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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Folds and cusps in Banach spaces with applications to nonlinear partial differential equations. II


Authors: M. S. Berger, P. T. Church and J. G. Timourian
Journal: Trans. Amer. Math. Soc. 307 (1988), 225-244
MSC: Primary 35J65; Secondary 47H15, 58C27
MathSciNet review: 936814
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Abstract: Earlier the authors have given abstract properties characterizing the fold and cusp maps on Banach spaces, and these results are applied here to the study of specific nonlinear elliptic boundary value problems. Functional analysis methods are used, specifically, weak solutions in Sobolev spaces. One problem studied is the inhomogeneous nonlinear Dirichlet problem

$\displaystyle \Delta u + \lambda u - {u^3} = g\quad {\text{on}}\;\Omega ,\qquad u\vert\partial \Omega = 0,$

where $ \Omega \subset {{\mathbf{R}}^n}(n \leqslant 4)$ is a bounded domain. Another is a nonlinear elliptic system, the von Kármán equations for the buckling of a thin planar elastic plate when compressive forces are applied to its edge.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0936814-8
PII: S 0002-9947(1988)0936814-8
Keywords: Nonlinear partial differential equations or systems, elliptic boundary value problem, nonlinear Dirichlet problem, fold map, cusp map, von Kármán equations, bifurcation, singularity theory in infinite dimensions
Article copyright: © Copyright 1988 American Mathematical Society