Issues in Nonlinear Elliptic Equations and Free Boundary Problems

非线性椭圆方程和自由边界问题中的问题

• 批准号: 0701016
• 负责人: Luis Silvestre
• 金额: $ 9.5万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2008-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701016&HistoricalAwards=false
• 关键词: Issues; Nonlinear; Elliptic; Equations; Free

Issues in Nonlinear elliptic Equations and Free Boundary ProblemsAbstract of Proposed Research Luis SilvestreThis project concerns the analysis of nonlinear problems in elliptic partial differential equations, integro-differential equations and some related free boundary problems. We will focus on the obstacle problem for the fractional Laplacian, the thin obstacle problem, nonlinear integro-differential operators, and optimal design of composites. The obstacle problem for the fractional Laplacian is closely linked to lower dimensional obstacle problems, as the PI has recently shown in collaborative work. Thus these two problems will be investigated together. The PI also intends to generalize the concepts of viscosity solutions, well-posedness, and regularity results from the theory of fully nonlinear elliptic equations to nonlinear integro-differential equations. The PI intends to apply ideas from free boundary problems to some questions about composite design in materials science. The microstructures that maximize a certain effective quantity, or a combination of such, may sometimes be found by solving a free boundary problem. Analysis of this problem will provide a better understanding of these optimal structures.The free boundary problems studied in this project arise in elasticity, financial mathematics, and composite material design. The obstacle problem is originally a model for the shape of an elastic membrane resting on a solid body. The same equation is used in financial mathematics for pricing American options. When stock prices have large jumps they may be modeled using discontinuous processes; this results in an obstacle problem for an integro-differential operator. The research proposed here studies the qualitative properties of the solutions of this problem as well as general nonlinear problems arising from stochastic games driven by random processes with jumps. Related ideas are also applied to the design of composite materials. Suitable free boundary problems enable the analysis of the microscopic structures that are optimal for certain purposes, such as to maximize the sum of the thermal and electric conductivity in a material.

非线性椭圆方程和自由边界问题拟议研究摘要 Luis Silvestre 该项目涉及椭圆偏微分方程、积分微分方程和一些相关自由边界问题中的非线性问题的分析。我们将重点关注分数拉普拉斯算子的障碍问题、薄障碍问题、非线性积分微分算子以及复合材料的优化设计。正如 PI 最近在协作工作中所表明的那样，分数拉普拉斯算子的障碍问题与低维障碍问题密切相关。因此，这两个问题将被放在一起研究。 PI 还打算将粘度解、适定性和正则性结果的概念从完全非线性椭圆方程理论推广到非线性积分微分方程。 PI 打算将自由边界问题的思想应用于材料科学中复合材料设计的一些问题。有时可以通过解决自由边界问题来找到使特定有效量最大化的微观结构或这些有效量的组合。对该问题的分析将有助于更好地理解这些最优结构。该项目研究的自由边界问题出现在弹性、金融数学和复合材料设计中。障碍问题最初是一个关于固体上弹性膜形状的模型。金融数学中也使用相同的方程来对美式期权进行定价。当股票价格大幅上涨时，可以使用不连续过程对其进行建模；这导致了积分微分算子的障碍问题。这里提出的研究研究了该问题的解决方案的定性特性以及由带有跳跃的随机过程驱动的随机博弈引起的一般非线性问题。相关思想也应用于复合材料的设计。 适当的自由边界问题可以分析针对某些目的而优化的微观结构，例如最大化材料中的导热率和导电率之和。