On a singular perturbation problem related to optimal lifting in BV-space Academic Article uri icon

abstract

  • We prove a Γ-convergence result for the family of functionals defined on H 1(Ω) by \({\mathcal J}_\varepsilon(\varphi)=\int_\Omega\big(\varepsilon|\nabla\varphi|^2 +\frac{1}{\varepsilon}|u(x)-e^{i\varphi}|^{2p}\big)\hbox{d}x, \forall\varepsilon > 0,\) for a given \(u\in BV(\Omega,S^1)\) and a parameter \(p\in[1,\infty)\). We show that in either of the two cases, p = 2 or \(u\in W^{1,1}(\Omega,S^1)\), any limit of the minimizers is an optimal lifting.

publication date

  • January 1, 2007