An Hadamard maximum principle for the biplacian on hyperbolic manifolds

Håkan Hedenmalm

Journées équations aux dérivées partielles (1999)

  • page 1-5
  • ISSN: 0752-0360

Abstract

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We prove the existence of a maximum principle for operators of the type Δ ω - 1 Δ , for weights ω with log ω subharmonic. It is associated with certain simply connected subdomains that are produced by a Hele-Shaw flow emanating from a given point in the domain. For constant weight, these are the circular disks in the domain. The principle is equivalent to the following statement. THEOREM. Suppose ω is logarithmically subharmonic on the unit disk, and that the weight times area measure is a reproducing measure (for the harmonic functions). Then the Green function for the Dirichlet problem associated with Δ ω - 1 Δ on the unit disk is positive.

How to cite

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Hedenmalm, Håkan. "An Hadamard maximum principle for the biplacian on hyperbolic manifolds." Journées équations aux dérivées partielles (1999): 1-5. <http://eudml.org/doc/93380>.

@article{Hedenmalm1999,
abstract = {We prove the existence of a maximum principle for operators of the type $\Delta \omega \{-1\}\Delta $, for weights $\omega $ with $\log \omega $ subharmonic. It is associated with certain simply connected subdomains that are produced by a Hele-Shaw flow emanating from a given point in the domain. For constant weight, these are the circular disks in the domain. The principle is equivalent to the following statement. THEOREM. Suppose $\omega $ is logarithmically subharmonic on the unit disk, and that the weight times area measure is a reproducing measure (for the harmonic functions). Then the Green function for the Dirichlet problem associated with $\Delta \omega ^\{-1\}\Delta $ on the unit disk is positive.},
author = {Hedenmalm, Håkan},
journal = {Journées équations aux dérivées partielles},
language = {eng},
pages = {1-5},
publisher = {Université de Nantes},
title = {An Hadamard maximum principle for the biplacian on hyperbolic manifolds},
url = {http://eudml.org/doc/93380},
year = {1999},
}

TY - JOUR
AU - Hedenmalm, Håkan
TI - An Hadamard maximum principle for the biplacian on hyperbolic manifolds
JO - Journées équations aux dérivées partielles
PY - 1999
PB - Université de Nantes
SP - 1
EP - 5
AB - We prove the existence of a maximum principle for operators of the type $\Delta \omega {-1}\Delta $, for weights $\omega $ with $\log \omega $ subharmonic. It is associated with certain simply connected subdomains that are produced by a Hele-Shaw flow emanating from a given point in the domain. For constant weight, these are the circular disks in the domain. The principle is equivalent to the following statement. THEOREM. Suppose $\omega $ is logarithmically subharmonic on the unit disk, and that the weight times area measure is a reproducing measure (for the harmonic functions). Then the Green function for the Dirichlet problem associated with $\Delta \omega ^{-1}\Delta $ on the unit disk is positive.
LA - eng
UR - http://eudml.org/doc/93380
ER -

References

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  1. [1] M. Engliš, A weighted biharmonic Green function, Glasgow Math. J., to appear. 
  2. [2] P. R. Garabedian, A partial differential equation arising in conformal mapping, Pacific J. Math. 1 (1951), 485-524. Zbl0045.05102MR13,735a
  3. [3] J. Hadamard, OEuvres de Jacques Hadamard, Vols. 1-4, Editions du Centre National de la Recherche Scientifique, Paris, 1968. Zbl0168.24101
  4. [4] H. Hedenmalm, A computation of the Green function for the weighted biharmonic operators Δ|z|-2΁Δ, with ΁ ˃ -1, Duke Math. J. 75 (1994), 51-78. Zbl0813.31001MR95k:31005
  5. [5] H. Hedenmalm, S. Jakobsson, S. Shimorin, An Hadamard maximum principle for biharmonic operators, submitted. Zbl0931.31002
  6. [6] H. S. Shapiro, The Schwarz function and its generalization to higher dimensions, University of Arkansas Lecture Notes in the Mathematical Sciences, 9, Wiley-Interscience, John Wiley & Sons, Inc., New York, 1992. Zbl0784.30036MR93g:30059

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