Calculus of Variations pp 47-79 | Cite as

# Variations

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## Abstract

In this chapter we discuss around \(t = 0\). If \(t \mapsto \mathscr {F}[u_t]\) is differentiable at \(t = 0\), then its derivative at \(t = 0\) must vanish because of the minimization property. This is analogous to the elementary fact that if \(g \in \mathrm {C}^1((0,T))\) takes its minimum at a point \(t_* \in (0,T)\), then \(g'(t_*) = 0\).

*variations*of functionals. The idea is the following: Let \(\mathscr {F}:\mathrm {W}^{1,p}_g(\varOmega ;\mathbb {R}^m) \rightarrow \mathbb {R}\) be a functional with minimizer \(u_* \in \mathrm {W}^{1,p}_g(\varOmega ;\mathbb {R}^m)\). Take a path \(t \mapsto u_t \in \mathrm {W}^{1,p}_g(\varOmega ;\mathbb {R}^m)\) (\(t \in \mathbb {R}\)) with \(u_0 = u_*\) and consider the behavior of the map$$ t \mapsto \mathscr {F}[u_t] $$

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