Mon, 30 May 2022
16:30 - 17:30
Francesco Leonetti
Universita degli Studi dell'Aquila

In the plane, we know that area of a set is monotone with respect to the inclusion but perimeter fails, in general. If we consider only bounded and convex sets, then also the perimeter is monotone. This property allows us to estimate the minimum number of convex components of a nonconvex set.

When studying integral functionals of the calculus of variations, convexity with respect to minors of the Jacobian matrix is a nice tool for proving existence and regularity of minimizers.

Sometimes it happens that the infimum of a functional on a set is less then the infimum taken on a dense subset: we usually refer to it as Lavrentiev phenomenon. In order to avoid it, convexity helps a lot.

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