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Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory by Francesco Maggi

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17

First and second variation of perimeter

One of the most fundamental ideas in the Calculus of Variations, giving the name to the discipline itself, is that of deriving necessary conditions for minimality from the basic rules of Calculus by looking at curves of competitors which “pass through” a given candidate minimizer. Indeed, this is the usual procedure used to derive the Euler–Lagrange equations (first order necessary minimality conditions) and stability inequalities (second order necessary minimality conditions). Let us examine this idea in the case of a perimeter minimizer E into some open set A. We construct a curve of competitors “passing through” E by fixing a compactly supported smooth vector field T (A; n), and noticing that, ...

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