A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties

Antonio De Rosa; Domenico Angelo La Manna

doi:10.3934/cpaa.2021059

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2021, Volume 20, Issue 5: 2101-2116. Doi: 10.3934/cpaa.2021059

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A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties

Antonio De Rosa^1, , and
Domenico Angelo La Manna^2,

1.
Department of Mathematics, University of Maryland, 4176 Campus Dr, College Park, MD 20742, USA
2.
Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014, Finland

^* Corresponding author
^* Corresponding author

Received: October 31, 2020

Revised: January 31, 2021

Early access: April 2021

Published: May 2021

The first author is partially supported by the NSF DMS Grant No. 1906451. The second author has been partially supported by the Academy of Finland grant 314227

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Abstract

We study a non local approximation of the Gaussian perimeter, proving the Gamma convergence to the local one. Surprisingly, in contrast with the local setting, the halfspace turns out to be a volume constrained stationary point if and only if the boundary hyperplane passes through the origin. In particular, this implies that Ehrhard symmetrization can in general increase the non local Gaussian perimeter taken into consideration.

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Mathematics Subject Classification: Primary: 49Q15, 49Q20.

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References

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