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The idea about the calculus of variations is to start from an optimal solution $y_i^\ast(p)$ (assuming there exists one) and to look at small perturbations of the form: $$ y_i(p) = y_i^\ast(p) + \varepsilon \eta(p). $$ Here $\eta(p)$ is a (smooth) continuous function of $p$ and $\varepsilon \in \mathbb{R}$. Depending on the boundary or shape conditions of $...

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