In recent years, the analysis and geometry of sub-Riemannian spaces has received increasing attention. The quintessential examples of sub-Riemannian settings are the so-called Carnot groups, whose fundamental role in analysis was first highlighted by E. M. Stein. They now occupy a central position not only in such mathematical areas as hypoelliptic partial differential equations, harmonic analysis, and CR geometric function theory, but also in the applied sciences (e.g., mathematical finance, mechanical engineering, neurophysiology of the brain). The most distinctive feature of sub-Riemannian spaces is that the metric structure can be viewed as a constrained geometry, where motion is only possible along a prescribed set of directions, changing from point to point. The principal investigator has a long-term project aimed at exploring geometric and analytic properties of these structures. More specifically, she proposes to continue her study of the Bernstein problem and of the regularity of minimal surfaces in Carnot groups, to investigate subelliptic boundary value problems, and to develop a regularity theory for fully nonlinear equations of Monge-Ampere type. Another area of interest in this project is the investigation of elliptic and parabolic free boundary problems naturally arising in the theory of flame propagation. The principal investigator also intends to study a class of minimization problems, in which the relevant functional is modeled after the one introduced by Alt and Caffarelli. One of the main objectives of the proposed research is to prove regularity properties of the free boundary. The necessary tools from harmonic analysis and partial differential equations for the study of these problems will be developed concurrently. Finally, motivated by the striking analogy between the theories of minimal surfaces and of free boundaries in the Euclidean setting, the PI plans to merge her different lines of research into a yet quite unexplored area, namely, the study of free boundary problems (both of obstacle and Alt-Caffarelli type) in Carnot groups. The principal investigator will integrate her research plan with several educational, mentoring, and outreach activities.
This project will conduct research that lies at the interface of calculus of variations, partial differential equations, and geometric measure theory. The focus is on the study of analytic and geometric properties of solutions to variational inequalities and partial differential equations involving a system of noncommuting vector fields. The problems under consideration not only arise in a variety of mathematical contexts (e.g., optimal control theory, mathematical finance, and geometry), but also are of interest in other fields such as mechanical engineering, robotics, and neurophysiology. Another proposed research area concerns free boundary problems, which naturally arise in physics and engineering when a conserved quantity or relation changes discontinuously across some value of the variables under consideration. The free boundary appears, for instance, as the interface between a fluid and the air, or between water and ice. Part of the project aims at studying regularity properties of the free boundary in burnt-unburnt mixtures. The results of this investigation will lead to a better understanding of the models, to the improvement of simulation methods, and ultimately to a precise description of how flames propagate in nonhomogeneous media. Several elements of this project find their motivations in the applied sciences. On the other hand, the solutions to these probelms involve an interplay of ideas from different areas of analysis and geometry. It is conceivable that all these different fields will benefit from this synergy. The principal investigator is committed to the training of future generations of mathematicians, and to increasing the representation of women in the scientific community, via the organization of a variety of educational and mentoring activities for graduate, undergraduate, and K-12 students.
Partial differential equations (PDEs) express relations between an unknown function and its derivatives. They are used to describe a wide variety of physical phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, and elasticity. The focus of this proposal is to study PDEs in certain geometric settings, called sub-Riemannian spaces. The most distinctive feature of these spaces is that motion is only possible along a prescribed set of directions, changing from point to point, but still ensuring global accessibility. These structures naturally arise in the study of optimal control and path planning when only constrained movements are possible. Typical examples are robotic arms and wheeled vehicles. The quintessential prototype of a sub-Riemannian space is the Heisenberg group, which occupies a central position not only in many areas of mathematics (e.g. hypoelliptic partial differential equations, harmonic analysis, CR geometry, roup representation), but also in a broad range of applied sciences, such as quantum mechanics, mathematical finance, materials sciences, and medicine. Thanks to recent developments in neuro-mathematics and visual cognition, we now have considerable evidence that perceptual processes are accomplished by the first layer of the visual cortex by actively integrating local information. The principal level of organization of the visual cortex is the columnar structure, which has been modeled as a structure locally equivalent to a Heisenberg-type group. Of particular relevance in this context is a class of geometric objects, the so-called minimal surfaces. In the classical Euclidean setting, these are surfaces that minimize the area, subject to prescribed boundary condition. They can also be equivalently characterized as solutions of a nonlinear PDE. The study of minimal surfaces has been one of the prime drivers of the developments of geometry and calculus of variations in the twentieth century. A central role in it has been played by the Bernstein problem: If the graph of a function defined on an (n-1)-dimensional space is a minimal surface (as an object in an n-dimensional space), is it true that the function is linear? This problem was first formulated in 1915 by N. S. Bernstein, who also solved it when n=3. Thanks to the outstanding work of several mathematicians (W. H. Fleming, E. De Giorgi, F. J. Almgren, J. Simons, E. Bombieri, E. Giusti) in the 1960s, we know that the answer to this question is affirmative when the dimension n is at most 8, and that this property instead does not necessarily hold true when the dimension n is 9 or higher. One of the main accomplishments of this award is the proof of an equivalent property for suitably defined minimal surfaces in the first (3-dimensional) Heisenberg group. In joint work with N. Garofalo, D. M. Nhieu, and S. Pauls, it was also established that the property fails when the Heisenberg group is of order 5 or higher. Despite the formal similarities between the Euclidean and sub-Riemannian formulations of the Bernstein problem, many new phenomena, which are not present in the former setting, are observed in the latter, and need to be dealt with. For instance, some of the classical analytic tools of differential and integral calculus are not compatible with the underlying geometry of the ambient space, and ad hoc sub-Riemannian counterparts had to be developed. The PI and her co-authors also established monotonicity properties of the relevant area functional, and family of inequalities which are essential to study regularity properties of sub-Riemannian minimal surfaces. The results from this project have been published in leading, peer-reviewed scientific journals. They have also been presented by the PI and her collaborators in lectures at national and international conferences, as well as in individual invited talks and colloquia. The audiences of such lectures include graduate students and junior mathematicians, who are thus exposed to new developments in this area of research and made aware of open problems. The PI has supervised two graduate students and two postdoctoral advisees during the time period covered by this award. This grant has funded the 4th and 5th Symposia in Analysis and PDEs, held at Purdue University respectively in 2009 and 2012, and co-organized by the PI. The purpose of these conferences is twofold: On the one hand, to introduce prospective and young researchers to a larger mathematical community, and to help them to establish professional connections with key figures in their areas of interest. On the other hand, to bring together leading experts in the themes of the Symposium, at different stages of their careers, to summarize the most recent progress in such topics, provide an opportunity to exchange ideas towards the solution of open questions, and formulate and develop new problems and avenues of research. The PI has also maintained her commitment to increase the participation of underrepresented groups in the mathematical sciences by organizing activities aimed at recruiting and retaining women and minorities into the profession.
