Optimal estimates from below for biharmonic Green functions
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- by Hans-Christoph Grunau, Frédéric Robert and Guido Sweers PDF
- Proc. Amer. Math. Soc. 139 (2011), 2151-2161 Request permission
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.References
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Additional Information
- 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
- Received by editor(s): June 11, 2010
- Published electronically: November 29, 2010
- Communicated by: Walter Craig
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2151-2161
- MSC (2010): Primary 35B51; Secondary 35J40, 35A08
- DOI: https://doi.org/10.1090/S0002-9939-2010-10740-2
- MathSciNet review: 2775393