Abstract
In this paper, we study the asymptotic behavior of minimizing solutions of a Ginzburg–Landau type functional with a positive weight and with convex potential near 0 and we estimate the energy in this case. We also generalize a lower bound for the energy of unit vector field given initially by Brezis–Merle–Rivière.
Highlights
Let G be a bounded, connected and smooth domain of R2, g : ∂G → S1 a smooth boundary data of degree d and p a smooth positive function on G
As a corollary of upper and lower bounds of the energy, we find an estimate of the mutual distances between bad discs approaching the same singularity bk
The proof of Theorem 1 consists of two main ingredients: the method of Struwe [23], as used in [3] in norder to locate o the “bad discs”, (i.e., a finite collection of discs of radius O(ε) which cover the set x : |uε ( x ) < 12 | ) and the generalization of a result of Brezis, Merle and Rivière [10] which will play an important role in finding the lower bound of the energy
Summary
Let G be a bounded, connected and smooth domain of R2 , g : ∂G → S1 a smooth boundary data of degree d and p a smooth positive function on G. P = 12 corresponding to the Ginzburg–Landau energy, was studied by several authors since the groundbreaking works of Béthuel-Brezis and Hélein More precisely they dealt with the case with boundary data satisfying d = 0 and d 6= 0 respectively in [5,6]. Due to the presence of a non-constant weight and a potential with zero of infinite order at t = 0, the energy cost of each vortex of degree dk > 1 is much less than the previous one. Let us point out that it could be interesting for our problem to give a precise asymptotic behavior of the term o ( I (|log ε|) sk )) in (7) At the moment, this question is not yet fully understood, since it is related to renormalized energy introduced in [8] (see [3]). As a corollary of upper and lower bounds of the energy, we find an estimate of the mutual distances between bad discs approaching the same singularity bk
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