On a fractional harmonic replacement

Serena Dipierro; Enrico Valdinoci

doi:10.3934/dcds.2015.35.3377

Article Contents

2015, Volume 35, Issue 8: 3377-3392. Doi: 10.3934/dcds.2015.35.3377

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On a fractional harmonic replacement

Serena Dipierro^1, and
Enrico Valdinoci^2,

1.
Maxwell Institute for Mathematical Sciences and School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD
2.
Weierstraß Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin

Received: October 31, 2014

Revised: October 31, 2014

Published: May 2017

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Abstract

Given $s\in(0,1)$, we consider the problem of minimizing the fractional Gagliardo seminorm in $H^s$ with prescribed condition outside the ball and under the further constraint of attaining zero value in a given set $K$.
We investigate how the energy changes in dependence of such set. In particular, under mild regularity conditions, we show that adding a set $A$ to $K$ increases the energy of at most the measure of $A$ (this may be seen as a perturbation result for small sets $A$).
Also, we point out a monotonicity feature of the energy with respect to the prescribed sets and the boundary conditions.

Keywords:

Mathematics Subject Classification: 31A05, 35R11, 46E35.

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References

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