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.

在物理和工程中的许多问题中，相关能量与速度的平方成正比。所得的方程是线性和模型与平衡相对小的变化。但是，通过考虑非二次能量来更好地建模更大的变化。 在这项建议中，我们研究了由该类型的功率定律给出的数学理论的一些方程式（或物理和工程学中的其他数量）所产生的方程式理论的某些方面。当能量是二次的（p = 2）时，所得方程是线性的，但是对于非二次能量，我们必然会导致非线性方程。最近，佩雷斯（Peres）和谢菲尔德（Sheffield）发现了随机的拔河游戏和p谐波功能之间的意外联系。在这些考虑因素上，我们介绍了p可以破坏性功能的类别，并探索了其与经典p谐波功能的关系。尤其是，p谐的功能是对p谐波功能的很好的离散近似，并建议解决一些长期存在的开放问题的新方法。对其他科学的可能影响来自p-harmonic和p-harmonious功能与随机游戏之间的联系。它可能会将新的光照射到一些优化问题中，这些优化问题可以比欧几里得空间（图，树，长度空间）更通用的空间中。对人力资源的影响更直接的影响。 PI将使用简单的图表使用拔河游戏的配方来指导几位使用计算机模拟运行这些游戏的新生学生，并在本科生的早期就接触了数学思维。 PI现任研究生将于2010年12月毕业。PI将在该项目的期限内指导由匹兹堡大学资助的分析博士后分析。