Communications on Pure and Applied Analysis (CPAA)

On a 1-capacitary type problem in the plane

Pages: 1319 - 1333, Volume 9, Issue 5, September 2010      doi:10.3934/cpaa.2010.9.1319

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Matteo Focardi - Dipartimento di Matematica "U. Dini", Università di Firenze, viale Morgagni 67/A, I-50139 Firenze, Italy (email)
Maria Stella Gelli - Dip. Mat. “L. Tonelli”, Università di Pisa, L.go B. Pontecorvo 5, I-56127 Pisa, Italy (email)
Giovanni Pisante - Dip. Matematica, Seconda Università di Napoli, V. Vivaldi 43, I-81100 Caserta, Italy (email)

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.

Keywords:   Perimeter, capacity, indecomposable sets.
Mathematics Subject Classification:  49J40, 49Q20.

Received: August 2009;      Revised: December 2009;      Available Online: May 2010.