On the Green's function for the biharmonic equation in an infinite wedge
Author:
Joseph B. Seif
Journal:
Trans. Amer. Math. Soc. 182 (1973), 241260
MSC:
Primary 31B30
MathSciNet review:
0325989
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The Green's function for the biharmonic equation in an infinite angular wedge is considered. The main result is that if the angle a is less than , then the Green's function does not remain positive; in fact it oscillates an infinite number of times near zero and near . The method uses a number of transformations of the problem including the Fourier transform. The inversion of the Fourier transform is accomplished by means of the calculus of residues and depends on the zeros of a certain transcendental function. The distribution of these zeros in the complex plane gives rise to the determination of the angle . A general expression for the asymptotic behavior of the solution near zero and near infinity is obtained. This result has the physical interpretation that if a thin elastic plate is deflected downward at a point, the resulting shape taken by the plate will have ripples which protrude above the initial plane of the plate.
 [1]
Louis
Bauer and Edward
L. Reiss, Block five diagonal matrices and the
fast numerical solution of the biharmonic equation, Math. Comp. 26 (1972), 311–326. MR 0312751
(47 #1307), http://dx.doi.org/10.1090/S00255718197203127519
 [2]
R.
J. Duffin, The maximum principle and biharmonic functions, J.
Math. Anal. Appl. 3 (1961), 399–405. MR 0144069
(26 #1617)
 [3]
R. J. Duffin and D. H. Shaffer, On the modes of vibration of a ringshaped plate, Bull. Amer. Math. Soc. 58 (1952), 652.
 [4]
P.
R. Garabedian, A partial differential equation arising in conformal
mapping, Pacific J. Math. 1 (1951), 485–524. MR 0046440
(13,735a)
 [5]
J. Hadamard, Sur un problème d'analyse rélatif a l'équilibre des plaquea élastiques encastrées, Mém. Acad. Sci. 33 (1908), no. 4.
 [6]
V.
A. Kondrat′ev, Boundaryvalue problems for elliptic equations
in conical regions, Dokl. Akad. Nauk SSSR 153 (1963),
27–29 (Russian). MR 0158157
(28 #1383)
 [7]
Stanley
Osher, On Green’s function for the biharmonic equation in a
right angle wedge, J. Math. Anal. Appl. 43 (1973),
705–716. MR 0324209
(48 #2561)
 [8]
G.
Szegö, On membranes and plates, Proc. Nat. Acad. Sci. U.
S. A. 36 (1950), 210–216. MR 0035629
(11,757g)
 [1]
 L. Bauer and E. Reiss, Block five diagonal metrics and the fast numerical computation of the biharmonic equation, Math. Comp. 26 (1972), 311326. MR 0312751 (47:1307)
 [2]
 R. J. Duffin, The maximum principle and biharmonic functions, J. Math. Anal. Appl. 3 (1961), 399405. MR 26 #1617. MR 0144069 (26:1617)
 [3]
 R. J. Duffin and D. H. Shaffer, On the modes of vibration of a ringshaped plate, Bull. Amer. Math. Soc. 58 (1952), 652.
 [4]
 P. R. Garabedian, A partial differential equation arising in conformal mapping, Pacific J. Math. 1 (1951), 485524. MR 13, 735. MR 0046440 (13:735a)
 [5]
 J. Hadamard, Sur un problème d'analyse rélatif a l'équilibre des plaquea élastiques encastrées, Mém. Acad. Sci. 33 (1908), no. 4.
 [6]
 V. A. Kondrat'ev, Boundaryvalue problems for elliptic equations in conical regions, Dokl. Akad. Nauk SSSR 153 (1963), 2729 = Soviet Math. Dokl. 4 (1963), 16001602. MR 28 #1383. MR 0158157 (28:1383)
 [7]
 S. Osher, On the Green's function for the biharmonic equation in a right angle wedge, J. Math. Anal. Appl. (to appear). MR 0324209 (48:2561)
 [8]
 G. Szegö, On membranes and plates, Proc. Nat. Acad. Sci. U. S. A. 36 (1950), 210216. MR 11, 757. MR 0035629 (11:757g)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
31B30
Retrieve articles in all journals
with MSC:
31B30
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197303259899
PII:
S 00029947(1973)03259899
Keywords:
Biharmonic equation,
Green's function,
asymptotic expansion
Article copyright:
© Copyright 1973
American Mathematical Society
