Descripción del proyecto
Las matemáticas avanzadas proporcionan información novedosa sobre las relaciones estables
Cuando Joseph Louis Lagrange derivó la ecuación de superficies mínimas en 1762, poco podía sospechar de la relación que tendría su ecuación con la teoría de las transiciones de fase desarrollada dos siglos después. Todos estamos familiarizados con algunas transiciones de fase como, por ejemplo, el hielo que se derrite en el agua. Pero existen docenas de casos y todos ellos son esenciales para la actividad diaria y la innovación humana: metales en una aleación, superconductividad, intervalos de confianza en finanzas, cristales líquidos, combustión, diseño óptimo de aislantes, entre muchos otros. A pesar de la naturaleza omnipresente y la importancia de las transiciones de fase, nuestra capacidad para analizar matemáticamente sus comportamientos estables es limitadísima. El proyecto StableIF, financiado con fondos europeos, combina avances recientes con herramientas clásicas de la teoría de superficies mínimas para perfeccionar el análisis matemático que mejorará la comprensión de las transiciones estables de fase.
Objetivo
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”.
Ámbito científico
Programa(s)
Régimen de financiación
ERC-STG - Starting GrantInstitución de acogida
8092 Zuerich
Suiza