Georgia Institute of Technology College of Sciences

For any nonnegative density f and radially decreasing interaction potential W, the celebrated Riesz rearrangement inequality shows the interaction energy E[f] = \int f(x)f(y)W(x-y) dxdy satisfies E[f] <= E[f^*], where f^* is the radially decreasing rearrangement of f. It is a natural question to look for a quantitative version of this inequality: if its two sides almost agree, how close must f be to a translation of f^*? Previously the stability estimate was only known for characteristic functions. I will discuss a recent work with Xukai Yan, where we found a simple proof of stability estimates for general densities. 

I will also discuss another work with Matias Delgadino and Xukai Yan, where we constructed an interpolation curve between any two radially decreasing densities with the same mass, and show that the interaction energy is convex along this interpolation. As an application, this leads to uniqueness of steady states in aggregation-diffusion equations with any attractive interaction potential for diffusion power m>=2, where the threshold is sharp.