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CORDIS

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

Project description

Seeing stable relationships in a new light with a little help from advanced mathematics

When Joseph Louis Lagrange in 1762 derived the equation of minimal surfaces, he could not possibly suspect its deep connections with the theory of phase transitions, which would only be developed two centuries later. We are all familiar with some phase transitions such as ice melting in water. But there are dozens of them which are critical to daily activity and human innovation: metals in an alloy, superconductivity, decision boundaries in finance, liquid crystals, combustion, optimal design of insulators, and many more. Despite the ubiquitous nature and significance of phase transitions, our ability to analyse their stable behaviours mathematically is astoundingly limited. The EU-funded StableIF project combines recent advances with classical tools from the theory of minimal surfaces to develop the mathematical analysis that will enhance our understanding of stable phase transitions.

Objective

One of the main drivers of development for the theory of nonlinear elliptic PDE during the second half of the XX century has been the mathematical analysis of physical models for “inter- faces”. Depending on the specific model, these “interfaces” are called minimal surfaces, phase transitions, free boundaries, etc. These models are very important in applications and, due to their strong geometric content and the interdisciplinary methods required for their study, also from a “pure mathematics” perspective. One of the simplest semilinear PDE exhibiting an interface is the classical Allen-Cahn equation. Originally proposed as a model for metal alloys, it gained mathematical notoriety due to its deep connection with the minimal surface equation and many other important PDE. It is very related to the Cahn-Hiliard equation (phase separation in binary fluids), to the Peierls-Nabarro equation (crystal dislocations), and to the Ginzburg-Landau theory (phase transitions, super-conductivity). In addition, it has similarities with other important models such as Bernoulli’s free boundary problem (flame propagation and shape optimization) or the Eriksen-Leslie system (liquid crystals). In the last four decades, outstanding works led to a very deep understanding of the structure of (absolute) energy minimizers for most of the previous models. Still, up to very few exceptions, almost nothing is known today on the structure of stable solutions —i.e. (roughly speaking) minimizers with respect to sufficiently small perturbations. Since stable solutions are “the ones observable in Nature”, their understanding is a fundamental question. Even though it is a very challenging mathematical problem, all the new analysis tools developed in the last decades plus some recent progress give us now an excellent opportunity to address it. In three words, the very ambitious goal of this ERC project is to “understand stable interfaces”.

Host institution

EIDGENOESSISCHE TECHNISCHE HOCHSCHULE ZUERICH
Net EU contribution
€ 1 348 125,00
Address
Raemistrasse 101
8092 Zuerich
Switzerland

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Region
Schweiz/Suisse/Svizzera Zürich Zürich
Activity type
Higher or Secondary Education Establishments
Links
Total cost
€ 1 348 125,00

Beneficiaries (2)