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Stable interfaces: phase transitions, minimal surfaces, and free boundaries

Periodic Reporting for period 2 - StableIF (Stable interfaces: phase transitions, minimal surfaces, and free boundaries)

Reporting period: 2022-07-01 to 2023-12-31

Countless natural phenomena are accurately described by Partial Differential Equations (PDEs). These equations often succeed in capturing the essence of complex phenomena in just a few concise lines – think of the Navier-Stokes equations for fluid motion or Einstein's equations for the evolution of spacetime. However, deriving these equations is just the beginning of the story.
The simplicity of the equations doesn't necessarily mean their solutions are simple – consider phenomena like turbulence and black holes!

This project delves into nonlinear PDEs that model "interfaces", with phase transitions - such as ice melting in water or of two immiscible fluids in equilibrium (similarly to a lava lamp) - being a prime example. The PDEs governing these interfaces are remarkably simple, yet they yield an overwhelming array of solutions – some strikingly intricate.
However, only a fraction of these solutions truly materializes in the natural world. This selection is guided by the criterion of stability: only equilibria that withstand (small) spontaneous disruptions endure long enough for us to observe them.

In essence, the core questions we seek to address are quite fundamental. What are the qualitative properties that characterize phase transitions? Does the transition surface exhibit smooth behavior, or are there instances of singularities? How can we describe the overall structure of solutions, and to what extent can they vary in complexity?

However, attempting to provide answers that cover all potential solutions, including both stable and unstable ones, would inevitably lead to vagueness, lack of precision, or even misleading conclusions. This is due to the fact that certain unstable solutions might possess features or pathologic behaviors that are impossible within the realm of stable solutions.

By focusing our attention on stable solutions, we expect to be able to give precise analytic answers to the previous questions that match our observations of the natural world.
But here's the catch – despite trying since the 1950s, mathematicians haven't been able to prove rigorously some of the most basic properties of stable solutions in our physical three-dimensional space.
We have very precise conjectures on what should happen but no mathematical proof.

This project is all about taking on this big question and finding the answers.
When dealing with a highly complex mathematical model, it's often helpful to start by looking at a simplified version, known as a "toy model”.

An effective toy model strikes a balance: it's complex enough to offer insights into the original problem, yet simple enough to allow focused progress. The valuable insights gained from this kind of analysis play a pivotal role in tackling the larger problem at hand

The initial phase of our work primarily focused on addressing vital inquiries:

1. What are the key difficulties impeding advancements in understanding stable phase transitions?
2. Which toy models, exhibit similar difficulties but are more manageable?
3. Can we establish analogous versions of the conjectures made for the main model within the context of these toy models?

However, the initial phase of our work wasn't solely centered around toy models of interfaces. In fact, have also addressed an iconic problem with roots dating back to the 19th century – the Stefan problem.

Named after the Slovenian physicist Joseph Stefan, this Partial Differential Equation describes phase changes, such as the melting of ice in water. Within this context, the interface (e.g. the surface between the regions of ice and water) is termed the "free boundary."

Renowned results by Abel Prize laureates Luis Caffarelli and Louis Nirenberg from the 1970s established that free boundaries exhibiting local flatness must be analytic.
However, how often this necessary flatness condition could be verified remained unclear.
Furthermore, known examples demonstrated that the free boundary could be wildly nonsmooth at specific times.

During these first 2.5 years of action, we have invested significant efforts in tackling the following question: "Is the set of singularities in the Stefan problem sparse, in some sense?" In other words: "Is the free boundary typically smooth?".
To do so, we have built on techniques introduced in 2020 by the Fields Medalist laureate Alessio Figalli and the PI.
After decades of stagnation in progress, we have recently achieved a groundbreaking result concerning the size of the singular set within the Stefan problem, particularly in three spatial dimensions. We established that the interface – the boundary between ice and water ----regions, for instance – remains smooth except for a negligible (measure zero) set of times.

The fact that this theorem garnered attention in the prestigious Quanta magazine underscores its profound significance.

On the other hand, we successfully tackled the regularity of the interface for stable solutions to non-local Allen-Cahn equations. The Allen-Cahn model is paradigmatic because it serves as the scalar version of the Ginzburg-Landau model and is also linked to the stationary states of the Cahn-Hilliard model. Its rich mathematical properties and connections to minimal surfaces have rendered it iconic within the field of Calculus of Variations. Its non-local variant is an excellent toy model.

It is a common pattern in mathematics that breakthroughs in one domain lead to advances in seemingly unrelated areas.
In this case, the insights garnered from our novel findings have unlocked fresh applications in Differential Geometry, particularly in the exploration of minimal submanifolds within closed Riemannian manifolds.
In fact, we have introduced a novel "non-local approximation" approach, enabling the theoretical construction of minimal surfaces in closed three-dimensional manifolds.

On the other hand, building on the insights gained in our non-local analysis we expect to develop a deep understanding of stable interfaces that goes well beyond the existing state-of-the-art knowledge on the topic previous to the beginning of our project.

More specifically, we expect to establish the regularity of the interface for stable solutions of the Allen-Cahn equation in three dimensions, or at least for its free boundary version.
If successful, this achievement would mark the first-ever three-dimensional regularity result for a stable interface that, on larger scales, resembles a minimal surface (i.e. a soap film).

We anticipate that this breakthrough could pave the way for similar results in other related models, all of which hold significant physical implications.