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




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* Corresponding author 1 EDP, Analyse ICJ - Institut Camille Jordan Villeurbanne 2 ICJ - Institut Camille Jordan Villeurbanne

Abstract : Let $\Omega$ be a smooth bounded simply connected domain in $R^2$. We investigate the existence of critical points of the energy $E \varepsilonu=1-2\int \Omega | abla u|^2+1-4\varepsilon^2\int \O 1-|u|^2^2$, 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.

keyword : Ginzburg-Landau prescribed degrees singular perturbation





Author: Xavier Lamy - Petru Mironescu -

Source: https://hal.archives-ouvertes.fr/



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