Boundary value problems for degenerate von Kármán equations
Author:
Robert G. Root
Journal:
Quart. Appl. Math. 57 (1999), 1-17
MSC:
Primary 74K20; Secondary 35Q72, 74B20, 74G20
DOI:
https://doi.org/10.1090/qam/1672163
MathSciNet review:
MR1672163
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Abstract: This article presents regularity results that admit a weak formulation for degenerate von Kármán boundary value problems modeling the deformation of clamped plates that lose stiffness in one direction. These boundary value problems are derived in the companion article, A Derivation of Degenerate von Kármán Equations for Strongly Anisotropic Plates, by the author. The equations are a fourth-order elliptic-parabolic system of weakly coupled nonlinear equations. The article includes the weak formulation and a brief description of the appropriate existence results for the formulation.
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M. S. Berger, von Kármán’s equations and the buckling of a thin elastic plate, I; The Clamped Plate, Comm. Pure Appl. Math. 20, 687–719 (1967)
M. L. Benevento, T. Bruno, and L. Castellano, Existence and uniqueness of generalized solutions to a Dirichlet problem for a class of fourth order elliptic-parabolic operators which degenerate in one or more directions, Richerche Mat. 25, 81–100 (1976) (Italian)
W. Bertiger and C. Cosner, Systems of second order equations with nonnegative characteristic form, Comm. Partial Differential Equations (7) 4, 701–737 (1979)
M. S. Berger and P. C. Fife, von Kármán’s equations and the buckling of a thin elastic plate, II; Plate with General Edge Conditions, Comm. Pure Appl. Math. 21, 227–241 (1968)
A. Canfora, Existence and uniqueness of solutions to a boundary value problem for an elliptic-parabolic equation of order 2m, Richerche Mat. 25, 246–304 (1976) (Italian)
C. Cosner, A maximum principle for weakly coupled systems of second order partial differential equations with nonnegative characteristic form, Rocky Mountain Journal of Mathematics (1) 11, 59–74 (1981)
V. Esposito, Some fourth order degenerate elliptic-parabolic equations, Rend. Accad. Sci. Fis. Mat. Napoli (4) 51, 117–128 (1984) (Italian)
V. Esposito, An existence and uniqueness theorem for some elliptic-parabolic equations of the fourth order, Rend. Accad. Sci. Fis. Mat. Napoli (4) 51, No. 2, 41–55 (1985) (Italian)
A. Friedman, Partial Differential Equations, Kreiger, Malabar, Fla., 1976
A. Kufner, O. John, and S. Fučǐk, Function Spaces, Noordhoff International, Leyden, 1977
P. Grisvard, Elliptic Problems on Nonsmooth Domains, Pitman, Boston, 1985
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, New York, 1973
A. V. Ivanov, Quasilinear Degenerate and Nonuniformly Elliptic and Parabolic Equations of the Second Order, Proceedings of the Steklov Institute of Mathematics; 1984, issue 1, American Mathematical Society, Providence, RI, 1984
J. J. Kohn and L. Nirenberg, Noncoercive boundary value problems, Comm. Pure Appl. Math. 18, 443–492 (1965)
J. J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of second order, Comm. Pure Appl. Math. 20, 797–872 (1967)
C.-S. Lin and K. Tso, On regular solutions of second-order degenerate elliptic-parabolic equations, Comm. Partial Differential Equations (9) 15, 1329–1360 (1990)
O. A. Oleĭnik, On the smoothness of solutions of degenerate elliptic and parabolic equations, Soviet Math. Dokl. 6, 972–975 (1965)
O. A. Oleĭnik and E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form, Plenum Press, New York, 1973
R. S. Philips and L. Sarason, Singular symmetric positive first order differential operators, J. Math. Mech. 15, 235–271 (1966)
R. G. Root, Boundary value problems for degenerate elliptic-parabolic fourth order equations, University of Delaware, doctoral dissertation, 1988
R. G. Root, Boundary value problems for degenerate elliptic-parabolic equations of the fourth order, Trans. Amer. Math. Soc. (1) 324, 109–134 (1991)
R. G. Root, Existence theory of higher order elliptic-parabolic equations with an application to elasticity, Journal of Mathematical Analysis and its Applications (1) 54 (1991)
R. G. Root, A derivation of degenerate von Kármán equations for strongly anisotropic plates, Quart. Appl. Math. 57, 19–36 (1999)
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1970
G. Talenti, Best constant in Sobolev inequality, Ann. Math. Pura Appl. 110, 353–372 (1976)
D. Tartakoff, On the regularity of nonunique solutions of degenerate elliptic-parabolic systems of partial differential equations, Comm. Pure Appl. Math. 24, 763–788 (1971)
G. Warnecke, On the homogeneous Dirichlet problem for nonlinear partial differential equations of Boussinesq type, Math. Meth. in the Appl. Sci. 9, 493–519 (1987)
R. J. Weinacht, Asymptotic distribution of eigenvalues for a class of degenerate elliptic operators of the fourth order, Rend. Mat. (7) 6, No. 1–2, 159–170 (1986)
R. J. Weinacht, Degenerate elliptic equations and spongy elastic plates, in Methoden und Verfahren der mathematischen Physik, Problems of Applied Analysis, Vol. 33, pp. 59–74, Lang, Frankfurt am Main, 1987
E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. II/B, Springer, New York, 1990
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© Copyright 1999
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