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On a 1-capacitary type problem in the plane

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  • We study a $1$-capacitary type problem in $R^2$: given a set $E$, we minimize the perimeter (in the sense of De Giorgi) among all the sets containing $E$ (modulo $H^1$) and satisfying an indecomposability constraint (according to the definition by [1]. By suitably choosing the representant of the relevant set $E$, we show that a convexification process characterizes the minimizers.
        As a consequence of our result we determine the $1$-capacity of (a suitable representant of) sets with finite perimeter in the plane.
    Mathematics Subject Classification: 49J40, 49Q20.


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