CAREER: Analytic and Geometric Aspects of Partial Differential Equations

Purdue University, West Lafayette IN

Investigators

Abstract

PI: Donatella Danielli, Purdue University DMS-0239771 Abstract: ******************************************** The research part of this proposal presents a collection of problems motivated by the study of elliptic and parabolic free boundary problems, calculus of variations, and geometric measure theory. The P.I intends to study free boundary problems of interest in flame propagation, and related to Lord Rayleigh's conjecture that among all clamped plates of a given area, the circular one gives the lowest principal frequency. One of the main objectives of the proposed research is to prove regularity properties of the free boundary. Another area of interest is the optimal regularity of the solution and of the free boundary in the subelliptic obstacle problem. The necessary tools from harmonic analysis and PDEs for the study of these problems will be developed concurrently. The P.I. has also a program aimed at developing the regularity theory of minimal surfaces in Carnot groups. Such program entails the study of several basic questions. Among these, we mention the existence and characterization of traces on lower dimensional manifolds of Sobolev or BV functions. This issue is instrumental also in the study of the Neumann problem for sub-Laplacians. In connection with questions arising in geometry, the P.I. intends to develop a regularity theory for subelliptic fully nonlinear equations modeled on the classical Monge-Ampere operator. This program involves establishing an appropriate version of the celebrated Alexandrov-Bakelman-Pucci maximum principle, which in turn requires the investigation of a suitable notion of convexity. The P.I. is also interested in studying the method of ``moving spheres" for so-called Weingarten hypersurfaces, and in its use to prove symmetry properties of solutions to fully nonlinear equations. The P.I. proposes to integrate this research plan with several educational activities. In particular, we mention the organization of an annual Summer Symposium at Purdue University. The P.I. will supervise undergraduate research projects as part of Purdue's REU program. At the K-12 level, the P.I. hopes to hook receptive young minds organizing fun, hands-on mathematics workshops at the local science museum, as well as in the framework of Expanding Your Horizons conferences. To increase the representation of women in the scientific community, the P.I. will also continue mentoring women in science. Free boundary problems 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 water and ice. One of the proposed projects 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 non-homogeneous media. The P.I. has also a research program 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 PDEs involving a system of non-commuting vector fields. The problems described in the proposal not only arise in a variety of mathematical context (e.g. optimal control theory, mathematical finance, and geometry), but are also of interest in other fields such as mechanical engineering and robotics. The P.I. 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 activities for graduate, undergraduate, and K-12 students.