Curvature-driven geometric evolution problems

The project aims to advance the mathematical understanding in challenging models in which surface tensions act as a principal driving force and normal velocity of the surface depends on position and local shape of the surface, in particular on the curvature. Such models characterize many processes in material sciences such as phase transformation, crystal growth, domain growth, grain growth and so on. In reality the velocity is influenced by many other factors such as pressure, temperature, chemistry, particle size, surrounding atmosphere, etc. however, an isolation of certain physical processes and study of them in experiments shed light in understanding the underlying phenomena and provide clues towards more sophisticated processes. The models we consider in this project are given by variational energies that are defined on a set of surfaces or, more generally, union of surfaces so-called surface junctions, and will be investigated both in the static and evolutionary case. In the static setting we look for a surface/surface junction which represents an equilibrium configuration of the energy and study regularity properties of such equilibrium surfaces. In the evolutionary framework we search for a family of surfaces/surface junctions depending on time parameter that evolves according to certain evolution equations, such as area-minimizing flows. The groundbreaking nature of the project is the examination of new research questions and the study of new research ideas in three main directions in curvature-driven evolution problems which are currently open and hot topics in Geometric Analysis, Calculus of Variations, Geometric Measure Theory and Partial Differential Equations. First, we focus on the weak curvature-driven flows and study their existence and regularity properties. In addition we investigate singularity formation in flow, long-time behavior, and consistency of our weak flow with classical solutions. The second focus of the project is related to graph-like minimizers of a least-area problem, and the third focus resides in finding Willmore surfaces and Willmore flows subject to some fixed boundary conditions. The methodology combines and/or modifies existing methods in Geometric Measure Theory, Calculus of Variations and Differential Equations. In particular, we employ results from the theory of sets of finite perimeter, minimal surfaces, gradient flows, minimizing movements, the regularity theory, convex analysis, Plateau-type problems as well as Willmore equation and Willmore boundary problems. Promising preliminary results have been already achieved.

The project aimed to advance the mathematical understanding of curvature-driven geometric evolutions, where surface tension is the principal driving force and the velocity depends on both the position and the local shape of the surface, particularly its curvature. Motivated by physical processes in materials science (such as crystal growth and phase transitions), the project focused on variational models formulated via energy functionals defined on surfaces or unions thereof (surface junctions). Both the static (energy minimization) and dynamic (evolutionary) aspects were studied in three primary directions, all of which were successfully addressed: 1. Weak curvature-related flows. We studied curvature flows of planar (polygonal) networks in the crystalline setting. Key achievements: existence and regularity of crystalline planar network flows; analysis of singularity formation and restart procedures. 2. Least-area minimizers with graphical structure. We investigated regularity of minimizers of perturbed (possibly anisotropic) area functionals. A notable result is the establishment of regularity for one-dimensional minimizers, partially resolving a De Giorgi conjecture. 3. Willmore-type problems. We studied the crystalline elastic (Willmore-type) flow of polygonal curves. Key results: existence and uniqueness of the crystalline elastic flow for immersed polygons; analysis of maximal-time behavior and restart mechanisms; long-time behavior of the flow and Lojasiewicz-Simon-type inequalities; a classification of stationary and translating solutions in the case of square anisotropy. The project largely followed the proposed work plan. One minor deviation concerned the scope of Willmore-type flows and the consistency between crystalline curvature flow of networks and the minimizing movements: while initial progress was made, these problems turned out to be technically more involved and will be pursued beyond the project period. The other objectives were completed more or less as anticipated, in some cases exceeding expectations in terms of depth and mathematical insight. Although theoretical in nature, the project clarified fundamental aspects of curvature-driven flows that model physically relevant phenomena in materials science. We expect that the techniques developed will contribute significantly to the mathematical foundation for modeling interfaces and will find further applications in areas such as image processing. The project also enhanced the visibility of the host institution in the fields of geometric analysis and the calculus of variations, as reflected in invitations to collaborate and increased peer recognition following the dissemination of results at conferences.