Free Boundaries, PDE's, and Geometric Measure Theory

主要研究者：Donatella Danielli，Purdue University DMS-0202801-摘要：这个建议提出了一个集合的问题，由椭圆和抛物自由边界问题，变分法，和几何测度论。作者提出研究火焰传播中一类感兴趣的自由边界问题。该模型是通过一个渐近的方法，简化了一个复杂的系统的守恒定律描述的燃烧过程的基础上，物理上健全的近似。这个问题的推导方式表明，我们可以把它看作是正则化问题的极限。所提出的研究的主要目标之一是确定条件下的极限解的逼近问题收敛到经典的解决方案，原来的，并证明最佳的自由边界的正则性。另一个感兴趣的领域是最佳的正则性的解决方案和自由边界的次椭圆障碍问题。从谐波分析和偏微分方程的研究这些问题的必要工具将同时开发。私家侦探也有一个计划，旨在发展的规律性理论的极小曲面卡诺集团，并在调查的有效性，伯恩斯坦财产在这种情况下。这样的计划需要研究几个基本问题。其中，我们提到的存在性和低维流形上的Sobolev或BV功能的痕迹的特征。这个问题也有助于研究次椭圆算子的边值问题。特别是，P.I.计划调查次拉普拉斯算子的诺依曼问题的可解性，并确定解的最佳正则性。自由边界问题在物理学和工程学中很自然地出现，当一个守恒量或关系在所考虑的变量的某个值上不连续地变化时。例如，自由边界表现为流体与空气或水与冰之间的界面。其中一个项目旨在研究燃烧-未燃烧混合物中自由边界的正则性。这项调查的结果将导致更好地理解模型，改进模拟方法，并最终描述火焰如何在非均匀介质中传播。私家侦探也有一个研究计划，位于变分法，偏微分方程和几何测量理论的接口。重点是研究涉及非交换向量场系统的变分不等式和偏微分方程解的解析和几何性质。该提案中描述的问题不仅出现在各种数学背景下（例如最优控制理论，数学金融和几何），而且在其他领域也很感兴趣，如机械工程和机器人。

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.