Free Boundaries, PDE's, and Geometric Measure Theory

Purdue University, West Lafayette IN

Investigators

Abstract

PI: Donatella Danielli, Purdue University DMS-0202801 ------------------------------------------------------------------------------ Abstract: 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 proposes to study a class of free boundary problems of interest in flame propagation. The model is obtained via an asymptotic method that simplifies a complicated system of conservation laws describing the process of combustion on the basis of physically sound approximations. The very way the problem is derived suggests viewing it as the limit of regularizing problems. One of the main objectives of the proposed research is to determine conditions under which limit solutions of the approximating problems converge to classical solutions to the original one, and to prove optimal 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 pde's 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, and at investigating the validity of the Bernstein property in this setting. 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 question is instrumental also in the study of boundary value problems for subelliptic operators. In particular, the P.I. plans to investigate the solvability of the Neumann problem for sub-Laplacians, and to determine the optimal regularity of solutions. 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 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 pde's 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.