Free Boundaries, PDE's, and Geometric Measure Theory

自由边界、偏微分方程和几何测度理论

• 批准号: 0202801
• 负责人: Donatella Danielli
• 金额: $ 9.15万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-15 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202801&HistoricalAwards=false
• 关键词: Free; Boundaries; PDE; Geometric; Measure

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.

PI：Donatella Danielli，普渡大学 DMS-0202801---------------------------------------- -------------------------------------- 摘要：该提案提出了一系列由以下问题引发的问题：研究椭圆和抛物线自由边界问题、变分法和几何测度理论。 P.I 提议研究一类与火焰传播有关的自由边界问题。该模型是通过渐近方法获得的，该方法基于物理合理的近似简化了描述燃烧过程的复杂守恒定律系统。这个问题的导出方式表明将其视为正则化问题的极限。该研究的主要目标之一是确定逼近问题的极限解收敛于原始问题的经典解的条件，并证明自由边界的最优正则性质。另一个令人感兴趣的领域是亚椭圆障碍问题中解和自由边界的最优规律性。研究这些问题所需的调和分析和偏微分方程工具将同时开发。 P.I.还有一个计划旨在发展卡诺群中最小曲面的正则理论，并研究伯恩斯坦性质在这种情况下的有效性。该计划需要研究几个基本问​​题。其中，我们提到了 Sobolev 或 BV 函数的低维流形上迹的存在和表征。这个问题对于研究亚椭圆算子的边值问题也很有帮助。特别是，P.I.计划研究亚拉普拉斯诺依曼问题的可解性，并确定解的最佳规律性。当守恒量或关系在所考虑的变量的某些值上不连续变化时，物理和工程学中自然会出现自由边界问题。例如，自由边界表现为流体与空气、或水与冰之间的界面。拟议项目之一旨在研究已燃-未燃混合物中自由边界的规律性特性。这项研究的结果将有助于更好地理解模型，改进模拟方法，并最终描述火焰如何在非均匀介质中传播。 P.I.还有一个研究项目涉及变分法、偏微分方程和几何测度理论。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。提案中描述的问题不仅出现在各种数学背景中（例如最优控制理论、数学金融和几何），而且在机械工程和机器人技术等其他领域也很有趣。