# American Institute of Mathematical Sciences

August  2015, 35(8): 3377-3392. doi: 10.3934/dcds.2015.35.3377

## On a fractional harmonic replacement

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

Received  November 2014 Revised  November 2014 Published  February 2015

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.
Citation: Serena Dipierro, Enrico Valdinoci. On a fractional harmonic replacement. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3377-3392. doi: 10.3934/dcds.2015.35.3377
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