Regions of positivity for polyharmonic Green functions in arbitrary domains

Grunau, Hans-Christoph; Sweers, Guido

doi:10.1090/S0002-9939-07-08851-X

Regions of positivity for polyharmonic Green functions in arbitrary domains

Authors: Hans-Christoph Grunau and Guido Sweers
Journal: Proc. Amer. Math. Soc. 135 (2007), 3537-3546
MSC (2000): Primary 35J65, 35B50, 35J40
DOI: https://doi.org/10.1090/S0002-9939-07-08851-X
Published electronically: July 3, 2007
MathSciNet review: 2336568
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Abstract: The Green function for the biharmonic operator on bounded domains with zero Dirichlet boundary conditions is in general not of fixed sign. However, by extending an idea of Z. Nehari, we are able to identify regions of positivity for Green functions of polyharmonic operators. In particular, the biharmonic Green function is considered in all space dimensions. As a consequence we see that the negative part of any such Green function is somehow small compared with the singular positive part.

[ADN] 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
[B] T. Boggio, Sulle funzioni di Green d'ordine m, Rend. Circ. Mat. Palermo 20, 97-135 (1905).
[C] 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, https://doi.org/10.1137/0513051
[CD] C. V. Coffman and R. J. Duffin, On the structure of biharmonic functions satisfying the clamped plate conditions on a right angle, Adv. in Appl. Math. 1 (1980), no. 4, 373–389. MR 603137, https://doi.org/10.1016/0196-8858(80)90018-4
[CG] Charles V. Coffman and Carole L. Grover, Obtuse cones in Hilbert spaces and applications to partial differential equations, J. Functional Analysis 35 (1980), no. 3, 369–396. MR 563561, https://doi.org/10.1016/0022-1236(80)90088-9
[DS1] 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
[DS2] 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
[DMS] 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
[D] R. J. Duffin, On a question of Hadamard concerning super-biharmonic functions, J. Math. Physics 27 (1949), 253–258. MR 0029021
[Ga] P. R. Garabedian, Partial differential equations, 2nd ed., Chelsea Publishing Co., New York, 1986. MR 943117
[GS1] 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
[GS2] Hans-Christoph Grunau and Guido Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann. 307 (1997), no. 4, 589–626. MR 1464133, https://doi.org/10.1007/s002080050052
[Ha1] 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
[Ha2] 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
[K] Ju. P. Krasovskiĭ, Isolation of the singularity in Green’s function, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 977–1010 (Russian). MR 0223740
[M] V. A. Malyshev, The Hadamard conjecture and estimates for the Green function, Algebra i Analiz 4 (1992), no. 4, 1–44 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 4, 633–666. MR 1190781
[N] 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
[O] Stanley Osher, On Green’s function for the biharmonic equation in a right angle wedge, J. Math. Anal. Appl. 43 (1973), 705–716. MR 324209, https://doi.org/10.1016/0022-247X(73)90286-2
[S] E. Sassone, Positivity for polyharmonic problems on domains close to a disk, Ann. Mat. Pura Appl. 186, 419-432 (2007).
[ST] Harold S. Shapiro and Max Tegmark, An elementary proof that the biharmonic Green function of an eccentric ellipse changes sign, SIAM Rev. 36 (1994), no. 1, 99–101. MR 1267051, https://doi.org/10.1137/1036005