We prove a lower bound for the value of the L1-relaxed area of the graph of the map u:Bl(0)∖{0}⊂R2→R2, u(x):=x/|x|, x≠0, for all values of the radius l>0. In the computation of the singular part of the relaxed area, for l in a certain range, in particular l not too large, a nonparametric Plateau-type problem with partial free boundary has to be solved. Our lower bound turns out to be optimal, in view of an upper bound proven in a companion paper.
The L1-relaxed area of the graph of the vortex map: Optimal lower bound
Bellettini G.;
2025-01-01
Abstract
We prove a lower bound for the value of the L1-relaxed area of the graph of the map u:Bl(0)∖{0}⊂R2→R2, u(x):=x/|x|, x≠0, for all values of the radius l>0. In the computation of the singular part of the relaxed area, for l in a certain range, in particular l not too large, a nonparametric Plateau-type problem with partial free boundary has to be solved. Our lower bound turns out to be optimal, in view of an upper bound proven in a companion paper.File in questo prodotto:
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