Optimal estimates from below for biharmonic Green functions

Grunau, Hans-Christoph; Robert, Frédéric; Sweers, Guido

doi:10.1090/S0002-9939-2010-10740-2

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
DOI: https://doi.org/10.1090/S0002-9939-2010-10740-2
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.

1. S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, https://doi.org/10.1002/cpa.3160120405
2. T. Boggio, Sulle funzioni di Green d'ordine , Rend. Circ. Mat. Palermo 20, 97-135 (1905).
3. Kai Lai Chung and Zhong Xin Zhao, From Brownian motion to Schrödinger’s equation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 312, Springer-Verlag, Berlin, 1995. MR 1329992
4. Anna Dall’Acqua, Christian Meister, and Guido Sweers, Separating positivity and regularity for fourth order Dirichlet problems in 2d-domains, Analysis (Munich) 25 (2005), no. 3, 205–261. MR 2232852, https://doi.org/10.1524/anly.2005.25.3.205
5. Anna Dall’Acqua and Guido Sweers, Estimates for Green function and Poisson kernels of higher-order Dirichlet boundary value problems, J. Differential Equations 205 (2004), no. 2, 466–487. MR 2092867, https://doi.org/10.1016/j.jde.2004.06.004
6. A. Dall’Acqua and G. Sweers, On domains for which the clamped plate system is positivity preserving, Partial differential equations and inverse problems, Contemp. Math., vol. 362, Amer. Math. Soc., Providence, RI, 2004, pp. 133–144. MR 2091495, https://doi.org/10.1090/conm/362/06609
7. Anna Dall’Acqua and Guido Sweers, The clamped-plate equation for the limaçon, Ann. Mat. Pura Appl. (4) 184 (2005), no. 3, 361–374. MR 2164263, https://doi.org/10.1007/s10231-004-0121-9
8. R. J. Duffin, Continuation of biharmonic functions by reflection, Duke Math. J. 22 (1955), 313–324. MR 79105
9. F. Gazzola, H.-Ch. Grunau, G. Sweers, Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains. Springer Lecture Notes in Mathematics 1991, Springer-Verlag, Berlin, 2010.
10. Hans-Christoph Grunau and Frédéric Robert, Positivity and almost positivity of biharmonic Green’s functions under Dirichlet boundary conditions, Arch. Ration. Mech. Anal. 195 (2010), no. 3, 865–898. MR 2591975, https://doi.org/10.1007/s00205-009-0230-0
11. Hans-Christoph Grunau and Guido Sweers, Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions, Math. Nachr. 179 (1996), 89–102. MR 1389451, https://doi.org/10.1002/mana.19961790106
12. Hans-Christoph Grunau and Guido Sweers, Regions of positivity for polyharmonic Green functions in arbitrary domains, Proc. Amer. Math. Soc. 135 (2007), no. 11, 3537–3546. MR 2336568, https://doi.org/10.1090/S0002-9939-07-08851-X
13. Jacques Hadamard, Œuvres de Jacques Hadamard. Tomes I, II, III, IV, Comité de publication des oeuvres de Jacques Hadamard: M. Fréchet, P. Levy, S. Mandelbrojt, L. Schwartz, Éditions du Centre National de la Recherche Scientifique, Paris, 1968 (French). MR 0230598
14. Alfred Huber, On the reflection principle for polyharmonic functions, Comm. Pure Appl. Math. 9 (1956), 471–478. MR 85355, https://doi.org/10.1002/cpa.3160090318
15. Ju. P. Krasovskiĭ, Investigation of potentials connected with boundary value problems for elliptic equations, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 587–640 (Russian). MR 0213727
16. Ju. P. Krasovskiĭ, Isolation of the singularity in Green’s function, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 977–1010 (Russian). MR 0223740
17. Zeev Nehari, On the biharmonic Green’s function, Studies in mathematics and mechanics presented to Richard von Mises, Academic Press Inc., New York, 1954, pp. 111–117. MR 0064992