Convex Geometric Potential Theory

最初，势能理论来自于重建物体内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。就分析而言，这相当于在给定集合边界上的函数值的情况下，表示集合内函数的值。回想一下，当电导率不变时，电流通量F与差梯度(U)=电势u中u的梯度成正比。因此，在不带电荷的集合S的最简单情况下，问题可以表示为求S中的div(grad(U))=0的解u的问题，受S的边界上的函数f所规定的u值的约束。但实际上，存在比div(grad(U))更复杂的形式--一种情况是幂定律的，其中F=|grad(U)|p-2grad(U)，在S中导出了满足u=f的p-拉普拉斯方程div(|grad(U)|p-2grad(U))=0，这在某些材料中已经观察到，在温度附近，材料成为超导材料，对于这种材料，p是温度的函数。*尽管如此，p-拉普拉斯函数上的位势理论的重要性在于研究p-调和函数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、概率和几何相互作用的一个位置。因此，位势理论在其发展过程中对这些领域做出了贡献并受到了刺激。*有趣的是，一-拉普拉斯(n-1)-1div(|grad(U)|-1grad(U))测量水平集在每个点的平均曲率，无穷大-拉普拉斯(grad2(U))-1grad(U)，|grad(U)|-1grad(U)&gt；表示沿最陡峭上升方向的二阶导数。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)&gt；，我们看到p-拉普拉斯可以看作是1-拉普拉斯和无穷拉普拉斯的加权和。这样的观察导致了对凸几何势理论的研究，它包括关于平衡势和变化量的五个目标。*1.Hardy-Morrey函数的Riesz势的Hardy-Morrey-Sobolev空间的限制问题*2.关于1的Minkowski/Yau型最小/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