On a fractional harmonic replacement

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

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, United Kingdom
2.	Weierstraß Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany

Received November 2014 Revised November 2014 Published February 2015

Abstract
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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: Harmonic replacement, fractional Sobolev spaces, energy estimates..

Mathematics Subject Classification: 31A05, 35R11, 46E3.

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

References:

[1]	I. Athanasopoulos, L. A. Caffarelli, C. Kenig and S. Salsa, An area-Dirichlet integral minimization problem, Commun. Pure Appl. Math., 54 (2001), 479-499. doi: 10.1002/1097-0312(200104)54:4<479::AID-CPA3>3.0.CO;2-2.
[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, available at http://www.sciencedirect.com/science/article/pii/S0294144914000389 doi: 10.1016/j.anihpc.2014.04.004.
[3]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[4]	S. Dipierro, A. Figalli and E. Valdinoci, Strongly nonlocal dislocation dynamics in crystals, Comm. Partial Differential Equations, 39 (2014), 2351-2387. doi: 10.1080/03605302.2014.914536.
[5]	J. Jost, Partial Differential Equations. 3rd Revised and Expanded ed, Graduate Texts in Mathematics, 214. Springer, New York, 2013. xiv+410 pp. doi: 10.1007/978-1-4614-4809-9.
[6]	Y. J. Park, Fractional Polya-Szegö inequality, J. Chungcheong Math. Soc., 24 (2011), 267-271.
[7]	X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9), 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.
[8]	R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. doi: 10.5565/PUBLMAT_58114_06.
[9]	E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., S$\vec e$MA, 49 (2009), 33-44.

show all references

References:

[1]	I. Athanasopoulos, L. A. Caffarelli, C. Kenig and S. Salsa, An area-Dirichlet integral minimization problem, Commun. Pure Appl. Math., 54 (2001), 479-499. doi: 10.1002/1097-0312(200104)54:4<479::AID-CPA3>3.0.CO;2-2.
[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, available at http://www.sciencedirect.com/science/article/pii/S0294144914000389 doi: 10.1016/j.anihpc.2014.04.004.
[3]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[4]	S. Dipierro, A. Figalli and E. Valdinoci, Strongly nonlocal dislocation dynamics in crystals, Comm. Partial Differential Equations, 39 (2014), 2351-2387. doi: 10.1080/03605302.2014.914536.
[5]	J. Jost, Partial Differential Equations. 3rd Revised and Expanded ed, Graduate Texts in Mathematics, 214. Springer, New York, 2013. xiv+410 pp. doi: 10.1007/978-1-4614-4809-9.
[6]	Y. J. Park, Fractional Polya-Szegö inequality, J. Chungcheong Math. Soc., 24 (2011), 267-271.
[7]	X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9), 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.
[8]	R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. doi: 10.5565/PUBLMAT_58114_06.
[9]	E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., S$\vec e$MA, 49 (2009), 33-44.

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