An Hadamard maximum principle for the biplacian on hyperbolic manifolds
Journées équations aux dérivées partielles, (1999), article no. 3, 5 p.

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.

@article{JEDP_1999____A3_0,
author = {Hedenmalm, H\aa kan},
title = {An Hadamard maximum principle for the biplacian on hyperbolic manifolds},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
publisher = {Universit\'e de Nantes},
year = {1999},
mrnumber = {1718958},
language = {en},
url = {http://www.numdam.org/item/JEDP_1999____A3_0}
}

Hedenmalm, Håkan. An Hadamard maximum principle for the biplacian on hyperbolic manifolds. Journées équations aux dérivées partielles,  (1999), article  no. 3, 5 p. http://www.numdam.org/item/JEDP_1999____A3_0/

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