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.
