本文考虑了由于$\mathbb{S}^d$上的许多局域电荷分布而引起的外场存在下，欧氏空间$\mathbb{R}^{d+1}$中单位球$\mathbb{S}^d$上的最小离散和连续能量问题，其中能量来自于临界Riesz参数$s = d - 2$的Riesz势$1/r^s$（$r$是欧氏距离）（如果$d \geq 3$）和对数势$\log(1/r)$（如果$d = 2$）.单独地，局部电荷分布是点电荷或假定为旋转对称的。解决弱场连续外场问题的极值测度被证明是球上的一致测度，但仅限于局部电荷分布周围的球冠外部。半径由产生电荷的相对强度决定。此外，我们表明，解决相关的离散外场问题的最小能量点仅限于此支持。对于$d-2\leq s<d$，我们表明，点源的球，平衡措施的支持补充工会指定的球冠的来源。数值例子来说明我们的结果。
英文摘要：
We consider the minimal discrete and continuous energy problems on the unit sphere $\mathbb{S}^d$ in the Euclidean space $\mathbb{R}^{d+1}$ in the presence of an external field due to finitely many localized charge distributions on $\mathbb{S}^d$, where the energy arises from the Riesz potential $1/r^s$ ($r$ is the Euclidean distance) for the critical Riesz parameter $s = d - 2$ if $d \geq 3$ and the logarithmic potential $\log(1/r)$ if $d = 2$. Individually, a localized charge distribution is either a point charge or assumed to be rotationally symmetric. The extremal measure solving the continuous external field problem for weak fields is shown to be the uniform measure on the sphere but restricted to the exterior of spherical caps surrounding the localized charge distributions. The radii are determined by the relative strengths of the generating charges. Furthermore, we show that the minimal energy points solving the related discrete external field problem are confined to this support. For $d-2\leq s<d$, we show that for point sources on the sphere, the equilibrium measure has support in the complement of the union of specified spherical caps about the sources. Numerical examples are provided to illustrate our results.
Sharp Riesz capacity inequalities and the Pólya--Szegő conjecture
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Tongji University), Ministry of Education, Tongji University（同济大学数学科学学院，智能计算与应用教育部重点实验室）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）
Global Existence and Nonlinear Stability of Finite-Energy Solutions of the Compressible Euler-Riesz Equations with Large Initial Data of Spherical Symmetry
Another look at regularity in transport-commutator estimates
Carnegie Mellon University（卡内基梅隆大学）；Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions (LJLL)（索邦大学、法国国家科学研究中心、巴黎大学、雅克-路易·利翁斯实验室）；Institut Universitaire de France（法国高等研究院）；Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）
Smale's 7-th problem concerns N-point configurations on the 2-dim sphere which minimize the logarithmic pair-energy V_0(r) = -ln r averaged over the pairs in a configuration; here, r is the chordal distance between the points forming a pair. More generally, V_0(r) may be replaced by the standardized Riesz pair-energy V_s(r)= (r^{-s} -1)/s, which becomes - ln r in the limit s to 0, and the sphere may be replaced by other compact manifolds. This paper inquires into the concavity of the map from the integers N>1 into the minimal average standardized Riesz pair-energies v_s(N) of the N-point confi…
