On the Green’s function for the biharmonic equation in an infinite wedge
HTML articles powered by AMS MathViewer
- by Joseph B. Seif PDF
- Trans. Amer. Math. Soc. 182 (1973), 241-260 Request permission
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 ${a_1} \cong 0.812\pi$, then the Green’s function does not remain positive; in fact it oscillates an infinite number of times near zero and near $\infty$. 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 ${a_1}$. 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.References
- 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 312751, DOI 10.1090/S0025-5718-1972-0312751-9
- R. J. Duffin, The maximum principle and biharmonic functions, J. Math. Anal. Appl. 3 (1961), 399–405. MR 144069, DOI 10.1016/0022-247X(61)90066-X R. J. Duffin and D. H. Shaffer, On the modes of vibration of a ring-shaped plate, Bull. Amer. Math. Soc. 58 (1952), 652.
- P. R. Garabedian, A partial differential equation arising in conformal mapping, Pacific J. Math. 1 (1951), 485–524. MR 46440, DOI 10.2140/pjm.1951.1.485 J. Hadamard, Sur un problème d’analyse rélatif a l’équilibre des plaquea élastiques encastrées, Mém. Acad. Sci. 33 (1908), no. 4.
- V. A. Kondrat′ev, Boundary-value problems for elliptic equations in conical regions, Dokl. Akad. Nauk SSSR 153 (1963), 27–29 (Russian). MR 0158157
- 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, DOI 10.1016/0022-247X(73)90286-2
- G. Szegö, On membranes and plates, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 210–216. MR 35629, DOI 10.1073/pnas.36.3.210
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 182 (1973), 241-260
- MSC: Primary 31B30
- DOI: https://doi.org/10.1090/S0002-9947-1973-0325989-9
- MathSciNet review: 0325989