Free Boundaries, PDE's, and Geometric Measure Theory

PI：多纳泰拉·丹尼利，普渡大学DMS-0202801------------------------------------------------------------------------------摘要：这项建议提出了一系列问题，这些问题的动机是研究椭圆和抛物线自由边界问题、变分法和几何测度论。P.I建议研究一类在火焰传播中感兴趣的自由边界问题。该模型是通过渐近方法得到的，该方法简化了描述燃烧过程的复杂的守恒定律系统，其基础是物理上合理的近似。这个问题的推导方式表明，它被视为正规化问题的极限。研究的主要目的之一是确定逼近问题的极限解在何种条件下收敛于原问题的经典解，并证明自由边界的最优正则性。另一个有趣的领域是次椭圆障碍问题解的最佳正则性和自由边界的最优正则性。将同时开发用于研究这些问题的调和分析和偏微分方程的必要工具。P.I.还有一个程序，旨在发展卡诺群中极小曲面的正则性理论，并在这种情况下研究Bernstein性质的有效性。这样的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刻画。这个问题在研究次椭圆算子的边值问题时也是有用的。特别是，P.I.计划研究次拉普拉斯人Neumann问题的可解性，并确定解的最佳正则性。在物理和工程中，当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，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。其中一个拟议的项目旨在研究燃烧-未燃烧混合物中自由边界的规律性。这项研究的结果将有助于更好地理解模型，改进模拟方法，并最终描述火焰在非均匀介质中的传播方式。P.I.也有一个研究项目，它位于变分、偏微分方程组和几何测度论的交界处。重点研究了涉及非对易向量场系统的变分不等式和偏微分方程组的解的解析性质和几何性质。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)，而且还涉及其他领域，如机械工程和机器人技术。

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.