In 1900, at the International Congress of Mathematicians in Paris, David Hilbert presented a list of 23 problems that, in his view, would guide twentieth-century mathematics. Problem 19 asked, in simple terms: are the solutions of the “well-behaved” problems of the calculus of variations always smooth functions? Half a century later, the question was still open for an important class of equations. In 1956 Nash heard about it from a colleague and decided to attack it.

The heat equation

Start with the simplest example. A very long metal bar has temperature $u(x,t)$ at the point $x$ and the instant $t$. Heat flows from hot to cold, and the temperature obeys Fourier’s heat equation:

$$\frac{\partial u}{\partial t}=\kappa\,\frac{\partial^2u}{\partial x^2},$$

where $\kappa>0$ is the diffusivity of the material. The equation says that the temperature at a point rises when it is colder than the average of its neighbours, and falls when it is hotter.

If at time zero all the thermal energy is concentrated at one point, the solution for $t>0$ is the Gaussian

$$G(x,t)=\frac1{\sqrt{4\pi\kappa t}}\,e^{-x^2/(4\kappa t)}.$$

Let us check. Differentiating once in $t$ and twice in $x$,

$$\frac{\partial G}{\partial t}=\left(-\frac1{2t}+\frac{x^2}{4\kappa t^2}\right)G,\qquad\kappa\frac{\partial^2G}{\partial x^2}=\left(-\frac1{2t}+\frac{x^2}{4\kappa t^2}\right)G.$$

They are equal. The bell curve widens and lowers over time, keeping area 1: the energy is conserved and spreads out. The width grows like $\sqrt t$, exactly like the distance travelled in Einstein’s Brownian motion. It is no coincidence: heat is the agitation of molecules, and the heat equation describes a random walk seen from far away.

When the material is irregular

The Gaussian has a remarkable property: even if the initial temperature is an infinitely concentrated peak, for any $t>0$ the solution is perfectly smooth. Diffusion smooths everything out.

But what if the material is not uniform? Imagine a plate made of pieces of copper, wood and glass, glued together any old way, each conducting heat differently. The equation becomes

$$\frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\left(a(x,t)\,\frac{\partial u}{\partial x}\right),$$

in which the conductivity $a(x,t)$ may jump abruptly from one piece to another. It is only required to stay between two positive bounds, $0<\lambda\leq a\leq\Lambda$. With such an irregular coefficient there is no formula for the solution, and it is not even clear that it has derivatives. The question is: can the temperature vary violently, or does heat still smooth things out?

The theorem

Nash proved, in 1957, that heat still smooths: in any dimension, the solutions of these equations are Hölder continuous, that is, there are constants $C$ and $0<\alpha\leq1$, depending only on $\lambda$, $\Lambda$ and the dimension, such that

$$\boxed{|u(x)-u(y)|\leq C\,|x-y|^{\alpha}.}$$

The temperature cannot jump, however disordered the material. Nash’s proof was deeply original: it used ideas from probability and entropy, treating heat as a mass that spreads out, to control the solution without any formula. With this result, Hilbert’s 19th problem was settled for this class of equations.

Exercise: for the Gaussian with κ = 1, what is the temperature at the centre at t = 1 and at t = 4?

At the centre, $x=0$, and $G(0,t)=\frac1{\sqrt{4\pi t}}$. At $t=1$, $G\approx\frac1{3.545}\approx0.282$. At $t=4$, $G=\frac1{\sqrt{16\pi}}\approx0.141$: half. Quadrupling the time halves the height and doubles the width, keeping the area.

Exercise: is the function u(x) = √|x| Hölder continuous? With what exponent?

Yes, with $\alpha=\frac12$: for $x,y\geq0$, $|\sqrt x-\sqrt y|\leq\sqrt{|x-y|}$, and the same holds in the other cases with $C=\sqrt2$. But $u$ has no derivative at $x=0$: the tangent line is vertical. Being Hölder is a weaker control than being differentiable, and it is exactly what the De Giorgi-Nash theorem guarantees.

Sources: John Nash, “Continuity of solutions of parabolic and elliptic equations”, American Journal of Mathematics 80 (1958) · Ennio De Giorgi, “Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari”, Memorie dell’Accademia delle Scienze di Torino (1957) · 2015 Abel Prize citation.

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