Minimality and Non-degeneracy of Degree-one Ginzburg-landau Vortex as a Hardy’s Type Inequality
نویسندگان
چکیده
We consider the unique radially symmetric degree-one solution w(x) = U(r)e of the Ginzbug-Landau equation in R ∆w + (1− |w|)w = 0, |w(x)| → 1 as |x| → +∞. We provide an elementary proof of its locally minimizing character and non-degeneracy up to the obvious invariance of the equation, in the natural Hilbert space for its second variation bilinear form. The proof reduces to an optimal vector-valued form of Hardy’s inequality. As a consequence, we establish a Fredholm alternative in this space for the associated linearized operator.
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