Restricted quadratic forms and their application to bifurcation and stability in constrained variational principles. (English) Zbl 0581.47049

Maddocks, John H.

SIAM J. Math. Anal. 16, 47-68 (1985).

The subjects of this investigation are the abstract properties and applications of restricted quadratic forms. The first part of the presentation resolves the following question: if L is a self-adjoint linear operator mapping a Hilbert space H into itself, and S is a subspace of H, when is the quadratic form \(<u,Lu>\) positive for any nonzero \(u\in S?\) In the second part of the presentation, restricted quadratic forms are further examined in the specific context of constrained variational principles; and the general theory is applied to obtain information on stability and bifurcation. Two examples are then solved: one is finite-dimensional and of an illustrative nature; the other is a longstanding problem in elasticity concerning the stability of a buckled rod. In addition to being a valuable analytical tool for isoperimetric problems in the calculus of variations, the tests described are amenable to numerical treatment.

MSC:

47J05	Equations involving nonlinear operators (general)
49J40	Variational inequalities
74H55	Stability of dynamical problems in solid mechanics
74G60	Bifurcation and buckling

Keywords:

restricted quadratic forms; stability of a buckled rod; isoperimetric problems in the calculus of variations