Effilement minimal en une singularité isolée de l’équation de Schrödinger et application au principe de Picard

Abstract

In this paper, the generalized Schrödinger equation (Δ – µ )u = 0 on the punctured unit disk Ω of ℝ² is investigated. If µ is rotation free and satisfies the Picard principle at the origin, it is shown that if a set E ⊂ Ω is minimal thin relatively to an extremal harmonic function h _µ with zero boundary values at $\{ |x| = 1\}$, there exists a sequence (r _n ) converging to zero such that ∂B(0,r _n ) ⊂ C E. Let e _µ be the µ-unit. It is proved that if a measure v satisfies ${{\smallint }_{{\Omega \backslash E}}}{{e}_{\mu }}{{h}_{\mu }}dv < \infty$, for a miniman thin, relatively to h _µ , set E then the Picard principle is valid for the measure µ + v.