Energy estimates for S1-valued maps
Fabrice Béthuel, Haïm Brézis, Frédéric Hélein · Birkhäuser Boston eBooks · 1994
Let G be a smooth, bounded and simply connected domain in ℝ 2 , and let ω i for i = 1,..., n, be open, smooth and simply connected subsets of G, with \( \overline {{\omega_i}} \subset G \) and \({\bar \omega _i} \cap {\bar \omega _j} = 0\). Let \( \Omega = G\backslash \mathop{ \cup }\limits_{{i = 1}}^n \overline {{\omega_i}} \) Consider the class of maps $$ \varepsilon = \left\{ {v \in {H^1}(\Omega; {S^1})\left| {_{{\deg (v,\partial {\omega_i}) = {d_i}\;for\;i = 1,2,...,n}}^{{\deg (v,\partial ) = d\quad and}}} \right.} \right\} $$ (1) where \( {d_i} \in Z \) are given and \( d = \sum\limits_{{i = 1}}^n {{d_i}} \).