Minimax Critical Points in Ginzburg-Landau Problems with Semi-Stiff Boundary Conditions: Existence and Bubbling

Leonid Berlyand; Petru Mironescu; Volodymyr Rybalko; Etienne Sandier

doi:10.1080/03605302.2013.851214

Minimax Critical Points in Ginzburg-Landau Problems with Semi-Stiff Boundary Conditions: Existence and Bubbling

Leonid Berlyand
, Petru Mironescu
, Volodymyr Rybalko
, Etienne Sandier

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

13 Scopus citations

Abstract

Let Ω ⊂ ℝ² be a smooth bounded simply connected domain. We consider the simplified Ginzburg-Landau energy (Formula Presented), where u: Ω → ℂ. We prescribe {pipe}u{pipe} = 1 and deg (u, ∂Ω) = 1. In this setting, there are no minimizers of E _ε{lunate}. Using a mountain pass approach, we obtain existence of critical points of E _ε{lunate} for large ε{lunate}. Our analysis relies on Wente estimates and on the study of bubbling phenomena for Palais-Smale sequences.

Original language	English (US)
Pages (from-to)	946-1005
Number of pages	60
Journal	Communications in Partial Differential Equations
Volume	39
Issue number	5
DOIs	https://doi.org/10.1080/03605302.2013.851214
State	Published - May 2014

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

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10.1080/03605302.2013.851214

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title = "Minimax Critical Points in Ginzburg-Landau Problems with Semi-Stiff Boundary Conditions: Existence and Bubbling",

abstract = "Let Ω ⊂ ℝ2 be a smooth bounded simply connected domain. We consider the simplified Ginzburg-Landau energy (Formula Presented), where u: Ω → ℂ. We prescribe \{pipe\}u\{pipe\} = 1 and deg (u, ∂Ω) = 1. In this setting, there are no minimizers of E ε\{lunate\}. Using a mountain pass approach, we obtain existence of critical points of E ε\{lunate\} for large ε\{lunate\}. Our analysis relies on Wente estimates and on the study of bubbling phenomena for Palais-Smale sequences.",

author = "Leonid Berlyand and Petru Mironescu and Volodymyr Rybalko and Etienne Sandier",

note = "Funding Information: The work of LB was supported by NSF grant DMS-1106666. The work of VR was partially supported by NSF grant DMS-1106666. PM was supported by the ANR project ANR-12-BS01-0013-03.",

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doi = "10.1080/03605302.2013.851214",

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T2 - Existence and Bubbling

AU - Berlyand, Leonid

AU - Mironescu, Petru

AU - Rybalko, Volodymyr

AU - Sandier, Etienne

N1 - Funding Information: The work of LB was supported by NSF grant DMS-1106666. The work of VR was partially supported by NSF grant DMS-1106666. PM was supported by the ANR project ANR-12-BS01-0013-03.

PY - 2014/5

Y1 - 2014/5

N2 - Let Ω ⊂ ℝ2 be a smooth bounded simply connected domain. We consider the simplified Ginzburg-Landau energy (Formula Presented), where u: Ω → ℂ. We prescribe {pipe}u{pipe} = 1 and deg (u, ∂Ω) = 1. In this setting, there are no minimizers of E ε{lunate}. Using a mountain pass approach, we obtain existence of critical points of E ε{lunate} for large ε{lunate}. Our analysis relies on Wente estimates and on the study of bubbling phenomena for Palais-Smale sequences.

AB - Let Ω ⊂ ℝ2 be a smooth bounded simply connected domain. We consider the simplified Ginzburg-Landau energy (Formula Presented), where u: Ω → ℂ. We prescribe {pipe}u{pipe} = 1 and deg (u, ∂Ω) = 1. In this setting, there are no minimizers of E ε{lunate}. Using a mountain pass approach, we obtain existence of critical points of E ε{lunate} for large ε{lunate}. Our analysis relies on Wente estimates and on the study of bubbling phenomena for Palais-Smale sequences.

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JF - Communications in Partial Differential Equations

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ER -

Minimax Critical Points in Ginzburg-Landau Problems with Semi-Stiff Boundary Conditions: Existence and Bubbling

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