During a volcanic eruption, a scorching, chaotic river of lava flows over the ground. But after hours or days (or perhaps even longer), it cools enough to enter a state of equilibrium. Its temperature is no longer changing from moment to moment, although it still varies from place to place across the vast expanse of space the lava covers.

Mathematicians describe situations like this using what are called elliptic PDEs. These equations represent phenomena that vary across space but not time, such as the pressure of water flowing through rock, the distribution of stress on a bridge, or the diffusion of nutrients in a tumor.

But solutions to elliptic PDEs are complicated. The solution to the lava PDE, for instance, describes its temperature at every point, given some initial conditions. It depends on a lot of interacting variables.

…

The key to proving that the solution to a PDE is regular is to show that it always changes in a controlled way. Mathematicians do this by looking at a special function that describes how fast the solution changes at each point. They want to show that this function, which is called the gradient, can’t get too big.

But just as it’s usually impossible to directly compute the solution to a PDE, it’s also usually impossible to calculate its gradient.

…

In a 2022 preprint, they were able to tame all these pieces well enough to show that most nonuniformly elliptic PDEs that satisfy Mingione’s inequality have to have regular solutions. But some PDEs were still missing. To prove the full conjecture, the mathematicians had to get even better bounds on the sizes of the gradient’s pieces. There was absolutely no wiggle room. This required starting over many times — “a never-ending game,” De Filippis said. But eventually, they were able to prove that the threshold Mingione had predicted decades earlier was exactly right(opens a new tab).