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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Optimal estimates from below for biharmonic Green functions


Authors: Hans-Christoph Grunau, Frédéric Robert and Guido Sweers
Journal: Proc. Amer. Math. Soc. 139 (2011), 2151-2161
MSC (2010): Primary 35B51; Secondary 35J40, 35A08
Published electronically: November 29, 2010
MathSciNet review: 2775393
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Abstract: Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary $ C^{4,\gamma}$-smooth domains. Maximum principles do not exist for fourth order elliptic equations, and the Green function may change sign. The lack of a maximum principle prevents using a Harnack inequality as for second order problems and hence complicates the derivation of optimal estimates. The present estimate is obtained by an asymptotic analysis. The estimate shows that this Green function is positive near the singularity and that a possible negative part is small in the sense that it is bounded by the product of the squared distances to the boundary.


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Hans-Christoph Grunau
Affiliation: Fakultät für Mathematik, Otto-von-Guericke-Universität, Postfach 4120, 39016 Magdeburg, Germany
Email: hans-christoph.grunau@ovgu.de

Frédéric Robert
Affiliation: Institut Élie Cartan, Université Henri Poincaré Nancy 1, BP 70239, 54506 Vandœuvre-lès-Nancy Cedex, France
Email: Frederic.Robert@iecn.u-nancy.fr

Guido Sweers
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany
Email: gsweers@math.uni-koeln.de

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10740-2
Received by editor(s): June 11, 2010
Published electronically: November 29, 2010
Communicated by: Walter Craig
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.