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

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.


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