Calculus of Variations and Geometric Measure Theory

The isoperimetric problem with a double density

Giorgio Saracco (Università di Firenze)

created by pinamonti on 09 Apr 2021
modified by saracco on 12 Apr 2021

13 apr 2021 -- 14:30   [open in google calendar]

Zoom seminar

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It is well-known that for any given volume, the sets that enclose said volume with the least perimeter are balls. What happens when one in place of the standard Euclidean volume and perimeter considers weighted counterparts? Given densities $f: \mathbb{R}^N \to \mathbb{R}^+$ and $h:\mathbb{R}^N \times \mathbb{S}^{N-1} \to \mathbb{R}^+$ to weigh, resp., the volume and the perimeter, we shall discuss under which hypotheses isoperimetric sets exist for all volumes. Furthermore, we shall introduce the $\varepsilon-\varepsilon^\beta$ property, which readily allows to prove boundedness. If time allows, some regularity results shall be discussed. Based on joint works with A. Pratelli (Università di Pisa).