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September  2010, 9(5): 1319-1333. doi: 10.3934/cpaa.2010.9.1319

On a 1-capacitary type problem in the plane

Matteo Focardi 1, , Maria Stella Gelli 2, and  Giovanni Pisante 3,

1. 

Dipartimento di Matematica "U. Dini", Università di Firenze, viale Morgagni 67/A, I-50139 Firenze
2. 

Dip. Mat. “L. Tonelli”, Università di Pisa, L.go B. Pontecorvo 5, I-56127 Pisa, Italy
3. 

Dip. Matematica, Seconda Università di Napoli, V. Vivaldi 43, I-81100 Caserta, Italy

Received  August 2009 Revised  December 2009 Published  May 2010

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.
Keywords: capacity, Perimeter, indecomposable sets..
Mathematics Subject Classification: 49J40, 49Q2.
Citation: Matteo Focardi, Maria Stella Gelli, Giovanni Pisante. On a 1-capacitary type problem in the plane. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1319-1333. doi: 10.3934/cpaa.2010.9.1319
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