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
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.  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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

Y. J. Park, Fractional Polya-Szegö inequality, J. Chungcheong Math. Soc., 24 (2011), 267-271. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

Y. J. Park, Fractional Polya-Szegö inequality, J. Chungcheong Math. Soc., 24 (2011), 267-271. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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