Vol. 30, No. 3, 1969

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Conjugate surfaces for multiple integral problems in the calculus of variations

Rene Felix Dennemeyer

Vol. 30 (1969), No. 3, 621–638

The Jacobi equation of the second variation for a multiple integral problem in the calculus of variations is a linear second order elliptic type partial differential equation provided certain hypotheses hold in the multiple integral problem. By means of the theory of quadratic forms in Hilbert space already present in the literature pertinent properties of solutions of such partial differential equations can be established. Here the pertinent property discussed is the vanishing of a solution on the boundary of a region, i.e. the existence of a conjugate surface of the differential equation. After developing the notion of focal point and stating the index theorems of the associated quadratic form, the existence of one parameter families of conjugate surfaces is shown, and illustrations of the theory are given.

Mathematical Subject Classification
Primary: 49.10
Received: 26 August 1968
Revised: 4 December 1968
Published: 1 September 1969
Rene Felix Dennemeyer