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Journal Articles Journal de Mathématiques Pures et Appliquées Year : 2014

Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains

Abstract

Let $\Omega$ be a smooth bounded simply connected domain in $R^2$. We investigate the existence of critical points of the energy $E_\varepsilon(u)=1/2\int_\Omega |\nabla u|^2+1/(4\varepsilon^2)\int_\Omega (1-|u|^2)^2$, where the complex map $u$ has modulus one and prescribed degree $d$ on the boundary. Under suitable nondegeneracy assumptions on $\Omega$, we prove existence of critical points for small $\varepsilon$. More can be said when the prescribed degree equals one. First, we obtain existence of critical points in domains close to a disc. Next, we prove that critical points exist in ''most'' of the domains.
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Dates and versions

hal-00784904 , version 1 (04-02-2013)
hal-00784904 , version 2 (22-02-2013)
hal-00784904 , version 3 (19-03-2013)
hal-00784904 , version 4 (28-10-2013)

Identifiers

  • HAL Id : hal-00784904 , version 4

Cite

Xavier Lamy, Petru Mironescu. Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains. Journal de Mathématiques Pures et Appliquées, 2014, 102 (2), pp.385-418. ⟨hal-00784904v4⟩
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