Generic bifurcation in the obstacle problem
Author:
John Mallet-Paret
Journal:
Quart. Appl. Math. 37 (1980), 355-387
MSC:
Primary 58E07; Secondary 35R35
DOI:
https://doi.org/10.1090/qam/564729
MathSciNet review:
564729
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Abstract: We consider a class of singularities, locally of the form ${y^2} = p\left ( x \right )$ near the origin in ${R^2}$, describing the shape of a free boundary curve arising from an elliptic free boundary value problem. The point of view taken is that of generic bifurcation, in particular with more than one parameter present. Of prime interest is a description of the unfoldings of such singularities, their normal forms, and generic conditions for one- and two-parameter unfoldings. The two simplest cases corresponding to perturbations of singularities ${y^2} = {x^n} + O\left ( {{x^{n + 1}}} \right ),n = 4, 5$ are treated in greater detail and the bifurcation diagram for a generic two-parameter unfolding is given. Our results do not rigorously concern the free boundary problem itself, but rather set down a formal framework, or model, for studying this problem in terms of bifurcation theory. We prove theorems describing this model. Nevertheless, our results have a bearing on any rigorous analysis of this problem since they form the necessary first step to such an analysis. The theory for computing the normal forms of solutions up to first order, for example, is given here.
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H. Brézis and D. Kinderlehrer, The smoothness of solutions to non-linear variational inequalities, Indiana Univ. Math. J. 23, 831–844 (1974)
L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139, 155–184 (1977)
L. A. Caffarelli and N. M. Riviére, Smoothness and analyticity of free boundaries in variational inequalities, Ann. Scuola Norm. Sup. Pisa, Ser. IV 3, 289–310 (1976)
S.-N. Chow, J. K. Hale and J. Mallet-Paret, Applications of generic bifurcation. I, Arch. Rat. Mech. Anal. 59, 159–188 (1975)
S.-N. Chow, J. K. Hale and J. Mallet-Paret, Applications of generic bifurcation. II, Arch. Rat. Mech. Anal. 62, 209–235 (1976)
S.-N. Chow and J. Mallet-Paret, The parameterized obstacle problem, submitted for publication.
J. K. Hale, Bifurcation with several parameters, in VII. Internationale Konferenz über nichtlineare Schwingungen, I, 1, pp. 321–332, Akademie-Verlag, Berlin 1977
J. K. Hale, Restricted generic bifurcation, in Nonlinear analysis, pp. 83–98, Academic Press, 1978
D. Kinderlehrer, The free boundary determined by the solution to a differential equation, Indiana Univ. Math. J. 25, 195–208 (1976)
D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems, Ann Scuola Norm. Sup. Pisa, Ser. IV4, 373–391 (1977)
H. Lewy and G. Stampacchia, On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math. 22, 153–188 (1969)
J. Mallet-Paret, Buckling of cylindrical shells with small curvature, Quart. Appl. Math. 35, 383–400 (1977)
D. G. Schaeffer, A stability theorem for the obstacle problem, Adv. Math. 17, 34–47 (1975)
D. G. Schaeffer, Some examples of singularities in a free boundary, Ann. Scuola Norm. Sup. Pisa, Ser. IV, 4, 133–144 (1977)
D. G. Schaeffer, One-sided estimates for the curvature of the free boundary in the obstacle problem, Adv. Math. 24, 78–98 (1977)
M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math. 23, 21–98 (1979)
J. Mallet-Paret, Generic unfoldings and normal forms of some singularities arising in the obstacle problem, Duke Math, J. 46, 645–683 (1979)
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