Cycles of least mass in a Riemannian manifold, described through the ``phase transition'' energy of the sections of a line bundle
type: Item
instance of: scholarly article
title: Cycles of least mass in a Riemannian manifold, described through the ``phase transition'' energy of the sections of a line bundle
zbMATH Open document ID: 0948.49024
DOI: 10.1007/PL00004631
author: Sisto Baldo
author: Giandomenico Orlandi
published in: Mathematische Zeitschrift
publication date: 1999-12-06T00:00:00Z
review text: The authors study the convergence of a family of variational problems towards a singular limit in analogy to previous work of \textit{L. Modica} and \textit{S. Mortola} [Boll. Unione Mat. Ital., V. Ser., B 14, 285-299 (1977; Zbl 0356.49008)]. Let \(M\) be a compact \(n\)-dimensional Riemannian manifold and \(\xi\) a real line bundle over \(M\) with structure group \(O(1)\). For sections \(u\) of \(\xi\) in the Sobolev class \(W^{1,2}\) the family of energies \(F_\varepsilon\) is defined as \[ F_\varepsilon(u)= \int_M \Biggl(\varepsilon|\nabla u(x)|^2+{1\over \varepsilon} ||u(x)|- 1|^{{3\over 2}}\Biggr) d\text{ vol}_M(x). \] The authors show that, given minimizers \(u_\varepsilon\) of \(F_\varepsilon\), there is a sequence \(\varepsilon_h\to 0\) \((h\to\infty)\) such that \(u_{\varepsilon_h}\) converges in \(L^1\) towards a section \(u_0\) of modulus 1 and of bounded variation. Moreover, for any such limit section \(u_0\) the jump set of \(u_0\) (i.e., the set where \(u_0\) fails to be continuous) represents a class in the homology group \(H_{n-1}(M,\mathbb{Z}_2)\) which is Poincaré dual to the first Stiefel-Whitney class \(w_1(\xi)\in H^1(M, \mathbb{Z}_2)\) of the bundle \(\xi\) and minimizes the \(\text{mod }2\) mass in its homology class. The proof combines \(\Gamma\)-convergence techniques and geometric measure theory with some topological arguments.
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 58E15
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 53C65
zbMATH DE Number: 1374308
zbMATH Keywords: cycles of least mass
zbMATH Keywords: phase transition energy
zbMATH Keywords: convergence
zbMATH Keywords: variational problems
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Poincaré dual
zbMATH Keywords: Stiefel-Whitney class
zbMATH Keywords: \(\Gamma\)-convergence
Wikidata QID: Q115390571
reviewed by: F. Tomi
MaRDI profile type: Publication
full work available at URL: pl00004631
OpenAlex ID: W2037050639
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description: scientific article; zbMATH DE number 1374308