On necessary conditions in the calculus of variations

Vladimir Jankovi\'c

Abstract: We consider weak and strong local solutions of the general isoperimetric problem. That problem differs from the classical calculus of variations in the fact that among constraints both constraints of the equality and of the inequality type appear. Necessary conditions (for both types of local solutions) are obtained, with no assumptions on integrand's phase variable. In the case of the simplest problem of calculus of variations necessary condition for $\hat x (\cdot)$ to be the weak local solution reduces here to the following equation $$ {d\over dt} [\hat L_{\dot x} (t)\dot{\hat x} (t) (t) -\hat L (t)] =\hat L_t (t),\quad t\in [t_0, t_1], $$ and necessary condition for $\hat x (\cdot)$ to be the strong local solution reduces here to the above differential equation together with the Weierstrass inequality.

Keywords: isoperimetric problem, weak local minimum, strong local minimum, Lagrange multipliers.

Classification (MSC2000): 49B10

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