Convex Geometric Potential Theory

最初，势理论来自于一个物理问题，即在给定物体边界上产生的电场的测量下，重建物体内部电荷的重新分配。在分析方面，这相当于在给定集合边界上的函数值的情况下表示集合内的函数值。回想一下，当电导率恒定时，电流通量F与差grad(u)= u在电势u中的梯度成正比。因此，在不带电荷的集合S的最简单情况下，问题可以表述为在S的边界上用函数f指定的u值，求S中div(grad(u))=0的解u。但是，实际上存在比div（grad(u)）更复杂的形式——一种情况是幂律，其中f =|grad(u)|p-2grad(u)，导致p-拉普拉斯方程div(|grad(u)|p-2grad (u))=0在S的边界上受u=f的约束-这在某些材料中已经观察到，在这些材料成为超导的温度附近，p作为温度的函数。******然而，势理论对p-拉普拉斯的重要性在于对p-调和函数的研究及其与许多领域的联系——事实上，它很好地填补了算子理论、复变量、偏微分方程、拓扑学、概率论和几何学的相互作用。因此，势理论在其发展过程中对这些领域作出了贡献，并受到了这些领域的刺激。******有趣的是，one-Laplacian (n-1)-1div(|grad（u)|-1grad(u)）测量每个点的水平集的平均曲率，而∞- laplacian (grad2(u))-1grad(u),|grad(u)|-1grad(u)>表示在最陡上升方向上的二阶导数。由*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))-1grad(u),|grad(u)|-1grad(u)>，*我们看到p-拉普拉斯算子可以看作是一元拉普拉斯算子和无穷拉普拉斯算子的加权和。这样的观察导致了对凸几何势理论（由p-拉普拉斯引起）的研究，该理论包括以下五个关于平衡势和变化能力的目标。**** *** 1。Hardy-Morrey- sobolev空间中Riesz势的约束问题。***2。一个Minkowski/Yau型的最小/最大问题

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))-1grad(u),|grad(u)|-1grad(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))-1grad(u),|grad(u)|-1grad(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