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

References:
[1]

Commun. Pure Appl. Math., 54 (2001), 479-499. doi: 10.1002/1097-0312(200104)54:4<479::AID-CPA3>3.0.CO;2-2.  Google Scholar

[2]

L. Caffarelli, O. Savin and E. Valdinoci, Minimization of a fractional perimeter-Dirichlet integral functional,, Ann. Inst. H. Poincaré Anal. Non Linéaire, ().  doi: 10.1016/j.anihpc.2014.04.004.  Google Scholar

[3]

Bull. Sci. math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[4]

Comm. Partial Differential Equations, 39 (2014), 2351-2387. doi: 10.1080/03605302.2014.914536.  Google Scholar

[5]

Graduate Texts in Mathematics, 214. Springer, New York, 2013. xiv+410 pp. doi: 10.1007/978-1-4614-4809-9.  Google Scholar

[6]

J. Chungcheong Math. Soc., 24 (2011), 267-271. Google Scholar

[7]

J. Math. Pures Appl. (9), 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[8]

Publ. Mat., 58 (2014), 133-154. doi: 10.5565/PUBLMAT_58114_06.  Google Scholar

[9]

Bol. Soc. Esp. Mat. Apl., S$\vec e$MA, 49 (2009), 33-44.  Google Scholar

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