# An Hadamard maximum principle for the biplacian on hyperbolic manifolds

Journées équations aux dérivées partielles (1999)

- page 1-5
- ISSN: 0752-0360

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topHedenmalm, 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

top- [1] M. Engliš, A weighted biharmonic Green function, Glasgow Math. J., to appear.
- [2] P. R. Garabedian, A partial differential equation arising in conformal mapping, Pacific J. Math. 1 (1951), 485-524. Zbl0045.05102MR13,735a
- [3] J. Hadamard, OEuvres de Jacques Hadamard, Vols. 1-4, Editions du Centre National de la Recherche Scientifique, Paris, 1968. Zbl0168.24101
- [4] H. Hedenmalm, A computation of the Green function for the weighted biharmonic operators Δ|z|-2Δ, with ˃ -1, Duke Math. J. 75 (1994), 51-78. Zbl0813.31001MR95k:31005
- [5] H. Hedenmalm, S. Jakobsson, S. Shimorin, An Hadamard maximum principle for biharmonic operators, submitted. Zbl0931.31002
- [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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