Distributions and Weak Derivatives

Physics and engineering have long used derivatives of functions that have none. The density of a point mass is "infinite at one point and zero elsewhere, with total mass $1$"; the derivative of a step is that density; the force on a beam from a concentrated load enters the beam equation as a spike. Paul Dirac's delta function and Oliver Heaviside's operational calculus worked, but they were not mathematics until Laurent Schwartz made them so in the late 1940s, with the theory of distributions.

The idea is to define a generalised function not by its values but by how it acts on smooth test functions, and to define its derivative by moving the derivative onto the test function, with a minus sign, as integration by parts would. With this definition every distribution is infinitely differentiable, the derivative of the step is the Dirac mass, and many functions that aren't differentiable in the classical sense have weak derivatives that are honest functions. Sobolev spaces (4A.9 Sobolev Spaces) are defined by such weak derivatives. This chapter is kept to what that needs, plus the one other idea that PDE uses everywhere: the fundamental solution, the response to a point source.

By the end of this chapter you will be able to:

• define test functions, distributions and distributional derivatives, and compute derivatives of functions with jumps and corners;
• decide whether a function has a weak derivative, and compute it;
• show that $\log|x|$ (in the plane) and $-\frac{1}{4\pi|x|}$ (in space) are fundamental solutions of the Laplacian;
• read the phrase "in the sense of distributions" and know what it permits.

8.1A point load on a beam

8.2Test functions and distributions

Let $\Omega \subseteq \mathbb{R}^n$ be open. Test functions are the smooth functions with compact support in $\Omega$, $C_c^\infty(\Omega)$; the bumps of 2B.6 Power Series, Exponentials and Bump Functions are examples, and there are plenty of them (3A.8 Convolution and Mollifiers). A sequence $\phi_k \to \phi$ in $C_c^\infty(\Omega)$ if all supports lie in one compact set and every derivative $\partial^\alpha\phi_k$ converges uniformly to $\partial^\alpha\phi$.

The examples:

• Functions. Every locally integrable $f$ (integrable on compact subsets) defines a distribution $\langle T_f, \phi\rangle = \int f\phi$. Two functions give the same distribution exactly when they are equal almost everywhere (by 3A.6 Modes of Convergence and Differentiation's differentiation theorem, or by density of test functions), so we identify $f$ with $T_f$ and say that distributions generalise functions.
• The Dirac mass $\langle\delta_a, \phi\rangle = \phi(a)$. It is not of the form $T_f$ for any function $f$ (3A.4 Measures, Probability and Weights).
• Measures. Any locally finite measure $\mu$ gives $\langle\mu, \phi\rangle = \int\phi\,d\mu$.
• Limits. If $f_k$ are functions and $\int f_k\phi$ converges for every test function $\phi$, the limit is a distribution. So the narrowing Gaussians of 3A.4 Measures, Probability and Weights converge to $\delta_0$ in $\mathcal{D}'$.

8.3Derivatives

If $f$ is continuously differentiable and $\phi$ is a test function, integration by parts has no boundary terms ($\phi$ vanishes near the boundary), so $\int\partial_jf\,\phi = -\int f\,\partial_j\phi$. The right side makes sense for any distribution, and that is the definition.

By the computation above, for $C^1$ functions it agrees with the classical derivative. And since $\partial_j\phi$ is again a test function, it can be repeated: every distribution has derivatives of every order, $\langle\partial^\alpha T, \phi\rangle = (-1)^{|\alpha|}\langle T, \partial^\alpha\phi\rangle$. Differentiation is also continuous: if $T_k \to T$ in $\mathcal{D}'$, then $\partial^\alpha T_k \to \partial^\alpha T$, so one can differentiate limits term by term, which classical analysis can never do (2B.5 Uniform Convergence and Arzelà–Ascoli).

Multiplying a distribution by a smooth function $a$ is also defined, $\langle aT, \phi\rangle = \langle T, a\phi\rangle$, and the product rule holds. But two distributions can't in general be multiplied: $\delta_0^2$ has no meaning. That limitation is why nonlinear PDE need more than distribution theory, namely spaces in which the solutions are functions.

8.4Weak derivatives

The cases that matter most are those where the distributional derivative of a function is again a function.

A weak derivative, if it exists, is unique almost everywhere. Examples on $(-1, 1)$:

• $|x|$ has weak derivative $\operatorname{sign}x$.
• More generally, a continuous function that is $C^1$ except at finitely many corners has its classical derivative (defined except at the corners) as a weak derivative (Exercise 8.7).
• $\operatorname{sign}x$ has no weak derivative: its distributional derivative is $2\delta_0$, which is not a function. Functions with jumps don't have weak derivatives.
• $|x|^\alpha$ for $0 < \alpha < 1$ has weak derivative $\alpha|x|^{\alpha-1}\operatorname{sign}x$, which is unbounded but integrable.

So having a weak derivative is a genuine regularity condition, between continuity and differentiability: corners are allowed, jumps are not. In several variables more interesting things happen. The function $\log\log(1 + 1/|x|)$ is unbounded at the origin of $\mathbb{R}^2$ but has weak derivatives in $L^2$ (4A.5 The Fourier Transform): a function with a weak gradient need not be continuous once $n \geq 2$.

Weak derivatives commute with mollification: $\partial_j(f * \phi_\varepsilon) = (\partial_jf) * \phi_\varepsilon$ (Exercise 8.10). So a function with weak derivatives in $L^p$ is the $L^p$ limit of smooth functions whose derivatives converge in $L^p$ too. That is the bridge to Sobolev spaces: "weak derivatives in $L^p$" and "limit of smooth functions in the norm $\|f\|_p + \|\nabla f\|_p$" turn out to be the same thing (4A.9 Sobolev Spaces).

8.5Fundamental solutions

A fundamental solution of a linear differential operator $L$ with constant coefficients is a distribution $E$ with $LE = \delta_0$: the response to a unit point source. Its use is that $u = E * f$ solves $Lu = f$, because $L(E * f) = (LE) * f = \delta_0 * f = f$. To solve the equation for every source it suffices to solve it for one point source.

The fundamental solution of the heat operator $\partial_t - \Delta$ is the heat kernel $H(x, t)$ of 3A.8 Convolution and Mollifiers, extended by $0$ for $t \leq 0$: it is the temperature after a unit of heat is released at the origin at time $0$ (6A.3 The Heat Equation on ℝⁿ).

8.6History

Oliver Heaviside used the step function and its derivative in his operational calculus for electrical circuits in the 1890s. Paul Dirac introduced the delta function in quantum mechanics in 1927 and in his 1930 book. Sergei Sobolev defined generalised functions and weak derivatives in 1935–36, in work on hyperbolic equations, and introduced the spaces now named after him. Laurent Schwartz built the general theory of distributions in 1945–1950 (Théorie des distributions, 1950–51), for which he received the Fields Medal in 1950. Macaulay's brackets for beam deflections date from 1919.

8.7Exercises

Solution

$x\delta_0 = 0$; $x\delta_0' = -\delta_0$; $\langle x^2\delta_0'', \phi\rangle = (x^2\phi)''(0) = 2\phi(0)$, so $x^2\delta_0'' = 2\delta_0$.

Solution

On $[0, \frac L2]$, $u''' = \frac{P}{2EI}$ (constant), $u''(0) = 0$, so $u'' = \frac{P}{2EI}x$, which is the moment diagram up to sign. Symmetry gives $u'(\frac L2) = 0$, so $u' = \frac{P}{4EI}(x^2 - \frac{L^2}{4})$, and $u(0) = 0$ gives $u = \frac{P}{4EI}(\frac{x^3}{3} - \frac{L^2x}{4})$. At $x = \frac L2$: $|u| = \frac{P}{4EI}\big(\frac{L^3}{8} - \frac{L^3}{24}\big) = \frac{PL^3}{48EI}$.

Solution

(a) $\partial_i r = x_i/r$, $\partial_i^2r = \frac1r - \frac{x_i^2}{r^3}$, summing to $\frac{n}{r} - \frac1r$. (b) $\int_{B_1}\frac1r\,dx = |S^{n-1}|\int_0^1r^{n-2}dr < \infty$ for $n \geq 2$. On $|x| = \varepsilon$ the terms $r\,\partial_\nu\phi$ and $\phi\,\partial_\nu r$ are bounded, and the sphere has area $|S^{n-1}|\varepsilon^{n-1} \to 0$.