Discontinuous solutions for a non-parametric variational problem
William T. Reid · Applicable Analysis · 1971
For a fixed end point problem in (n+1)-space with integrand function there are discussed certain necessary conditions that are satisfied by a minimizing arc having one or more corners. Firstly, there is clarified the type of minimum for which the continuity of the function across corners is a necessary condition, since various authors have erroneously stated that this is a necessary condition for a weak relative minimum. There are discussed two methods by which one may establish the necessity of the continuity of this function in the case of a strong relative minimum. The more comprehensive method is that employed many years ago by the author in the study of discontinuous solutions for the non-parametric problem of Mayer. For the problem herein considered there is presented the second order condition involving the non-negativeness of the second variation along a non-singular extremaloid on the class of so-called generalized admissible variations vanishing at the end-values, including a discussion of conjugate points in the extended sense as introduced in the earlier paper on discontinuous solutions for the Mayer problem.