I am writing seminar in Calculus of Variations and have some difficulties with Weierstrass-Erdmann conditions of Broken Extrema. The thing that bothers me is why second Weierstrass-Erdmann condition ($y'\frac{\partial L}{\partial y'}-L$ is continuous in every corner point) is necessary only for strong extrema, when I haven't used a fact that im working only with strong minimum? http://yima.csl.illinois.edu/psfile/ECE553/Lectures7-10.pdf page 59
2026-03-28 09:56:49.1774691809
Calculus of Variations, Weierstrass Erdmann conditions
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If you have a weak extremum then you exclude some of the pertubations which are produced by $\Delta x$, because of the $1$-norm. So you need the strong minima assumption for the argument to work.
You can however reproduce the argument exluding the $\Delta x$ pertubations and recover the other nessesary condition.