega) : \mbox{ $u$ has compact support in $\Omega$} \}. \]

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

\[ \int_\Omega D^\alpha u \varphi \, dx = (-1)^{|\alpha|} \int_\Omega u D^\alpha \varphi \, dx \qquad \forall \, \varphi \in {\cal D}(\Omega). \]

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

\[ u(\varphi) := \left< u, \varphi \right>_{{\cal D}^\prime \times {\cal D}} := \int_\Omega u \varphi \, dx. \]

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

\[ L_1^{loc} (\Omega) = \{ u : u_K \in L_1(K) \; \forall \mbox{ compact } K \subset \Omega \}. \]

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

\[\begin{split} u(x) = \left\{ \begin{array}{cl} 1+x & \quad x \leq 0 \\ 1-x & \quad x > 0 \end{array} \right\} \end{split}\]

Then,

\[\begin{split} g(x) = \left\{ \begin{array}{cl} 1 & \quad x \leq 0 \\ -1 & \quad x > 0 \end{array} \right\} \end{split}\]

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:

\[\begin{align*} \left< D_g u, \varphi \right> & = -\left< u, \varphi^\prime \right> = -\int_{-1}^1 u(x) \varphi^\prime(x) \, dx \\ & = -\int_{-1}^0 (1+x) \varphi^\prime(x) \, dx - \int_0^1 (1-x) \varphi^\prime(x) \, dx \end{align*}\]

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

\[\begin{align*} \left< D_g u, \varphi \right> & = \int_{-1}^0 (1+x)^\prime \varphi(x) dx - (1+x) \varphi(x) |_{x=-1}^{x=0} \\ & + \int_{0}^1 (1-x)^\prime \varphi(x) dx - (1-x) \varphi(x) |_{x=0}^{x=1} \end{align*}\]

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:

\[\begin{align*} \left< D_g u, \varphi \right> &= \int_{-1}^0 1 \, \varphi(x) dx - 1 \varphi(0) + \int_0^1 (-1) \varphi(x) dx + 1 \varphi(0) \\ &= \int_{-1}^1 g(x) \varphi(x) \, dx \end{align*}\]

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

\[ \left< h, \varphi \right> = -2 \varphi(0) \qquad \forall \, \varphi \in {\cal D} \]

Proof: The second derivative of $u$ is the first derivative of $g$:

\[\begin{align*} \left< D^2_g u, \varphi\right> = \left< D_g g, \varphi \right> & = -\int_{-1}^1 g(x) \varphi^\prime \, dx \\ &= -\int_{-1}^0 1 \varphi^\prime \, dx - \int_{0}^1 (-1) \varphi^\prime \, dx \\ &= -\varphi(0) - \varphi(0) = -2 \varphi(0) \end{align*}\]

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

\[ \left< D_g^3 u, \varphi \right> = 2 \varphi^\prime(0) \]

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 $g$.