Geodesic
Existence of discontinuous solutions for simplest semidefinite problems of variational calculus
Matematičeskie zametki, Tome 7 (1970) no. 1, pp. 69-78.

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An existence theorem is proved for discontinuous solutions of a semidefinite quasiregular variational problem concerned with determining $\inf J[y]$, in which the expression under the integral sign is of a special form. 
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     author = {V. N. Koshelev and S. F. Morozov},
     title = {Existence of discontinuous solutions for simplest semidefinite problems of variational calculus},
     journal = {Matemati\v{c}eskie zametki},
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     year = {1970},
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V. N. Koshelev; S. F. Morozov. Existence of discontinuous solutions for simplest semidefinite problems of variational calculus. Matematičeskie zametki, Tome 7 (1970) no. 1, pp. 69-78. http://geodesic.mathdoc.fr/item/MZM_1970_7_1_a7/