Aviles-Giga Conjecture, Differential Inclusions and Rigidity

The project aims to advance understanding of some key problems in the field of Calculus of Variations, specifically the Aviles-Giga conjecture, and more broadly, how restrictions on gradients of functions imply rigidity, stability, and compactness properties. The Aviles-Giga conjecture is a central open problem in the Calculus of Variations, modeling phenomena such as thin film blistering and micromagnetics. The conjecture seeks to provide a mathematical justification for a scaling law observed in physics, leading to more accurate modeling of certain physical phenomena. Part of the conjecture involves sharp regularity estimates for a well-studied class of equations known as Eikonal equations, which arise in liquid crystal models and optics. These estimates are valuable for numerically solving such equations and are of broad mathematical interest. The Aviles-Giga theory is closely connected to the theory of scalar conservation laws, and its methods are being applied to understand a class of solutions of scalar conservation laws that arise in probability, specifically the large deviation conjecture. The project also aims to propagate its outcomes through seminars, lectures, graduate student recruitment, and the research produced. The project consider problems in Calculus of Variations. The first problem is the Aviles-Giga conjecture, where the main open problem is showing that the energy concentrates, as it is not even known if the measure representing the limiting energy is singular. Achieving this goal would lead to a complete understanding of the regularizing properties of the Eikonal equation on the Besov scale. The second problem deals with quantitative rigidity for non-elliptic differential inclusions and builds on a previous result for rotation matrices and an optimal generalization to connected 1D elliptic curves in the space of two-by-two matrices. One of the purpose of this work is a more general regularity/rigidity theory for non-elliptic curves. The third project studies compensated compactness and conservation laws in higher dimensions. Reformulating regularity and uniqueness questions of PDEs as differential inclusions has led to the solution of a number of outstanding conjectures. This part of the research focuses on further developing methods initiated by the principal investigator and collaborators to study the differential inclusion problem related to regularity and uniqueness questions for conservation laws in higher dimensions. The final project on gamma-convergence for the Bellettini-Bertini-Mariani-Novaga functional considers a proposed gamma-limit related to certain conjectures in large deviation theory. The project focuses on a special case of this conjecture. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.