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

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


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

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 \(g\).