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 setE ⊂ Ω is minimal thin relatively to an extremal harmonic functionh μ with zero boundary values at {|x|=1}, there exists a sequence (r n ) converging to zero such that ∂B(O,r n ) ⊂C E. Lete μ be the μ-unit. It is proved that if a measure ν satisfies ∫Ω\E e μ h μdν<∞, for a minimal thin, relatively toh μ, setE then the Picard principle is valid for the measure μ + ν.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bliedtner, J. and Hansen, W.:Potential Theory. An Analytic and Probabilistic Approach to Balayage, Universitext, Springer-Verlag, 1986.
Boukricha, A. and Hueber, H.: ‘The Poisson Spacec P x for Δu =cu with Rotation Freec’,Acad. Roy. Belg. Bull. Cl. Sci. (5)LXIV(10), (1978), 651–658.
Boukricha, A.: ‘Das Picard-Prinzip und verwandte Fragen bei Störung von harmonischen Räumen’,Math. Ann. 239 (1979), 247–270.
Boukricha, A.: Principe de Picard et comportement des solutions continues de l'équation de Schrödinger, au voisinage d'une singularité isolée, BiBos preprint 247, University of Bielefeld (1987).
Brelot, M.: ‘Etude de l'équation de la chaleur Δu=c(M)u(M),c(M)≥0, au voisinage d'un point singulier du coefficient’,Ann. Sci. Ecole Norm. Sup. 48 (1931), 153–246.
Brelot, M.:On Topoligies and Boundaries in Potential Theory, Lecture Notes in Math. 175, Springer-Verlag, 1971.
Constaninescu, C. and Cornea, A.:Potential Theory on Harmonic Spaces, Springer-Verlag, 1972.
Helms, L.:Introduction to Potential Theory, Wiley-Interscience, 1969.
Jassenn, K.: ‘On the Existence of a Green Function for Harmonic Spaces’,Math. Ann. 208 (1974), 295–303.
Nakai, M.: ‘A Test for Picard Principle’,Nagoya Math. J. 56 (1974), 105–119.
Nakai, M.: ‘A Test for Picard Principle for Rotation Free Densities’,J. Math. Soc. Japan 27 (1975), 412–431.
Nakai, M.: ‘Picard Principle for Finite Densities’,Nagoya Math. J. 70 (1978), 7–24.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Haouala, E. Effilement minimal en une singularité isolée de l'équation de Schrödinger et application au principe de Picard. Potential Anal 3, 133–143 (1994). https://doi.org/10.1007/BF01047840
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01047840