论文标题
阈值凝结以单数支持Riesz平衡问题
Threshold condensation to singular support for a Riesz equilibrium problem
论文作者
论文摘要
我们计算尺寸D = S+4的平衡度量与Riesz S-Kernel相互作用与由Euclidean Norm的功率给出的外部场相关联。我们的研究表明，平衡度量可以是连续部分和奇异部分的混合物。根据功率的值，发生阈值现象，并由奇异部分的尺寸降低或缩合组成。特别是，在对数情况下s = 0（d = 4），当外部磁场的功率变得二次时，在特殊半径的球体上存在凝结。这与先前研究的情况D = S+3的情况形成对比，这表明平衡度量是完全尺寸并在球上支撑的。我们的方法除其他工具外，使用了Frostman或Euler-Lagrange差异表征，Funk-Hecke公式，Gegenbauer正交多项式以及超几何特殊功能。
We compute the equilibrium measure in dimension d=s+4 associated to a Riesz s-kernel interaction with an external field given by a power of the Euclidean norm. Our study reveals that the equilibrium measure can be a mixture of a continuous part and a singular part. Depending on the value of the power, a threshold phenomenon occurs and consists of a dimension reduction or condensation on the singular part. In particular, in the logarithmic case s=0 (d=4), there is condensation on a sphere of special radius when the power of the external field becomes quadratic. This contrasts with the case d=s+3 studied previously, which showed that the equilibrium measure is fully dimensional and supported on a ball. Our approach makes use, among other tools, of the Frostman or Euler-Lagrange variational characterization, the Funk-Hecke formula, the Gegenbauer orthogonal polynomials, and hypergeometric special functions.
