On a 1-capacitary type problem in the plane

Matteo Focardi; Maria Stella Gelli; Giovanni Pisante

doi:10.3934/cpaa.2010.9.1319

Article Contents

2010, Volume 9, Issue 5: 1319-1333. Doi: 10.3934/cpaa.2010.9.1319

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On a 1-capacitary type problem in the plane

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
3.
Dip. Matematica, Seconda Università di Napoli, V. Vivaldi 43, I-81100 Caserta

Received: July 31, 2009

Revised: November 30, 2009

Published: August 2010

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Abstract

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.

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Mathematics Subject Classification: 49J40, 49Q20.

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