Dimension reduction for Riesz energies with power-law external fields beyond the characterized parameter range

Determine whether dimension reduction of the support of the equilibrium measure occurs, in the sense that the support becomes a sphere, for Riesz energy problems with radial power-law external fields V(x) = c‖x‖^α when −2 < s < d and α > max{0, −s} outside the parameter values for which spherical support has been characterized.

Background

The paper studies Riesz equilibrium problems on ℝ^d with radial external fields and characterizes when the equilibrium measure is supported on a sphere, focusing in particular on power-law external fields V(x) = (γ/α)‖x‖^α. For −2 < s < d − 3 and sufficiently large α (α ≥ α_{s,d}), the authors establish that the uniform measure on a sphere is the equilibrium measure.

In the discussion of related works, the authors explicitly note that their characterization for spherical support in the power-law case leaves unresolved whether a similar dimension reduction of the equilibrium support occurs for other combinations of the Riesz parameter s and the power α within the admissible regime −2 < s < d and α > max{0, −s}. They highlight that available results suggest a negative answer in some cases but do not settle the general question.