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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711107
• 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. From p-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，势能理论来自于重建物体内部电荷重新分配的物理问题，给出了对物体边界上产生的电场的测量。在分析方面，这相当于表达一个函数在一个集合内的值，给定该函数在集合边界上的值。回想一下，当电导率恒定时，电流通量F正比于差格拉德（u）= u在电势u中的梯度。因此，在一个不带电荷的集合S的最简单情况下，问题可以表述为在S中找到div（格拉德（u））=0的解u，u的值由S边界上的函数f规定。但是，实际上存在比div（格拉德（u））更复杂的形式-一种情况是幂律，其中F=|格拉德（u）|2019 - 02 - 13 00：00：00 00：00（|格拉德（u）|在S中p-2grad（u））=0，在S的边界上u=f-这在某些材料中已经在接近材料变得超导的温度下观察到，其中p作为温度的函数。 然而，潜在的理论的重要性超过p-Laplacian在于研究p-调和函数和它的联系，以许多领域-事实上-它很好地填补了一个位置，在相互作用的算子理论，复变量，偏微分方程，拓扑，概率和几何。因此，潜势理论在其发展过程中，对这些领域做出了贡献，并受到了这些领域的刺激。 2016 - 01 - 12 00：01：01 00：01 00：01 00：00|格拉德（u）|-1grad（u））测量每个点处水平集的平均曲率，无穷大拉普拉斯算子（grad 2（u））表示最陡上升方向上的二阶导数。从 p-1|格拉德（u）|2-PDIV（|格拉德（u）|p-2grad（u））=p-1|格拉德（u）|（|格拉德（u）|-1grad（u））+（1-p-1）（grad2（u））， 我们看到，p-Laplacian可以被认为是一个Laplacian和无穷大Laplacian的加权和。这样的观察导致凸几何势理论（由p-Laplacian引起）的调查，包括以下五个目标的平衡势和变化能力。 1. Hardy-Morrey函数的Riesz势的Hardy-Morrey-Sobolev空间的一个限制问题。 2.一个Minkowski/Yau型最小/最大问题，