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 Δω-1Δ, for weights ω with logω 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 Δω -1 Δ on the unit disk is positive.

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

[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. | MR | Zbl

[3] J. Hadamard, OEuvres de Jacques Hadamard, Vols. 1-4, Editions du Centre National de la Recherche Scientifique, Paris, 1968. | Zbl

[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. | MR | Zbl

[5] H. Hedenmalm, S. Jakobsson, S. Shimorin, An Hadamard maximum principle for biharmonic operators, submitted.

[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. | MR | Zbl