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arXiv · 2507.20725

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

Daniele Bartolucci·Aleks Jevnikar·Juncheng Wei·Ruijun Wu
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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.

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Daniele Bartolucci, Aleks Jevnikar, Juncheng Wei, Ruijun Wu. 2025-07-28. Classification of singular limits for free boundary and singularly perturbed elliptic problems: the Dancer-Yan spikes revisited. https://doi.org/10.1093/imrn%2Frnag187

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