Generic bifurcation in the obstacle problem

Mallet-Paret, John

doi:10.1090/qam/564729

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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