On a singular perturbation problem related to optimal lifting in BV-space

Poliakovsky, Arkady
April 2007
Calculus of Variations & Partial Differential Equations;Apr2007, Vol. 28 Issue 4, p411
Academic Journal
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.


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