Limiting behavior of the Ginzburg-Landau functional

Robert L. Jerrard; Halil Mete Soner

doi:10.1006/jfan.2001.3906

Limiting behavior of the Ginzburg-Landau functional

Robert L. Jerrard, Halil Mete Soner

Research output: Contribution to journal › Article › peer-review

33 Scopus citations

Abstract

We continue our study of the functional for u ε H¹ (U; ℝ²), where U is a bounded, open subset of ℝ². Compactness results for the scaled Jacobian of u^e are proved under the assumption that E_e(u^e) is bounded uniformly by a function of e. In addition, the Gamma limit of E_e(u^e)/(ln e)² is shown to be where v is the limit of j(u^e)/ ln e , j(u^e) := u^e x Du^e, and ∥ · ∥is the total variation of a Radon measure. These results are applied to the Ginzburg-Landau functional with external magnetic field h_ext ≈ H ln e . The Gamma limit of F_e/(ln e)² is calculated to be where v is as before, and a is the limit of A^e/ ln e .

Original language	English (US)
Pages (from-to)	524-561
Number of pages	38
Journal	Journal of Functional Analysis
Volume	192
Issue number	2
DOIs	https://doi.org/10.1006/jfan.2001.3906
State	Published - Jul 10 2002
Externally published	Yes

All Science Journal Classification (ASJC) codes

Analysis

Keywords

BnV
Compactness
Gamma limit
Ginzburg-Landau functional

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10.1006/jfan.2001.3906

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title = "Limiting behavior of the Ginzburg-Landau functional",

abstract = "We continue our study of the functional for u ε H1 (U; ℝ2), where U is a bounded, open subset of ℝ2. Compactness results for the scaled Jacobian of ue are proved under the assumption that Ee(ue) is bounded uniformly by a function of e. In addition, the Gamma limit of Ee(ue)/(ln e)2 is shown to be where v is the limit of j(ue)/ ln e , j(ue) := ue x Due, and ∥ · ∥ is the total variation of a Radon measure. These results are applied to the Ginzburg-Landau functional with external magnetic field hext ≈ H ln e . The Gamma limit of Fe/(ln e)2 is calculated to be where v is as before, and a is the limit of Ae/ ln e .",

keywords = "BnV, Compactness, Gamma limit, Ginzburg-Landau functional",

author = "Jerrard, {Robert L.} and Soner, {Halil Mete}",

note = "Funding Information: 1Partially supported by the National Science Foundation Grant DMS-96-00080. Part of this work was completed during visits of the first author to the Max-Planck-Institute for Mathematics in the Science in Leipzig, Germany, the National Center for Theoretical Science in Hsinchu, Taiwan, and Koc University in Istanbul. 2Partially supported by the National Science Foundation Grant DMS-98-17525. 3To whom correspondence should be addressed.",

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AU - Jerrard, Robert L.

AU - Soner, Halil Mete

N1 - Funding Information: 1Partially supported by the National Science Foundation Grant DMS-96-00080. Part of this work was completed during visits of the first author to the Max-Planck-Institute for Mathematics in the Science in Leipzig, Germany, the National Center for Theoretical Science in Hsinchu, Taiwan, and Koc University in Istanbul. 2Partially supported by the National Science Foundation Grant DMS-98-17525. 3To whom correspondence should be addressed.

PY - 2002/7/10

Y1 - 2002/7/10

N2 - We continue our study of the functional for u ε H1 (U; ℝ2), where U is a bounded, open subset of ℝ2. Compactness results for the scaled Jacobian of ue are proved under the assumption that Ee(ue) is bounded uniformly by a function of e. In addition, the Gamma limit of Ee(ue)/(ln e)2 is shown to be where v is the limit of j(ue)/ ln e , j(ue) := ue x Due, and ∥ · ∥ is the total variation of a Radon measure. These results are applied to the Ginzburg-Landau functional with external magnetic field hext ≈ H ln e . The Gamma limit of Fe/(ln e)2 is calculated to be where v is as before, and a is the limit of Ae/ ln e .

AB - We continue our study of the functional for u ε H1 (U; ℝ2), where U is a bounded, open subset of ℝ2. Compactness results for the scaled Jacobian of ue are proved under the assumption that Ee(ue) is bounded uniformly by a function of e. In addition, the Gamma limit of Ee(ue)/(ln e)2 is shown to be where v is the limit of j(ue)/ ln e , j(ue) := ue x Due, and ∥ · ∥ is the total variation of a Radon measure. These results are applied to the Ginzburg-Landau functional with external magnetic field hext ≈ H ln e . The Gamma limit of Fe/(ln e)2 is calculated to be where v is as before, and a is the limit of Ae/ ln e .

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KW - Gamma limit

KW - Ginzburg-Landau functional

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Limiting behavior of the Ginzburg-Landau functional

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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