Calculus of Variations and Geometric Measure Theory

F. Bethuel - H. Brezis - G. Orlandi

Asymptotics for the Ginzburg-Landau equation in arbitrary dimensions

created on 14 Dec 2001


Published Paper

Inserted: 14 dec 2001

Journal: J. Funct. Anal.
Volume: 186
Number: 2
Pages: 432-520
Year: 2001


Let $\Omega$ be a bounded, simply connected, regular domain of $R^N$, $N\ge 2$. For $0<\varepsilon<1$, let $u_\varepsilon :\Omega\to C$ be a smooth solution of the Ginzburg-Landau equation in $\Omega$ with Dirichlet boundary condition $g_\varepsilon$, i.e., $$ \cases{ -\Delta u\varepsilon ={1\over \varepsilon2} u\varepsilon (1-
2) & {\rm in } \Omega,\cr u\varepsilon= g\varepsilon & {\rm on } \partial\Omega. \cr} $$ We are interested in the asymptotic behavior of $u_\varepsilon$ as $\varepsilon$ goes to zero under the assumption that $E_\varepsilon(u_\varepsilon)\le M_0
$ and some conditions on $g_\varepsilon$ which allow singularities of dimension $N-3$ on $\partial\Omega$.

Keywords: Ginzburg-Landau equation