Analysis of the p-Laplacian

p-拉普拉斯分析

• 批准号: 1001179
• 负责人: Juan Manfredi
• 金额: $ 12.2万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-15 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001179&HistoricalAwards=false
• 关键词: Analysis; Laplacian

In many problems in physics and engineering the relevant energies are proportional to the square of the velocity. The resulting equations are linear and model very well small variations from equilibrium. But more substantial variations are better modeled by considering non-quadratic energies. In this proposal we study some aspects of the mathematical theory of the equations that result from the minimization of energies (or other quantities in physics and engineering) that are given by power laws of the type "Energy is approximatley the sum of the absolute values of velocity raised to the pth power", for p different from 2. The minimizers of these non-quadratic energies are p-harmonic functions. When the energy is quadratic (p=2) the resulting equations are linear, but for non-quadratic energies we are necessarily led to non-linear equations. Recently Peres and Sheffield found an unexpected connection between random Tug-of-War games and p-harmonic functions. Motivated by these considerations we introduce the class of p-harmonious functions and explore its relationship to the classical p-harmonic functions. In particular, p-harmonious functions serve as very good discrete approximations to p-harmonic functions and suggest novel approaches to some long-standing open problems.The possible impact on other sciences comes from the connection between p-harmonic and p-harmonious functions and stochastic games. It may shine new light into some optimization problems that can be formulated in spaces quite more general than Euclidean space (graphs, trees, length spaces). The more immediate impact will be on human resources. The PI will use the formulation of Tug-of-War games in simple graphs to mentor several freshman students who used computer simulation to run these games, and are exposed to mathematical thinking early in their undergraduate career. The PI current graduate student will graduate in December 2010. The PI will mentor a postdoc in Analysis funded by the University of Pittsburgh for the duration of this project.

在许多物理和工程问题中，相关能量与速度的平方成正比。由此产生的方程是线性的，并且很好地模拟了平衡态的微小变化。但更实质性的变化更好地模拟考虑非二次能量。 在这个建议中，我们研究了方程的数学理论的某些方面，这些方程是由能量（或物理学和工程学中的其他量）最小化引起的，这些能量由幂律给出，幂律类型为“能量近似为速度绝对值的p次方之和”，p不同于2。这些非二次能量的极小值是p-调和函数。当能量是二次的（p=2），得到的方程是线性的，但对于非二次能量，我们必然导致非线性方程。最近佩雷斯和谢菲尔德发现了一个意想不到的连接之间的随机拔河比赛和p-调和函数。基于这些考虑，我们引入了p-调和函数类，并探讨了它与经典p-调和函数的关系。特别是，p-调和函数作为p-调和函数的一个很好的离散逼近，为一些长期存在的问题提供了新的解决途径。p-调和函数和p-调和函数与随机博弈之间的联系可能对其他科学产生影响。它可能会给一些优化问题带来新的启发，这些优化问题可以在比欧氏空间更一般的空间（图、树、长度空间）中表达。更直接的影响将是人力资源。PI将使用简单图形中的拔河游戏的公式来指导几位使用计算机模拟运行这些游戏的大一学生，并在他们的本科生涯早期接触到数学思维。PI目前的研究生将于2010年12月毕业。PI将在本项目期间指导匹兹堡大学资助的分析博士后。