Convex Geometric Potential Theory

凸几何势理论

• 批准号: RGPIN-2017-05036
• 负责人: Xiao, Jie
• 金额: $ 1.46万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759175
• 关键词: Convex; Geometric; Potential; Theory

Originally, potential theory comes from the physical problem of reconstructing a repartition of electric charges inside a body, given a measuring of the electrical field created on the boundary of this body. In terms of analysis this amounts to expressing the values of a function inside a set given the values of the function on the boundary of the set. Recall that current flux F is proportional to difference grad(u)=the gradient of u in electric potential u whenever conductivity is constant. Thus, in the simplest case of a set S without electric charges the problem can be formulated as that of finding a solution u to div(grad(u))=0 in S subject to u's values prescribed by a function f on the boundary of S. But, in reality there exist more complicated forms than div(grad(u)) - one situation is of power-law where F=|grad(u)|p-2grad(u), leading to the p-Laplace equation div(|grad(u)|p-2grad (u))=0 in S subject to u=f on the boundary of S - this has been observed in certain materials near the temperatures where the material becomes super-conductive for which p acts as a function of temperature. Nevertheless, the importance of potential theory over p-Laplacian lies in the study of p-harmonic functions and its links to many areas - in fact - it nicely fills up a position at the interaction of operator theory, complex variables, partial differential equations, topology, probability and geometry. Therefore, potential theory has contributed to and received stimulus from these areas, in its developments.Interestingly, one-Laplacian (n-1)-1div(|grad(u)|-1grad(u)) measures the mean curvature of the level set at each point and infinity-Laplacian (grad2(u)) represents the second derivative in the direction of steepest ascent. Fromp-1|grad(u)|2-pdiv(|grad(u)|p-2grad(u))=p-1|grad(u)|div(|grad(u)|-1grad(u))+(1-p-1)(grad2(u)), we see that p-Laplacian may be regarded as a weighted sum of one-Laplacian and infinity-Laplacian. Such an observation leads to an investigation of the convex-geometric-potential-theory (induced by p-Laplacian) that comprises the following five objectives on equilibrium potential and variation capacity. 1. A restriction problem for the Hardy-Morrey-Sobolev space of Riesz potentials of Hardy-Morrey functions.2. A Minkowski/Yau type minimum/maximum problem for the 1

最初，潜在理论来自重建体内电荷的重建电荷的物理问题，因为测量了该体内边界上产生的电场。在分析方面，这等于在集合边界上的函数值给定函数的值表示集合中的函数值。回想一下，当电势u恒定时，电流f与差级= u的梯度成正比。因此，在一组无电荷的集合的最简单情况下，可以将问题提出为将解决方案u（grad（u））在s中的u（grad（u））= 0的= 0，但受到u的值的规定。 div（| grad（u）| p-2grad（u））在s的s边界上受u = f的约为0-在某些材料附近的某些材料中观察到了该材料变得超导性的某些材料，其中P充当温度的函数。然而，潜在理论对p -laplacian的重要性在于对p谐波功能的研究及其与许多领域的联系 - 实际上 - 它很好地填补了操作者理论，复杂变量，部分微分方程，拓扑，拓扑，拓扑，概率和几何形状的相互作用。因此，潜在的理论在其发展中从这些领域的发展中贡献并接受了刺激。从而，单拉帕（N-1）-1div（| Grad（U）（U）| -1Grad（U））测量在每个点和Infinity-Laplacian（grad2（u））的平均水平集合的平均曲率代表第二个衍生性的次级速度较高的次数。从P-1 | GRAD（U）| 2-PDIV（| Grad（U）| P-2Grad（U））= P-1 | Grad（U）| Div（| Grad（U）| -1Grad（U）| -1Grad（u））+（1-P-1）（Grad2（U）），我们认为Plaplacian可以看作是一位加权的单拉普拉斯和Infrinity-Infrinity-Laplacian。这样的观察结果导致对凸线几何学理论（由p-laplacian诱导）的研究，该理论构成了以下五个目标，其平衡势和变异能力。 1。硬质 - 莫里函数Riesz电位的Hardy-Morrey-Sobolev空间的限制问题2。 Minkowski/Yau类型的最小/最大问题1