Positivity cases, estimates and asymptotic expansions for condenser capacities

Alain R. Bonnafé (1, 2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 INSA Toulouse - Institut National des Sciences Appliquées - Toulouse

We study positivity cases, estimates and asymptotic expansions of condenser p-capacities of points and segments in a bounded domain, having in mind application fields, such as imaging, requiring detection and quanti fication of zero measure sets. As preliminary tools, we provide estimates of capacities when the internal part of the condenser has a non-empty interior. The study of the point and its approximation follow. Our main contribution is then to introduce equidistant condensers and to establish the positivity cases for d-dimensional condenser capacities of segments in a new way bringing out the relationship with the d-dimensional and more significantly with the (d-1)-dimensional condenser capacities of points. We discuss how equidistant condensers might allow to obtain by induction the positivity cases for compact submanifolds of higher dimensions. For estimation purposes, we then introduce elliptical condensers and in particular provide an estimate and the asymptotic expansion for the condenser capacity of a segment in the harmonic case p = 2.

Mots clés: Elliptical condenser; p-Laplace equation; Nonlinear potential theory; Sets of zero measure; Asymptotic analysis; Topological gradient; Imaging.; Imaging; Condenser p- capacity; Equidistant condenser

Domaines: Equations aux dérivées partielles [math.AP]

Alain R. Bonnafé. Positivity cases, estimates and asymptotic expansions for condenser capacities. 2012. ⟨hal-00702308v2⟩