Issues in Nonlinear Elliptic Equations and Free Boundary Problems

非线性椭圆型方程和自由边界问题的问题建议研究摘要路易斯·西尔维斯特此项目涉及椭圆型偏微分方程，积分微分方程和一些相关的自由边界问题的非线性问题的分析。我们将集中讨论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和复合材料的优化设计。分数拉普拉斯算子的障碍问题与低维障碍问题密切相关，正如PI最近在合作工作中所示。因此，这两个问题将一起研究。PI还打算将粘性解，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有关复合材料设计的一些问题。最大化某个有效量的微结构或其组合有时可以通过求解自由边界问题来找到。对这个问题的分析将提供对这些最优结构的更好理解。本项目中研究的自由边界问题出现在弹性力学、金融数学和复合材料设计中。障碍物问题最初是一个模型的形状的弹性膜上的固体。金融数学中也用同样的公式来为美式期权定价。当股票价格有很大的跳跃，他们可以使用不连续的过程建模，这导致在一个障碍问题的积分微分算子。这里提出的研究，研究这个问题的解决方案的定性性质，以及一般的非线性问题所产生的随机游戏驱动的随机过程与跳跃。相关的思想也被应用到复合材料的设计中。 合适的自由边界问题使得能够分析对于某些目的最佳的微观结构，例如最大化材料中的热导率和电导率的总和。

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.