Regularity of minimizers and pattern formation in geometric minimization problems

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1. Distance related energies. Energy driven evolutions are ubiquitous. Such a physical system generally evolve to a ground state, and if the involved energy is sufficiently regular, then ground states exhibit a high degree of regularity and periodicity. For instance, ground states are often made of congruent copies of a basic cell, or “crystal”, placed along a regular lattice (see e.g. [15]). Such a process is known as “crystallization”, which is among the most relevant and investigated processes. Several results are available in 2D, e.g. [6, 21, 43], while few such results are available in 3D. Among the simplest such energies, are the distance related energies. These energies are used in a wide range of problems, e.g. data analysis, machine learning, vector quantization, image processing, etc.. The aim of this plan is to study classic problems about minimizers of some distance related energy, the pattern formation often observed for such minimizers, and their applications. Short term goals include investigating the geometry of the ground states in 3D. Since the techniques developed to analyze ground states are quite general, as a long term goal, several classic problems, e.g. the optimal foam problem, and sphere packing, can be studied using the same techniques. A deeper understanding (or a full solution) of these mathematical problems will be beneficial to their applications, potentially allowing for faster, more accurate, more efficient models.******2. PDEs in material sciences. Evolution processes, in several fields, are often modeled by Partial Differential Equations (PDEs). One such area is epitaxy: it has received a lot of attention in recent years since as one of the key industrial processes to produce high quality crystals, crucial for semiconductors and nanotechnology. Such a process is highly nonlinear in nature (see e.g. [7, 27, 38]). There is a plethora of PDEs modeling several different epitaxial processes, but our present knowledge is still far from complete. Another area receiving much attention is liquid crystals, due to their widespread applications, and relevance to biology. A common issue is that the PDEs are highly nonlinear, which makes their treatment (both theoretical and numerical) quite difficult. As short term goal, we plan to investigate PDEs arising from material sciences, since many are still poorly understood due to the strong nonlinearities. As long term goal, we aim to develop new techniques to study gradient flows of “badly behaved” energies, since so many arise from hotly studied areas of material sciences. Rigorously proving the well-posedness of such PDEs will potentially give a predictive theory, crucial to manufacturing processes since it allows to accurately predict the behavior of materials under different conditions. Moreover, quantitative estimates are likely to be required, which will be useful for the numerical analysis of these PDEs.