Issues in Nonlinear Elliptic Equations and Free Boundary Problems

非线性椭圆型方程中的问题和自由边界问题建议研究摘要本项目主要研究椭圆型偏微分方程组、积分-微分方程组和一些相关的自由边界问题中的非线性问题。我们将集中讨论分数阶拉普拉斯的障碍问题，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。分数拉普拉斯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表明的那样。因此，这两个问题将被一起调查。PI还打算将完全非线性椭圆型方程理论中的粘性解、适定性和正则性的概念推广到非线性积分-微分方程组。PI打算将自由边界问题的思想应用于材料科学中关于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某一有效数量最大化的微观结构，或者这种结构的组合。对这一问题的分析将提供对这些优化结构的更好的理解。本项目研究的自由边界问题涉及到弹性力学、金融数学和复合材料设计。障碍物问题最初是一个固定在固体上的弹性膜形状的模型。金融数学中也使用了同样的公式来为美式期权定价。当股票价格有很大的跳跃时，可以使用不连续的过程来建模；这导致了积分-微分操作符的障碍问题。本文研究了这一问题的解的定性性质，以及由带跳的随机过程驱动的随机对策所产生的一般非线性问题。相关思想也被应用到复合材料的设计中。适当的自由边界问题能够分析对某些目的最优的微观结构，例如使材料的导热系数和导电率之和最大化。

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.