Classification of singular limits for free boundary and singularly perturbed elliptic problems: the Dancer-Yan spikes revisited

TL;DR

This paper classifies the singular limits of a free boundary problem in plasma physics in two dimensions, revealing a richer variety of spike behaviors than previously understood, including detailed asymptotic descriptions.

Contribution

It provides a detailed classification of singular limits, especially the Dancer-Yan spikes, in a 2D free boundary problem, expanding understanding beyond higher dimensions.

Findings

Identification of multiple asymptotic behaviors including Dancer-Yan spikes

Detailed description of the difference between spike maxima and free boundary

Revealing richer spike structures in 2D compared to higher dimensions

Abstract

We classify the singular limits relative to a free boundary problem arising in plasma physics in dimension $d=2$, under suitable natural integral bounds. It turns out that one of the asymptotic behaviors allowed corresponds to the Dancer-Yan spikes (J. London Math. Soc. ({\bf 78}) 2008, 639--662). Interestingly enough, roughly speaking and unlike the higher dimensional case, it is not true that any solution in the limit is a Dancer-Yan spike. Indeed, the spiking structure is more rich and we succeed in a detailed description of the singular behavior by a careful analysis, from local to global, of the tiny difference between the maximum value of the spikes and their ``vanishing level'' defining the free boundary.