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The interval of resolvent-positivity
for the biharmonic operator


Author: Michael Ulm
Journal: Proc. Amer. Math. Soc. 127 (1999), 481-489
MSC (1991): Primary 34L40, 35B50, 47A10
MathSciNet review: 1468205
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Abstract: For an operator $A$ on a Banach lattice we examine the interval on the real line for which the resolvent $\left( \lambda - A \right)^{-1}$ is positive. This positivity interval is then explicitly calculated for the biharmonic operator $A f = - f''''$ with three different boundary conditions.


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  • 1. Wolfgang Arendt, Charles J. K. Batty, and Derek W. Robinson, Positive semigroups generated by elliptic operators on Lie groups, J. Operator Theory 23 (1990), no. 2, 369–407. MR 1066813
  • 2. Charles V. Coffman, On the structure of solutions Δ²𝑢=𝜆𝑢 which satisfy the clamped plate conditions on a right angle, SIAM J. Math. Anal. 13 (1982), no. 5, 746–757. MR 668318, 10.1137/0513051
  • 3. Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 2, Springer-Verlag, Berlin, 1988. Functional and variational methods; With the collaboration of Michel Artola, Marc Authier, Philippe Bénilan, Michel Cessenat, Jean Michel Combes, Hélène Lanchon, Bertrand Mercier, Claude Wild and Claude Zuily; Translated from the French by Ian N. Sneddon. MR 969367
  • 4. Günther Greiner, Jürgen Voigt, and Manfred Wolff, On the spectral bound of the generator of semigroups of positive operators, J. Operator Theory 5 (1981), no. 2, 245–256. MR 617977
  • 5. H.-C. Grunau and G. Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, TWI report 95-56, TU Delft, 1995; Math. Ann. 307 (1997), 589-626. CMP 97:16
  • 6. V. A. Kozlov, V. A. Kondrat′ev, and V. G. Maz′ya, On sign variability and the absence of “strong” zeros of solutions of elliptic equations, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 2, 328–344 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 2, 337–353. MR 998299

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Additional Information

Michael Ulm
Affiliation: Abteilung Mathematik V, Universität Ulm, D-89069 Ulm, Germany
Email: ulm@mathematik.uni-ulm.de

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04556-6
Keywords: Biharmonic operator, resolvent-positivity
Received by editor(s): April 19, 1996
Received by editor(s) in revised form: May 22, 1997
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society