Sign change for the green function and for the first eigenfunction of equations of clamped-plate type

Archive for Rational Mechanics and Analysis

Volumn 150, Issue 2, 1999, Pages 179-190

  Grunau, Hans Christoph ^a   Sweers, Guido ^b  

^a UNIVERSITY OF BAYREUTH   (Germany)

^b DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033275620     PISSN: 00039527     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002050050185     Document Type: Article

References (28)

1. SH. AGMON, A. DOUGLIS & 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.
2. T. BOGGIO, Sull'equilibrio delle piastre elastiche incastrate. Rend. Acc. Lincei 10 (1901), 197-205.
3. T. BOGGIO, Sulle funzioni di Green d'ordine m. Rend. Circ. Mat. Palermo 20 (1905), 97-135.
4. C. V. COFFMAN & R. J. DUFFIN, On the structure of biharmonic functions satisfying the clamped plate conditions on a right angle. Adv. Appl. Math. 1 (1980), 373-389.
5. C. V. COFFMAN & C. L. GROVER, Obtuse cones in Hilbert spaces and applications to partial differential equations, J. Funct. Anal. 35 (1980), 369-396.
6. 2u = λu which satisfy the clamped plate conditions on a right angle. SIAM J. Math. Anal. 13 (1982), 746-757.
7. R. J. DUFFIN, On a question of Hadamard concerning super-biharmonic functions. J. Math. Phys. 27 (1949), 253-258.
8. P. R. GARABEDIAN, A partial differential equation arising in conformal mapping. Pacific J. Math. 1 (1951), 485-524.
9. D. GILBARG & N.S. TRUDIGER, Elliptic partial differential equations of second order. Springer-Verlag, 1983.
10. H.-CH. GRUNAU & G. SWEERS, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions. Math. Ann. 307 (1997), 589-626.
11. H.-CH. GRUNAU & G. SWEERS, Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions. Math. Nachr. 179 (1996), 89-102.
12. H.-CH. GRUNAU & G. SWEERS, The maximum principle and positive principal eigenfunctions for polyharmonic equations. In: G. CARISTI & E. MITIDIERI eds, Reaction Diffusion systems, Marcel Dekker, 1997, pp. 163-182.
13. H.-CH. GRUNAU & G. SWEERS, Positivity properties of elliptic boundary value problems of higher order. Nonlinear Analysis, T.M.A. 30 (1997), 5251-5258 (Proc. Second World Congress of Nonlinear Analysts).
14. J. HADAMARD, Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées. In Œuvres de Jaques Hadamard, Tome II, pp. 515-641, Centre National de la Recherche Scientifique: Paris, 1968, reprint of: Mémoires présentés par divers savants à l'Académie des Sciences 33 (1908), 1-128.
15. J. HADAMARD, Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées. In Œuvres de Jaques Hadamard, Tome II, pp. 515-641, Centre National de la Recherche Scientifique: Paris, 1968, reprint of: Mémoires présentés par divers savants à l'Académie des Sciences 33 (1908), 1-128.
16. J. HADAMARD, Sur certains cas intéressants du problème biharmonique. In: Œuvres de Jaques Hadamard, Tome III, pp. 1297-1299, Centre National de la Recherche Scientifique: Paris, 1968, reprint of: Atti IVe Congr. Intern. Mat. Rome (1908), 12-14.
17. J. HADAMARD, Sur certains cas intéressants du problème biharmonique. In: Œuvres de Jaques Hadamard, Tome III, pp. 1297-1299, Centre National de la Recherche Scientifique: Paris, 1968, reprint of: Atti IVe Congr. Intern. Mat. Rome (1908), 12-14.
18. P. JENTZSCH, Über Integralgleichungen mit positivem Kern. J. Reine Angew. Math. 141 (1912), 235-244.
19. V. A. KOZLOV, V. A. KONDRAT'EV & V. G. MAZ'YA, On sign variation and the absence of "strong" zeros of solutions of elliptic equations. Math. USSR Izvestiya 34 (1990), 337-353.
20. YU. P. KRASOVSKIǏ, Isolation of singularities of the Green's function. Math. USSR Izvestiya 1 (1967), 935-966.
21. M. G. KREǏN & M. A. RUTMAN, Linear operators leaving invariant a cone in a Banach space. Uspehi Mat. Nauk N. S. 3 (1948), 3-95. (Translation in: Amer. Math. Soc. Transl. (1950)).
22. M. G. KREǏN & M. A. RUTMAN, Linear operators leaving invariant a cone in a Banach space. Uspehi Mat. Nauk N. S. 3 (1948), 3-95. (Translation in: Amer. Math. Soc. Transl. (1950)).
23. M. H. PROTTER, Unique continuation for elliptic equations. Trans. Amer. Math. Soc. 95 (1960), 81-91.
24. H. H. SCHAEFER, Banach Lattices and Positive Operators. Springer, 1974.
25. J. SCHRÖDER, Operator Inequalities. Academic Press, 1980.
26. H. S. SHAPIRO, M. TEGMARK, An elementary proof that the biharmonic Green function of an eccentric ellipse changes sign. SIAM Rev. 36 (1994), 99-101.
27. J. SERRIN, private communication.
28. G. TALENTI On the first eigenvalue of the clamped plate. In: H. BREZIS & J. L. LIONS eds., Nonlinear partial differential equations and their applications, Collège de France Seminar, Vol. VI, Pitman Research Notes Math. 109 (1984), 309-313.