$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\supp}{\operatorname{supp}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ #[Chapter 11. Distributions and weak solutions](id:chapter-11) > In this Chapter we extend notion of function. These new "functions" (actually most of them are not functions at all) are called *distributions* and are very useful for PDE (and not only). We define them as linear forms on the *test functions* which are some nice functions. For usual function $f$ such form is \begin{equation\*} f(\varphi)=\int f(x)\varphi(x)\,dx. \end{equation*} We also extend the notion of solution. ##[11.1. Distributions](id:sect-11.1) ____________________________ > 1. [Test functions](#sect-11.1.1) > 2. [Distributions](#sect-11.1.2) > 3. [Operations on distributions](#sect-11.1.3) ###[Test functions](id:sect-11.1.1) We introduce three main spaces of test functions: **[Definition 1.](id:definition-11.1.1)** Let a. $\mathcal{D}=C\_0^\infty$ is a space of infinitely smooth functions with compact support. It means that for each function $\varphi$ exists $a$ such that $\varphi(x)=0$ as $|x|\ge a$; b. $\mathcal{E}=C^\infty$ is a space of infinitely smooth functions; c. $\mathcal{S}$ is a space of infinitely smooth functions which decay at infinity (with all their derivatives faster than any power: \begin{equation} |\partial ^m \varphi| (1+|x|)^k \le M\_{mk} \qquad \forall x\, \forall m,k. \label{eq-11.1.1} \end{equation} Loran Schwartz who provided the first systematic theory of distributions used these notations and they became traditional. However we need to explain what does it mean *convergence of test function*: **[Definition 2.](id:definition-11.1.2)** a. $\varphi\_n \to \varphi $ in $\mathcal{D}$ iff $\max |\partial ^m(\varphi\_n -\varphi)|\to 0$ as $n\to \infty$ for all $m$ and also there exist $a$ such that $\varphi\_n(x)=0$ as $|x|\ge a$ for all $n$; b. $\varphi\_n \to \varphi $ in $\mathcal{E}$ iff $\max\_{|x|\le a} |\partial ^m(\varphi\_n -\varphi)|\to 0$ as $n\to \infty$ for all $m$, $a$; c. $\varphi\_n \to \varphi $ in $\mathcal{S}$ iff \begin{equation} |\partial ^m (\varphi\_n-\varphi)| (1+|x|)^k \to 0 \qquad \forall x\, \forall m,k. \label{eq-11.1.2} \end{equation} **[Theorem 1.](id:thm-11.1.1)** \begin{equation} \mathcal{D} \subset \mathcal{S} \subset \mathcal{E} \label{eq-11.1.3} \end{equation} where $\mathcal{K}\_1\subset\mathcal{K}\_2$ means not only that all elements of $\mathcal{K}\_1$ are also elements of $\mathcal{K}\_2$ but also that $\varphi\_n\overset{\mathcal{K}\_1}{\to}\varphi$ implies that $\varphi\_n\overset{\mathcal{K}\_2}{\to}\varphi$. Also in (\ref{eq-11.1.3}) each smaller space $\mathcal{K}\_1$ is *dense* in the larger one $\mathcal{K}\_2$: for each $\varphi\in \mathcal{K}\_2$ there exists a sequence $\varphi\_n\in \mathcal{K}\_1$ converging to $\varphi$ in $\mathcal{K}\_2$. **[Remark 1.](id:remark-11.1.2)** Those who studies Real Analysis heard about *Topological Vector Spaces* but we are not going to introduce topology (which is ridiculously complicated on $\mathcal{D}$), just convergence is sufficient for all needs. The same approach is also used in the very advanced cources. ###[Distributions](id:sect-11.1.2) **[Definition 3.](id:definition-11.1.3)** a. *Distribution* $f$ (over $\mathcal{K}$) is a continuous linear form on $\mathcal{K}$: $f:\mathcal{K}\to \mathbb{C}$ such that \begin{gather\*} f(\alpha\_1 \varphi\_1 + \alpha\_2 \varphi\_2)= \alpha\_1 f(\varphi\_1)+\alpha\_2 f(\varphi\_2)\qquad \forall \varphi\_1,\varphi\_2\in \mathcal{K}\; \forall \alpha\_1,\alpha\_2\in \mathbb{C};\\\\ \varphi\_n\overset{\mathcal{K}}{\to}\varphi \implies f(\varphi\_n)\to f(\varphi). \end{gather\*} b. The space of such linear forms is denoted by $\mathcal{K}'$. [Theorem 1](#thm-11.1.1) immediately implies **[Theorem 2.](id:thm-11.1.2)** \begin{equation} \mathcal{D}' \supset \mathcal{S}' \supset \mathcal{E}' \label{eq-11.1.4} \end{equation} where $\mathcal{K}'\_1\supset\mathcal{K}'\_2$ means not only that all elements of $\mathcal{K}\_2$ are also elements of $\mathcal{K}\_1$ but also that $f\_n\overset{\mathcal{K}'\_2}{\to} f$ implies that $f\_n\overset{\mathcal{K}'\_2}{\to}f$. Also in (\ref{eq-11.1.4}) each smaller space $\mathcal{K}'\_2$ is *dense* in the larger one $\mathcal{K}'\_1$: for each $f\in \mathcal{K}'\_1$ there exists a sequence $f\_n\in \mathcal{K}'\_2$ converging to $f$ in $\mathcal{K}'\_1$. So far we have not introduced the convergence of distributions, so we do it right now: **[Definition 4.](id:definition-11.1.4)** $f\_n\overset{\mathcal{K}'}{\to}f$ iff $f\_n(\varphi)\to f(\varphi)$ for all $\varphi\in \mathcal{K}$. **[Remark 2.](id:remark-11.1.2)** a. $\mathcal{E}'$ consists of distributions with compact support: $f\in \mathcal{D}'$ belongs to $\mathcal{E}'$ iff there exists $a$ such that $f(\varphi)=0$ for all $\varphi$ such that $\varphi (x)=0$ as $|x|\le a$. b. $\mathcal{S}'$ consists off *temperate distributions*. c. For $f\in L^1\_{\mathrm{loc}}$ we can define action $f(\varpi)$ on $\mathcal{D}$ \begin{equation} f(\varphi)=\int f(x)\varphi(x)\,dx \label{eq-11.1.5} \end{equation} where integral is always understood as integral over the whole line $\mathbb{R}$ (or a whole space $\mathbb{R}^d$) and $L^1\_{\mathrm{loc}}$ consists of *locally integrable functions* (notion from the Real Analysis which means that $\int\_{|x|\le a} |f(x)|\,dx <\infty$ for all $a$ but integral is a *Lebesgue integral* which is a natural extension of Riemann integral). One can prove that this form is continuous and thus $f\in \mathcal{D}'$. Due to this we sometimes non–rigorously will write (\ref{eq-11.1.5}) even for distributions which are not ordinary functions. **[Example 1.](id:example-11.1.1)** $\delta:=\delta (x)$ is an element of $\mathcal{E}'$ defined as $\delta(\varphi)=\varphi(0)$. It is traditionally called *$\delta$-function* or *Dirac $\delta$-function* despite not being a function but a distribution. ###[Operations on distributions](id:sect-11.1.3) We introduce operations on distributions as an extension of operations on ordinary functions as long as they make sense. **[Definition 5.](id:definition-11.1.5)** *Linear operations*: \begin{equation} (\alpha\_1 f\_1 +\alpha\_2 f\_2)(\varphi)= \alpha\_1 f\_1(\varphi) +\alpha\_2 f\_2(\varphi) \label{eq-11.1.6} \end{equation} as $\alpha\_1,\alpha\_2\in \mathbb{C}$. **[Exercise 1.](id:exercise-11.1.1)** Check that for ordinary functions $f\_1,f\_2$ we get a standard definition of $\alpha\_1 f\_1 +\alpha\_2 f\_2$ (in the framework of (\ref{eq-11.1.5})). **[Definition 6.](id:definition-11.1.6)** *Shift*. Let $T\_a$ denote a shift of $\varphi$: $(T\_a\varphi)(x) =\varphi (x-a)$. Then \begin{equation} (T\_a f)(\varphi)= f(T\_{-a}\varphi). \label{eq-11.1.7} \end{equation} We will write $T\_af$ as $f(x-a)$. **[Exercise 2.](id:exercise-11.1.2)** a. Check that for ordinary function $f$ we get a standard definition of $f(x-a)$ (in the framework of (\ref{eq-11.1.5})). b. Check that for $\delta$ we $\delta\_a(x):=\delta(x-a)$ is defined as $\delta\_a (\varphi)= \varphi (a)$. **[Definition 7.](id:definition-11.1.7)** *Linear change of variables*. Let $R\_A$ denote a linear change of variables: $(R\_A\varphi )(x)= \varphi(Ax)$ where $A$ is a non-degenerate linear transformation. Then \begin{equation} (R\_A f)(\varphi)= |\det A|^{-1} f(R\_{A^{-1}}\varphi) \label{eq-11.1.8} \end{equation} We will write $R\_Af$ as $f(Ax)$. **[Exercise 3.](id:exercise-11.1.3)** a. Check that for ordinary function $f$ we get a standard definition of $R\_A f$ (in the framework of (\ref{eq-11.1.5})). b. Check that for $\delta$ we get $ \delta (Ax)= |\det A|^{-1} \delta(x)$. In particular as $|\det A|=1$ we have $\delta(Ax)=\delta(x)$ and as $Ax=\lambda x$ (uniform dilatation) $\delta (\lambda x)=\lambda^{-d}\delta(x)$ where $d$ is a dimension. Therefore $\delta$ is *spherically symmetric* and *positively homogeneous of degree $-d$*. **[Definition 8.](id:definition-11.1.8)** *Derivative*. Then \begin{equation} (\partial f)(\varphi)= -f (\partial \varphi) \label{eq-11.1.9} \end{equation} where $\partial $ is a first order derivative. **[Exercise 4.](id:exercise-11.1.4)** a. Check that for ordinary function $f$ we get a standard definition of $\partial f$ (in the framework of (\ref{eq-11.1.5})). Use integration by parts. b. Check that for $\delta$ we get $\delta '$: $\delta'_a (\varphi)=-\varphi'(a)$ (in one dimension and similarly in higher dimensions). c. Check that if $\theta(x)$ is a *Heaviside function*: $\theta(x)=1$ as $x\>0$ and $\theta(x)=0$ as $x\le 0$ then $\theta' (x)=\delta(x)$. d. Check that if $f(x)$ is a smooth function as $x<\ a$ and as $x \> 0$ but with a jump at $a$ then $f'=\overset{\circ}{f}{}'+ (f(a+0)-f(a-0))\delta (x-a)$ where $f'$ is understood in the sense of distributions and $\overset{\circ}{f}{}'(x)$ is an ordinary function equal to derivative of $f$ as $x\ne a$. f. Prove that if $f=\ln |x|$ then $f'(\varphi)= pv \int x^{-1}\varphi (x)\,dx $ where integral is understood as a principal value integral. Let $g\in C^\infty$. Observe that for $g\varphi \in \mathcal{D}$ and $g\varphi \in \mathcal{E}$ for $\varphi \in \mathcal{D}$ and $\varphi \in \mathcal{E}$ respectively. Therefore the following definition makes sense: **[Definition 9.](id:definition-11.1.9)** *Multiplication by a function*. Let either $f\in \mathcal{D}'$ or $f\in \mathcal{E}'$. Then $gf\in \mathcal{D}'$ or $gf\in \mathcal{E}'$ respectively is defined as \begin{equation} (g f)(\varphi)= f (g \varphi). \label{eq-11.1.10} \end{equation} **[Exercise 5.](id:exercise-11.1.5)** a. Check that for ordinary function $f$ we get a standard definition of $g f$ (in the framework of (\ref{eq-11.1.5})). b. Prove that $g \delta\_a= g(a)\delta\_a$ (use definitions); c. Calculate $g\delta '\_a$, $g\delta ''\_a$ (use definitions!). We cannot define in general the product of two distributions. However in some cases it is possible, f.e. when distributions are of different arguments. **[Definition 10.](id:definition-11.1.10)** *Direct product*. Let $f,g$ be distributions. Then $f(x)g(y)$ (also denoted as $f\otimes g$) is defined as \begin{equation} (f g) (\varphi) = f( g(\varphi)) \label{eq-11.1.11} \end{equation} where $\varphi=\varphi (x,y)$, then applying $g$ to it we get $\psi (x):=g(\varphi)$ a test function, and then applying $f$ we get a number. Similarly we get the same $fg$ if we apply first $f$ and then $g$. **[Exercise 6.](id:exercise-11.1.6)** a. Check that for ordinary functions $f,g$ we get a standard definition of $fg$ (in the framework of (\ref{eq-11.1.5})). b. Prove that $ \delta\_{a\_1}(x\_1)\cdots \delta\_{a\_d}(x\_d)= \delta_a(x)$ with $a=(a\_1,\ldots,a\_d)$, $x=(x\_1,\ldots,x\_d)$ and we have on the left product of $1$-dimensional $\delta$-functions and on the right $n$-dimensional. --------------- [$\Leftarrow$](../Chapter10/S10.A.html)  [$\Uparrow$](../contents.html)  [$\Rightarrow$](./S11.2.html)