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A semigroup version of the isoperimetric inequality. (English) Zbl 1093.47042
The isoperimetric inequality in $${\mathbb R}^n$$ states the following principle. Let $$A$$, $$B$$ be subsets of $${\mathbb R}^n$$ with the same volume, $$B$$ a Euclidean ball and denote by $$| {\partial}A|$$, $$| {\partial}B|$$ the respective measure of the surfaces $${\partial}A$$ and $${\partial}B$$. Then the inequality $$| {\partial}A|\geq |{\partial}B|$$ holds. In the present paper, the author develops a semigroup version of this inequality. For this purpose, the notion of perimeter of a Cacciopoli set (introduced by E. De Giorgi) yields the appropriate measure theoretical background. In the proofs, the author uses properties of the heat semigroup as well as its explicit integral representation.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 49Q20 Variational problems in a geometric measure-theoretic setting
##### Keywords:
heat semigroup; isoperimetric inequality; Cacciopoli sets
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