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Coulomb gas ensembles and Laplacian growth

Hedenmalm, Håkan and Makarov, Nikolai (2013) Coulomb gas ensembles and Laplacian growth. Proceedings of the London Mathematical Society, 106 (4). pp. 859-907. ISSN 0024-6115. doi:10.1112/plms/pds032.

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We consider weight functions Q : ℂ→ℝ that are locally in a suitable Sobolev space and impose a logarithmic growth condition from below. We use Q as a confining potential in the model of one-component plasma (2-dimensional Coulomb gas) and study the configuration of the electron cloud as the number n of electrons tends to infinity, while the confining potential is rescaled: we use mQ in place of Q and let m tend to infinity as well. We show that if m and n tend to infinity in a proportional fashion, with n/m→t, where 0<t<+∞ is fixed, then the electrons accumulate on a compact set St, which we call the droplet. The set S_t can be obtained as the coincidence set of an obstacle problem, if we remove a small set (the shallow points). Moreover, on the droplet St, the density of electrons is asymptotically ΔQ. The growth of the droplets St as t increases is known as the Laplacian growth. It is well known that Laplacian growth is unstable. To analyse this feature, we introduce the notion of a local droplet, which involves removing part of the obstacle away from the set S_t. The local droplets are no longer uniquely determined by the time parameter t, but at least they may be partially ordered. We show that the growth of the local droplets may be terminated in a maximal local droplet or by the droplets' growing to infinity in some direction (‘fingering’).

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Additional Information:© 2012 London Mathematical Society. Received 11 May 2011; revised 13 February 2012; published online 26 October 2012. We thank Kurt Johansson for helpful comments in connection with the previous arXiv preprint, and Serguei Shimorin for help with the proofreading. The first author was supported by the Göran Gustafsson Foundation (KVA) and Vetenskapsrådet (VR). The second author was supported by NSF Grant no. 0201893
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Göran Gustafsson FoundationUNSPECIFIED
Issue or Number:4
Classification Code:2010 Mathematics Subject Classification: 15A52, 31B20, 35R35, 62E20, 82B44 (primary), 76M40, 76D27, 31A05 (secondary).
Record Number:CaltechAUTHORS:20130625-102609096
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:39076
Deposited On:25 Jun 2013 18:36
Last Modified:09 Nov 2021 23:42

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