We consider a variational problem involving competition between surface tension and charge repulsion. We show that, as opposed to the case of weak (short-range) interactions where we proved ill-posedness of the problem in a previous paper, when the repulsion is stronger the perimeter dominates the capacitary term at small scales. In particular we prove existence of minimizers for small charges as well as their regularity. Combining this with the stability of the ball under small C^{1,\gamma} perturbations, this ultimately leads to the minimality of the ball for small charges. We cover in particular the borderline case of the capacity where both terms in the energy are of the same order.

Michael Goldman, M.N. (In stampa/Attività in corso). Rigidity of the ball for an isoperimetric problem with strong capacitary repulsion. JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, First on line, 1-41 [10.4171/JEMS/1451].

Rigidity of the ball for an isoperimetric problem with strong capacitary repulsion

Berardo Ruffini
In corso di stampa

Abstract

We consider a variational problem involving competition between surface tension and charge repulsion. We show that, as opposed to the case of weak (short-range) interactions where we proved ill-posedness of the problem in a previous paper, when the repulsion is stronger the perimeter dominates the capacitary term at small scales. In particular we prove existence of minimizers for small charges as well as their regularity. Combining this with the stability of the ball under small C^{1,\gamma} perturbations, this ultimately leads to the minimality of the ball for small charges. We cover in particular the borderline case of the capacity where both terms in the energy are of the same order.
In corso di stampa
Michael Goldman, M.N. (In stampa/Attività in corso). Rigidity of the ball for an isoperimetric problem with strong capacitary repulsion. JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, First on line, 1-41 [10.4171/JEMS/1451].
Michael Goldman, Matteo Novaga, Berardo Ruffini
File in questo prodotto:
File Dimensione Formato  
gnr4_final (1).pdf

accesso aperto

Tipo: Postprint
Licenza: Licenza per accesso libero gratuito
Dimensione 472.17 kB
Formato Adobe PDF
472.17 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/963921
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact